When Can One Detect Overdominant Selection in the Infinite Alleles Model?

By Paul Joyce^{1}, Stephen M. Krone^{1}, and Thomas G. Kurtz^{2} University of Idaho, University of Idaho, and University of Wisconsin

———————————————————————–

Abstract

One of the goals of this paper is to show that the infinite alleles model with overdominant selection “looks like” the neutral infinite alleles model when the selection intensity and mutation rate get large together. This rather surprising behavior was noticed by Gillespie (1999) in simulations. To make rigorous and refine Gillespie’s observations, we analyze the limiting behavior of the likelihood ratio of the stationary distributions for the model under selection and neutrality, as the mutation rate and selection intensity go to infinity together in a specified manner. In particular, we show that the likelihood ratio tends to one as the mutation rate goes to infinity, provided the selection intensity is a multiple of the mutation rate raised to a power less than 3/2. (Gillespie’s simulations correspond to the power 1.) This implies that we cannot distinguish between the two models in this setting. Conversely, if the selection intensity grows like a multiple of the mutation rate raised to a power greater than 3/2, selection can be detected; i.e., the likelihood ratio tends to 0 under neutrality and infinity under selection. We also determine the nontrivial limit distributions in the case of the critical exponent 3/2. We further analyze the limiting behavior when the exponent is less than 3/2 by determining the rate at which the likelihood ratio converges to one and by developing results for the distributions of finite samples.

November 27, 2001

Short title: Detecting Selective Overdominance

AMS 1991 subject classification. Primary 62F05; secondary 60G42, 60F15

Key words and phrases: infinite alleles model, overdominant selection, homozygosity, mutation rate, likelihood ratio, Ewens sampling formula, Poisson–Dirichlet, weak limits.

1Research supported in part by NSF grants DMS-96-26764 and DMS-00-72198.

2Research supported in part by NSF grant DMS-99-71571.

§1. Preliminaries and Main Results.

In this paper we attempt to explain and refine some observations of Gillespie (1999) which were based on simulations of an infinite alleles model with selective overdominance (as well as other models). Gillespie made the surprising observation that, if the selection intensity and mutation rate get large together, the behavior looks like that of a neutral model. Gillespie argued that for a K allele model, high mutation rate and high selection intensity tend to push the population toward equal frequencies, implying that the population frequencies will tend to (1/K, …, 1/K) under both selection and neutrality. This argument does not extend to the infinite alleles model, so simulations were used. His simulations show that various measures of gene diversity look increasingly similar as the mutation rate and selection intensity are increased. By analyzing the infinite alleles diffusion, we are able to justify Gillespie’s claims and, in fact, to go quite a bit further.

Our approach is based on likelihood ratios for the stationary measures under selection and neutrality. We begin with a description of the diploid infinite alleles model with selective overdominance (i.e., heterozygote advantage). This is a diffusion process obtained via a large-population limit as follows. Consider a diploid population of M individuals in which each of the 2M genes is assigned a type x ∈ [0, 1]. Suppose the genes in the current generation are labelled x_{1}, …, x_{2M}. To determine the genotype of a new individual in the next generation, a pair of parental genes is selected at random from the population. The probability that the ith and jth genes are selected is

\frac{w_{M} (x_{i}, x_{j})}{∑_{1≤l,m≤2M} w_{M} (x_{l}, x_{m})},

where w_{M} (x, y) ≥ 0 is a symmetric function specifying the fitness of an individual with genotype (x, y). To model selective overdominance, we give homozygotes a fitness disadvantage by setting

w_{M} (x, y) = 1 − \frac{σ}{4M} δ_{x,y},

where σ > 0 is a constant (the selection intensity) and δ_{x,y} is the Kronecker delta. With probability u_{M} = θ/(4M), one of the genes in the new individual will be selected at random to mutate, and the new mutant type will be chosen according to a uniform distribution on [0, 1]; with probability one, it will be different from all previous types. With probability 1 − u_{M}, there is no mutation and the genotype of the new individual is determined by the parental genes. Letting M → ∞ and keeping track of the proportions of the different types of genes, one obtains a (measure-valued) diffusion process known as the infinite alleles model with overdominant selection. [Cf. Ethier and Kurtz (1994) for more details on the above construction and process.] Our interest here is in the stationary distribution of this process.

Consider the ordered infinite simplex

∇ ≡ {(x_{1}, x_{2}, …) : x_{1} ≥ x_{2} ≥ ⋯ ≥ 0, ∑_{i=1}^{∞} x_{i} = 1}.

Here, we think of x_{i} as representing the relative frequency of the ith most frequent allele. When these frequencies are considered to be random, we will write X = (X_{1}, X_{2}, …) ∈ ∇ for the random vector of allele relative frequencies in the population, in descending order. Let μ_{σ} be the stationary distribution for X in the infinite alleles model with selection intensity σ, and 𝔼_{σ} the corresponding expectation. In particular, μ_{0} and 𝔼_{0} give the stationary distribution and expectation in the neutral case, σ = 0.

In the neutral case, if we keep track of the proportions of the alleles in descending order, the stationary distribution, μ_{0}, is given by the Poisson–Dirichlet distribution with parameter θ (cf. Kingman 1977). This is a probability measure on ∇ and, to make explicit the dependence on θ, we write X_{θ} ∼ μ_{0} when X_{θ} ∈ ∇ has this distribution.

The form of the stationary distribution μ_{σ}, in the case of overdominant selection with intensity σ, follows as a special case of Theorem 4.4 in Ethier and Kurtz (1994): for integrable f,

∫_{∇} f(x) μ_{σ}(dx) = ∫_{∇} f(x) \frac{e^{−σH(x)}}{Z(σ, θ)} μ_{0}(dx),  (1)

where

H(x) = H(x_{1}, x_{2}, …) ≡ ∑_{i=1}^{∞} x_{i}^{2}  (2)

is the population homozygosity when the allele frequencies are given by x = (x_{1}, x_{2}, …) ∈ ∇, and

Z(σ, θ) = ∫_{∇} e^{−σH(x)} μ_{0}(dx)  (3)

is a normalizing constant. Thus \frac{dμ_{σ}}{dμ_{0}} (x) = \frac{e^{−σH(x)}}{Z(σ, θ)} is the Radon–Nikodym derivative (or likelihood ratio) for the stationary measure under selection with respect to the stationary measure under neutrality.

We are interested in the situation where the selection intensity σ goes to infinity like a power of θ. As we will see, the exponent 3/2 plays a very special role. Thus, we write σ = cθ^{3/2+γ}, where c is a positive constant and γ ∈ (−∞, ∞) is also held constant. We will write Y_{θ}^{(γ)} ∼ μ_{σ} when Y_{θ}^{(γ)} ∈ ∇ has this distribution. Sometimes, when dealing with the case γ < 0, we will instead write Y_{θ}^{(−α)} ∼ μ_{σ}, σ = cθ^{3/2−α}, etc., where α > 0. Our first results concern the asymptotic behavior of the above likelihood ratio. Here and throughout the paper, we denote weak convergence by ⇒.

Theorem 1 Suppose X_{θ} = (X_{1,θ}, X_{2,θ}, …) ∼ μ_{0} and σ = cθ^{3/2+γ}, where c > 0 is a constant. Then, as θ → ∞,

\frac{dμ_{σ}}{dμ_{0}} (X_{θ}) = \frac{e^{−σH(X_{θ})}}{𝔼_{0} (e^{−σH(X_{θ})})} ⇒ \{ \begin{matrix}1, & if γ < 0 \\ exp(cZ_{2} − c^{2}), & if γ = 0 \\ 0, & if γ > 0 \end{matrix}

where Z_{2} ∼ N(0, 2).

Theorem 2 Suppose X_{θ} = (X_{1,θ}, X_{2,θ}, …) ∼ μ_{0}, Y_{θ}^{(γ)} = (Y_{1,θ}^{(γ)}, Y_{2,θ}^{(γ)}, …) ∼ μ_{σ}, where σ = cθ^{3/2+γ} and c > 0 is a constant. Then, as θ → ∞,

\frac{dμ_{σ}}{dμ_{0}} (Y_{θ}^{(γ)}) = \frac{e^{−σH(Y_{θ}^{(γ)})}}{𝔼_{0} (e^{−σH(X_{θ})})} ⇒ \{ \begin{matrix}1, & if γ < 0 \\ exp(cZ_{2}^{∗} − c^{2}), & if γ = 0 \\ ∞, & if γ > 0 \end{matrix}

where Z_{2}^{∗} ∼ N(2c, 2).

These two results combine to say that, if the selection intensity grows like θ to a power greater than 3/2, then under neutrality, the likelihood ratio \frac{dμ_{σ}}{dμ_{0}} (X_{θ}) goes to zero, while under selection, \frac{dμ_{σ}}{dμ_{0}} (Y_{θ}) goes to infinity. This says that, for large θ, one can distinguish between a neutral population and a population with selection intensity very large relative to the mutation rate. If the selection intensity grows like θ to a power less than 3/2, one cannot distinguish between the neutral and selective models when θ is large; i.e., one cannot detect selective overdominance. Loosely speaking, the model with σ = cθ^{β} undergoes a phase transition as the exponent β crosses 3/2. Notice that in the critical case, σ = cθ^{3/2}, the limiting likelihood ratio has a lognormal distribution under both neutrality and selection. Notice also that Z_{2}^{∗} is larger on average than Z_{2}. As expected, in the limit, the likelihood ratio tends to be smaller under neutrality than under selection.

When σ = cθ^{3/2+γ}, with γ < 0, selection is masked by a high mutation rate. In the limit, one cannot distinguish between neutrality and selection and the likelihood ratio tends to one, regardless of whether the population frequencies were generated from neutral or selective models. The next theorem gives the rate at which the likelihood ratio tends to one in this case.

Theorem 3 Suppose X_{θ} = (X_{1,θ}, X_{2,θ}, …) ∼ μ_{0}, Y_{θ}^{(−α)} = (Y_{1,θ}^{(−α)}, Y_{2,θ}^{(−α)}, …) ∼ μ_{σ}, where σ = cθ^{3/2−α}, and α > 0 and c > 0 are constants. Then, as θ → ∞,

θ^{α} (\frac{dμ_{σ}}{dμ_{0}} (X_{θ}) − 1) ⇒ cZ_{2}  (4)

and

θ^{α} (\frac{dμ_{σ}}{dμ_{0}} (Y_{θ}^{(−α)}) − 1) ⇒ cZ_{2},  (5)

where Z_{2} ∼ N(0, 2).

The case in which σ grows like cθ = cθ^{3/2−1/2} in the above theorems corresponds to the simulations in Gillespie (1999). To make clear the connection between this paper and Gillespie (1999), recall that θ = 4Mu, where u is the probability of mutation per individual, and w = 1 − σ/(4M) is the fitness for homozygotes. Therefore, as M → ∞, both θ and σ = (1 − w)4M tend to infinity; moreover,

σ = \frac{1 − w}{u} θ.

In Gillespie’s simulations, u and w were fixed as M → ∞. This implies that σ increases linearly in θ, with c = (1 − w)/u. The above results make the point that, even if the entire population were observable, the neutral model is indistinguishable from the selective overdominance model when the selection intensity is not sufficiently larger than θ.

In practice one cannot observe the entire population. Instead, one must make inferences based on information in a finite sample. In the next two theorems we see that, if we restrict attention to a finite sample when σ = cθ^{3/2+γ} and γ < 0, then the likelihood ratio converges to 1 faster than in the case in which the entire population is observed. To consider a finite sample of genes drawn from a population that has evolved under the infinite alleles model, it is convenient to use the notion of a partition structure.

Let a = (a_{1}, a_{2}, …, a_{n}) denote an allelic partition of a sample of n genes; i.e., a_{i} (i = 1, …, n) gives the number of distinct alleles each of which is represented exactly i times in the sample. Clearly, a_{i} ≥ 0, ∑_{i=1}^{n} ia_{i} = n, and ∑_{i=1}^{n} a_{i} ≡ k gives the number of distinct alleles in the sample. We write A_{n} = (A_{1}, A_{2}, …, A_{n}) for the random partition obtained by independently sampling n genes according to the relative frequencies of the different alleles in the population. If the allele proportions are given by x = (x_{1}, x_{2}, …) ∈ ∇, then the conditional sampling probability ℙ(A_{n} = a|X = x) is given by the multinomial sampling function (cf. Kingman (1977)):

ℙ(A_{n} = a|X = x) = ϕ_{a}(x) ≡ \frac{n!}{∏_{i=1}^{n}(i!)^{a_{i}}} ∑_{ν∈A_{a}} x_{1}^{ν_{1}}x_{2}^{ν_{2}} ⋯,  (6)

where A_{a} ≡ {ν = (ν_{1}, ν_{2}, …) ∈ ℤ_{+} × ℤ_{+} × ⋯: #(i : ν_{i} = j) = a_{j}} represents the set of allele counts which are consistent with the partition a. In the neutral infinite alleles model, the stationary probability of observing the allelic partition a in a sample of size n,

ℙ_{0}(A_{n} = a) = ∫_{∇} ϕ_{a}(x) μ_{0}(dx) = 𝔼_{0} (ϕ_{a}(X_{θ})) ,  (7)

is given by the Ewens sampling formula

ℙ_{0}(A_{n} = a) = ESF(θ, a) ≡ \frac{n!}{θ_{(n)}} ∏_{j=1}^{n} (\frac{θ}{j} )^{a_{j}} \frac{1}{a_{j}!},  (8)

where θ_{(n)} = θ(θ + 1) ⋯ (θ + n − 1). In the selective case, σ > 0, the stationary sampling distribution is given by

ℙ_{σ}(A_{n} = a) = ∫_{∇} ϕ_{a}(x) μ_{σ}(dx) = 𝔼_{0} (ϕ_{a}(X_{θ})\frac{dμ_{σ}}{dμ_{0}} (X_{θ})) .  (9)

Theorem 4 Let c > 0 and α ≥ 0 be constants and suppose σ = cθ^{3/2−α}. If a = (a_{1}, a_{2}, …, a_{n}) is an allelic partition for a sample of size n with a_{1} ≠ n, then

lim_{θ→∞} θ^{α+1/2} |\frac{ℙ_{σ}(A_{n} = a)}{ℙ_{0}(A_{n} = a)} − 1| = 2c ∑_{i=2}^{n} a_{i}\binom{i}{2}.  (10)

Remarks. Note that the sum ∑_{i=2}^{n} a_{i}\binom{i}{2} appearing in the limit gives the number of homozygous pairs that can be formed from the n alleles in the sample. The above theorem says that, if we restrict attention to a finite sample, the convergence is of order \sqrt{θ} faster than that given in Theorem 3. It also says that, unlike the situation in Theorem 3, the likelihood ratio converges to one in the case of the critical exponent 3/2 (as well as when the exponent is less than 3/2). This is perhaps not too much of a surprise since it is harder to distinguish the selective model from the neutral model when observing only a finite sample (as opposed to knowing the allele frequencies for the whole population).

Again, the case with α = 1/2 covers the Gillespie conjecture in the context of random samples. Returning to Gillespie’s notation, let θ = 4Mu and σ = 4M(1 − w). As the effective population size M becomes large, the likelihood ratio of a sample under selection relative to a sample under neutrality converges to one at a rate proportional to 1/M.

The case a = (n, 0, …, 0), not covered by Theorem 4, corresponds to all alleles in the sample being distinct. This is the sample configuration with the lowest possible homozygosity. Since large values of θ and σ in both the neutral and selective models give increasingly higher probability to samples for which a_{1} = n, it is natural to expect that the two models will look even more similar at this most probable sample configuration; cf. (8). This suggests that the rate of convergence of ℙ_{σ}(A_{n} = a)/ℙ_{0}(A_{n} = a) to one is even faster in this case. This is the content of the next theorem.

Theorem 5 Let α > 0 and c > 0 be constants. Suppose σ = cθ^{3/2−α} and let a_{1} = (n, 0, …, 0) be the allelic partition corresponding to a sample of n distinct alleles. Then

lim_{θ→∞} θ^{α+3/2} |\frac{ℙ_{σ}(A_{n} = a_{1})}{ℙ_{0}(A_{n} = a_{1})} − 1| = 2c.  (11)

The rest of the paper is organized as follows. In Section 2, we give some asymptotic normality results for various sampling distributions under neutrality and selection. In Section 3, we prove Lemma 4, which is used to give the forms of the limits in Theorems 1 and 2. In order to apply Lemma 4 in the proof of Theorems 1 and 2, as well as for some other results, we must establish uniform integrability for certain families of scaled homozygosities. This is done in Section 4. In Section 5, these ideas are extended to rescaled versions of the likelihood ratios and sampling distributions. Finally, in Section 6 we apply these results to prove the five main theorems.

§2. Some Normal Limits

Most of the results in this paper depend on normal limit theorems for population homozygosities and sampling functions. In this section we recall some such results from Joyce, Krone, and Kurtz (2001), and prove some needed extensions.

If the decreasing allele frequencies in the population are given by x = (x_{1}, x_{2}, …), define for m ≥ 2, the mth-order population homozygosity by

H_{m}(x) ≡ ∑_{i=1}^{∞} x_{i}^{m}.  (12)

This can be thought of as the probability that all individuals in a sample of size m are of the same type, given that the vector of population allele frequencies is x. Thus, (12) is a special case of (6). Of course, H_{2}(x) = H(x) is the usual population homozygosity.

Next, define for m ≥ 2 the following scaled versions of the mth-order population homozygosities when the ordered allele frequencies are distributed according the stationary distributions under neutrality and selection, respectively. If X_{θ} ∼ μ_{0}, put

Z_{m,θ} ≡ \sqrt{θ} (\frac{θ^{m−1}H_{m}(X_{θ}) − (m − 1)!}{(m − 1)!}) ,  (13)

and if Y_{θ}^{(γ)} ∼ μ_{σ}, where σ = cθ^{3/2+γ}, put

Z_{m,θ}^{(γ)} ≡ \sqrt{θ} (\frac{θ^{m−1}H_{m}(Y_{θ}^{(γ)}) − (m − 1)!}{(m − 1)!}) .  (14)

Note that, instead of centering by the means and dividing by the standard deviations in the definitions of Z_{m,θ} and Z_{m,θ}^{(γ)}, we scale by quantities that are asymptotically the same (when θ is large), but easier to work with.

It is easy to see that, for any γ ∈ (−∞, ∞) and σ = cθ^{3/2+γ},

\frac{dμ_{σ}}{dμ_{0}} (X_{θ}) = \frac{e^{−σH(X_{θ})}}{𝔼_{0} (e^{−σH(X_{θ})})} = \frac{exp\{−cθ^{γ}Z_{2,θ}\}}{𝔼_{0}(exp\{−cθ^{γ}Z_{2,θ}\})}  (15)

and

\frac{dμ_{σ}}{dμ_{0}} (Y_{θ}^{(γ)}) = \frac{e^{−σH(Y_{θ}^{(γ)})}}{𝔼_{0} (e^{−σH(X_{θ})})} = \frac{exp\{−cθ^{γ}Z_{2,θ}^{(γ)}\}}{𝔼_{0}(exp\{−cθ^{γ}Z_{2,θ}\})}.  (16)

In the next lemma we show that Z_{m,θ} and Z_{m,θ}^{(γ)} have normal limit laws as θ → ∞. These weak limits will play a crucial role in the proofs of the theorems described above.

Lemma 1 Let (Z_{2}, Z_{3}, …, Z_{n}) be multivariate normal with mean 0 and

Cov(Z_{i}, Z_{j}) = \frac{(i + j − 1)! − i!j!}{(i − 1)!(j − 1)!}, i, j ∈ \{2, 3, …, n\},  (17)

and let Z_{m,θ} and Z_{m,θ}^{(γ)} be defined as in (13) and (14). Then, as θ → ∞, we have the following limits:

(i) If X_{θ} ∼ μ_{0}, then

(Z_{2,θ}, Z_{3,θ}, …, Z_{n,θ}) ⇒ (Z_{2}, Z_{3}, …, Z_{n}).  (18)

(ii) If Y_{θ}^{(−α)} ∼ μ_{σ}, with σ = cθ^{3/2−α}, then

(Z_{2,θ}^{(−α)}, Z_{3,θ}^{(−α)} ⋯, Z_{n,θ}^{(−α)}) ⇒ \{ \begin{matrix}(Z_{2}^{∗}, Z_{3}^{∗}, ⋯, Z_{n}^{∗}),  if α = 0, \\ (Z_{2}, Z_{3}, …, Z_{n}),  if α > 0, \end{matrix}  (19)

where Z^{∗} = Z − cb and b = (2, 3 · 2, …, n(n − 1)).

Proof. The neutral case, given by equation (18), is proved in Joyce, Krone, and Kurtz (2001). To establish (19) in the selective case with σ = cθ^{3/2}, we can use (18) together with (15), with γ = 0, to conclude for any z^{∗} ∈ ℝ^{n−1} that

ℙ_{σ}(Z_{θ}^{(0)} ≤ z^{∗}) = 𝔼_{0} (\frac{dμ_{σ}}{dμ_{0}} (X_{θ})1\{Z_{θ} ≤ z^{∗}\})

= 𝔼_{0} (\frac{exp\{−cZ_{2,θ}\}}{𝔼_{0}(exp\{−cZ_{2,θ}\})})1\{Z_{θ} ≤ z^{∗}\})

→ 𝔼 (\frac{exp\{−cZ_{2}\}}{𝔼_{0}(exp\{−cZ_{2}\})}1\{Z ≤ z^{∗}\}) ,

as θ → ∞. To get convergence inthe last step, it is not enough that Z_{θ} ⇒ Z as θ → ∞. In addition, we need to show that

𝔼_{0}(exp{−cZ_{2,θ}}) ⇒ 𝔼(exp{−cZ_{2}}).  (20)

This requires a uniform integrability argument, and will be given in Lemma 9, in Section 4.

Assuming this to hold, the rest of the proof in the case α = 0 follows from standard algebraic manipulations involving multivariate normal distributions. Indeed, let Σ be the covariance matrix for Z and write c′ = (c, 0, ⋯, 0), the prime denoting transpose. Then

𝔼 (\frac{exp\{−cZ_{2}\}}{𝔼_{0}(exp\{−cZ_{2}\})}1\{Z ≤ z^{∗}\}) = \frac{exp\{−c^{2}\}}{(2π)^{n/2}\sqrt{det Σ}} ∫_{z≤z^{∗}} exp\{−cz_{2}\} exp \{\frac{−z′Σ^{−1}z}{2}\} dz

= \frac{exp\{−c^{2}\}}{(2π)^{n/2}\sqrt{det Σ}} ∫_{z≤z^{∗}} exp\{−c′z\} exp \{\frac{−z′Σ^{−1}z}{2}\} dz

= \frac{1}{(2π)^{n/2}\sqrt{det Σ}} ∫_{z≤z^{∗}} exp \{−\frac{1}{2}(2c′z + z′Σ^{−1}z + c′Σc)\} dz.

= \frac{1}{(2π)^{n/2}\sqrt{det Σ}} ∫_{z≤z^{∗}} exp \{−\frac{1}{2}(z′ + c′Σ)Σ^{−1}(z + Σc)\} dz

= ℙ (Z^{∗} ≤ z^{∗}) .

The remaining case is much simpler. Use γ = −α < 0 in (15) and compute the limit as above with the change

\frac{dμ_{σ}}{dμ_{0}} (X_{θ}) = \frac{exp\{−cθ^{−α}Z_{2,θ}\}}{𝔼_{0}(exp\{−cθ^{−α}Z_{2,θ}\})} ⇒ \frac{e^{0}}{𝔼e^{0}} = 1,  (21)

as θ → ∞.

The next lemma is a restatement of Theorem 2 in Joyce, Krone, and Kurtz (2001). It is essentially the analogue of (18) for finite sampling distributions.

Lemma 2 Suppose X_{θ} ∼ μ_{0} and (Z_{2}, …, Z_{n}) is as in Lemma 1, and let a be a partition corresponding to a sample of size n drawn from a population with ordered frequencies X_{θ}.

(i) If a ≠ (n, 0, …, 0), then as θ → ∞,

\sqrt{θ} (\frac{ϕ_{a}(X_{θ}) − ESF(θ, a)}{ESF(θ, a)}) ⇒ ∑_{i=2}^{n} a_{i}Z_{i}.  (22)

(ii) If a_{1} = (n, 0, …, 0) and a_{2} = (n − 2, 1, 0, …, 0) are the allelic partitions with n singletons and n − 2 singletons, respectively, then as θ → ∞,

\sqrt{θ} (\frac{ϕ_{a_{1}}(X_{θ}) − ESF(θ, a_{1})}{ESF(θ, a_{2})}) ⇒ −Z_{2}.  (23)

§3. Two Basic Lemmas

The following lemmas, which may apply in a variety of settings, will be used to prove Theorems 1 and 2. More specifically, Lemma 3 is used to prove the last part of Lemma 4, and the latter provides the form of the limits in Theorems 1 and 2.

Lemma 3 Suppose (Y_{n}) is a sequence of random variables satisfying Y_{n} ⇒ Y, where Y is a continuous random variable whose density is strictly positive on all of (−∞, ∞). Suppose further that 𝔼(e^{−sY_{n}}) < ∞ for all s ∈ (0, ∞). Then, for any α > 0,

lim_{n→∞} \frac{𝔼(\sqrt{e^{−n^{α}Y_{n}}})}{\sqrt{𝔼(e^{−n^{α}Y_{n}})}} = 0.  (24)

Proof. We begin by remarking that the exponential moment condition in the hypotheses is simply to ensure that all the expected values in (24) are finite. Let F_{n}(x) = ℙ(Y_{n} ≤ x) and define F_{n}^{−1}(y) = min{x : F_{n}(x) ≥ y} for y ∈ [0, 1]. It is easy to see that ℙ(Y_{n} ≤ F_{n}^{−1}(y)) ≥ y. Defining −y_{n} = F_{n}^{−1}(1/n), it is clear that −y_{n} → −∞ as n → ∞, and that ℙ(Y_{n} ≤ −y_{n}) ≥ 1/n. Therefore, by the Cauchy–Schwarz inequality,

\frac{𝔼(e^{−n^{α}Y_{n}/2})}{\sqrt{𝔼(e^{−n^{α}Y_{n}})}} = \frac{𝔼(e^{−n^{α}Y_{n}/2} 1\{|Y_{n}| ≤ \sqrt{y_{n}}\})}{\sqrt{𝔼(e^{−n^{α}Y_{n}})}} + \frac{𝔼(e^{−n^{α}Y_{n}/2} 1\{|Y_{n}| > \sqrt{y_{n}}\})}{\sqrt{𝔼(e^{−n^{α}Y_{n}})}}

≤ \frac{exp(n^{α}\sqrt{y_{n}}/2)}{\sqrt{𝔼(exp(−n^{α}Y_{n}))}} + \sqrt{ℙ(|Y_{n}| > \sqrt{y_{n}})}

≤ \frac{exp(n^{α}\sqrt{y_{n}}/2)}{\sqrt{𝔼(exp(−n^{α}Y_{n})1\{Y_{n} ≤ −y_{n}\})}} + \sqrt{ℙ(|Y_{n}| > \sqrt{y_{n}})}

≤ \frac{exp(n^{α}\sqrt{y_{n}}/2)}{\sqrt{exp(n^{α}y_{n})ℙ(Y_{n} ≤ −y_{n}})} + \sqrt{ℙ(|Y_{n}| > \sqrt{y_{n}})}

≤ \frac{exp(n^{α}\sqrt{y_{n}}/2)}{exp(n^{α}y_{n}/2)\sqrt{(1/n)}} + \sqrt{ℙ(|Y_{n}| > \sqrt{y_{n}})}

≤ \sqrt{n} exp(−n^{α}\sqrt{y_{n}}/2) + \sqrt{ℙ(|Y_{n}| > \sqrt{y_{n}})} → 0,

as n → ∞.

Lemma 4 Suppose (Y_{n}) is a sequence of random variables satisfying Y_{n} ⇒ Y, where Y is a continuous random variable whose density is strictly positive on all of (−∞, ∞). Suppose further that 𝔼(e^{−sY_{n}}) < ∞ for all s ∈ (0, ∞), and that sup_{n} 𝔼(e^{−tY_{n}}) < ∞ for some t > 1. Then, as n → ∞,

\frac{e^{−n^{γ}Y_{n}}}{𝔼(e^{−n^{γ}Y_{n}})} ⇒ \{ \begin{matrix}1, & if γ < 0 \\ \frac{e^{−Y}}{𝔼(e^{−Y})}, & if γ = 0 \\ 0, & if γ > 0. \end{matrix}  (25)

Proof. Starting with the case γ = 0, note that e^{−Y_{n}} ⇒ e^{−Y} by the continuous mapping theorem. The hypothesis, sup_{n} 𝔼(e^{−tY_{n}}) < ∞ for some t > 1, implies uniform integrability of {e^{−Y_{n}}}, and this implies lim_{n→∞} 𝔼(e^{−Y_{n}}) = 𝔼(e^{−Y}). If γ < 0, then n^{γ} → 0, and hence n^{γ}Y_{n} ⇒ 0. The continuous mapping theorem implies exp(−n^{γ}Y_{n}) ⇒ 1. In addition, γ < 0 implies exp(−n^{γ}Y_{n}) ≤ max{exp(−Y_{n}), 1}. So, 𝔼[exp(−n^{γ}Y_{n})] → 1 follows from the dominated convergence theorem. Finally, in the case γ > 0, set U_{n} = e^{−n^{γ}Y_{n}}/𝔼(e^{−n^{γ}Y_{n}}). An appeal to Lemma 3 yields 𝔼(\sqrt{U_{n}}) → 0, as n → ∞. Therefore \sqrt{U_{n}} ⇒ 0, implying U_{n} ⇒ 0.

§4. Uniform Integrability.

This section contains uniform integrability results which are essential for proving the main theorems in Section 1. Since this is a long and technical section, it is probably a good idea for the reader to simply look at the statements of Lemmas 5 and 9, and then move on to the rest of the paper to see how they are used before coming back to digest the proofs.

To help motivate this section, we begin by alerting the reader that the condition, sup_{n} 𝔼(e^{−tY_{n}}) < ∞ for some t > 1, in Lemma 4, is an important one. To illustrate this point for the case γ = 0, consider the following example for which this condition does not hold. Let Y_{n} be defined via its density

f_{n}(y) = (1 − \frac{1}{n}) \frac{e^{−y^{2}/2}}{\sqrt{2π}} + \frac{1}{n} \frac{e^{−(y+log n)^{2}/2}}{\sqrt{2π}} , y ∈ (−∞, ∞).

Then Y_{n} converges in distribution to the standard normal Z, and hence e^{−Y_{n}} converges to the lognormal e^{−Z}. Now, using moment generating functions of normal random variables, we can see that

𝔼(e^{−tY_{n}}) = (1 − \frac{1}{n}) e^{t^{2}/2} + \frac{1}{n} e^{t log n+t^{2}/2} = (1 − \frac{1}{n}) e^{t^{2}/2} + n^{t−1}e^{t^{2}/2}.

So, even though 𝔼(Y_{n}^{r}) → 𝔼(Z^{r}) for all r ≥ 1, we have 𝔼(e^{−Y_{n}}) → 2e^{1/2} ≠ 𝔼(e^{−Z}). The missing ingredient, which would have allowed for the interchange of limit with expectation, is uniform integrability of the family {e^{−Y_{n}}}; this would have followed if sup_{n} 𝔼(e^{−tY_{n}}) < ∞, for some t > 1. However, the above calculation shows that 𝔼(e^{−tY_{n}}) → ∞, as n → ∞, for all t > 1.

Thus motivated, we seek to establish uniform integrability for certain functions of the Z_{m,θ}’s, the uniformity being in θ. First we show sup_{θ>1} 𝔼(Z_{m,θ}^{r}) < ∞ , for all r ≥ 1 (Lemma 5), and then use this to prove sup_{θ>θ_{0}} 𝔼(exp(−tZ_{m,θ})) < ∞, for some θ_{0} = θ_{0}(t), when t > 1 is fixed (Lemma 9). This will allow us to apply Lemma 4 in the proofs of Theorems 1 and 2. Note that Lemma 9 was also needed in the proof of Lemma 1. In the next section we will use these results to prove uniform integrability for the conditional sampling function ϕ_{a}(X_{θ}) (Lemma 11). This will be used, together with Lemma 2, to prove Theorems 4 and 5.

It is well known that, if V_{1}, V_{2}, … are i.i.d. Beta (1, θ) random variables with density given by

f(v) = θ(1 − v)^{θ−1}1_{(0,1)}(v),

then a “size-biased” version of the Poisson–Dirichlet distribution with parameter θ is given by

\begin{aligned}W_{1,θ} & = V_{1} \\ W_{k,θ} & = (1 − V_{1})(1 − V_{2}) ⋯ (1 − V_{k−1})V_{k}, & for k ≥ 2. \end{aligned}

The W_{k,θ}’s are just a relabelling of the X_{k,θ}’s, and so H_{m} ≡ H_{m}(X_{θ}) = ∑_{k} X_{k,θ}^{m} = ∑_{k} W_{k,θ}^{m}. It is easy to see, and was established by Griffiths (1988), that one can write

H_{m} =^{L} V^{m} + (1 − V)^{m}H̃_{m},

where V ∼ Beta (1, θ), and H̃_{m} is a random variable which is independent of V and has the same distribution as H_{m}. A version of this recursion for Z_{m,θ} is

(m − 1)!Z_{m,θ} = \sqrt{θ} (θ^{m−1}H_{m} − (m − 1)!)

=^{L} \sqrt{θ} (θ^{m−1}[V^{m} + (1 − V)^{m}H̃_{m}] − (m − 1)!)
= \sqrt{θ} (θ^{m−1}V^{m} + (m − 1)!(1 − V)^{m} − (m − 1)!)
+(1 − V)^{m}\sqrt{θ} (θ^{m−1}H̃_{m} − (m − 1)!)
= a_{m}(V) + (1 − V)^{m}(m − 1)!Z̃_{m,θ},  (26)

where

a_{m}(V) ≡ \sqrt{θ} (θ^{m−1}V^{m} − (m − 1)! [1 − (1 − V)^{m}]) ,  (27)

V ∼ Beta (1, θ), and Z̃_{m,θ} is a random variable which is independent of V and has the same distribution as Z_{m,θ}.

We record here a simple moment formula that will be used a number of times in the paper. If V ∼ Beta (1, θ), then for any nonnegative integers m and n,

𝔼 [V^{m}(1 − V)^{n}] = θ \frac{Γ(m + 1)Γ(n + θ)}{Γ(m + n + θ + 1)} = \frac{m! θ}{(n + θ)(n + θ + 1) ⋯ (n + θ + m)}.  (28)

We will also need the following bounds for a_{m}(V):

\frac{1}{\sqrt{θ}} (θ^{m}V^{m} − m!2^{m}θV) ≤ a_{m}(V) ≤ \frac{1}{\sqrt{θ}} (θ^{m}V^{m} + m!2^{m}θV) .  (29)

The second inequality follows from the fact that 0 < V < 1 with probability 1, and hence

0 < 1 − (1 − V)^{m} ≤ 2^{m}V.  (30)

The first inequality follows similarly. Also, (29) implies

|a_{m}(V)| ≤ \frac{1}{\sqrt{θ}} (θ^{m}V^{m} + m!2^{m}θV)  (31)

and so, for k ≥ 1,

𝔼|a_{m}(V)|^{k} ≤ θ^{−k/2}𝔼 (θ^{m}V^{m} + m! 2^{m}θV)^{k}
≤ θ^{−k/2} 2^{k} [𝔼 (θV)^{mk} + (m! 2^{m})^{k}𝔼 (θV)^{k}]  (32)
≤ (2/\sqrt{θ})^{k} [(mk)! + (m!2^{m})^{k}k!] ∼ O(θ^{−k/2}),

where the last inequality follows from

0 ≤ 𝔼(θV)^{j} = \frac{j! θ^{j}}{(θ + 1) ⋯ (θ + j)} ≤ j!  (j ≥ 1).  (33)

It is worth remarking that one of the reasons for writing the bounds in (29) and (31) the way we did is that θV converges in distribution to an Exp(1) random variable, as θ → ∞. Note also that we are suppressing the dependence on θ in the notation for V ∼ Beta (1, θ).

Lemma 5 Let Z_{m,θ} be defined by equation (13). Then, for all integers r ≥ 1 and m ≥ 2,

sup_{θ>1} 𝔼_{0} (|Z_{m,θ}|^{r}) < ∞.  (34)

Proof. Since 𝔼(|Z_{m,θ}|^{r}) ≤ \sqrt{𝔼_{0}(Z_{m,θ}^{2r})}, the lemma will follow if we prove

sup_{θ>1} 𝔼_{0} (Z_{m,θ}^{r}) < ∞,  (35)

for all integers r ≥ 1. It is easy to see, using the Ewens sampling formula, that

𝔼_{0}(Z_{m,θ}) = \sqrt{θ} [\frac{θ^{m−1}}{(θ + 1) ⋯ (θ + m − 1)} − 1] ∼ O(1/\sqrt{θ})  (36)

and hence (35) holds when r = 1. In the last step, we used the fact that one can express

\frac{θ^{m−1}}{(θ + 1) ⋯ (θ + m − 1)} = 1 − ϕ_{m}(θ),  (37)

where |ϕ_{m}(θ)| < \binom{m+1}{2}θ^{−1} + O(θ^{−2}).

We now proceed by induction on r (with m fixed), assuming that sup_{θ>1} 𝔼_{0}(Z_{m,θ}^{k}) < ∞ holds for 1 ≤ k ≤ r − 1. Using (26), we can write

[(m − 1)!Z_{m,θ}]^{r} =^{L} (a_{m}(V) + (1 − V)^{m}(m − 1)!Z̃_{m,θ})^{r}

= ∑_{k=0}^{r} \binom{r}{k}a_{m}^{k}(V)(1 − V)^{m(r−k)} [(m − 1)!Z̃_{m,θ}]^{r−k}

= (1 − V)^{mr} [(m − 1)!Z̃_{m,θ}]^{r} + ∑_{k=1}^{r} \binom{r}{k}a_{m}^{k}(V)(1 − V)^{m(r−k)} [(m − 1)!Z̃_{m,θ}]^{r−k} .

Taking expectation on both sides, using the independence of V and Z̃_{m,θ}, and setting

μ_{r,θ} ≡ ((m − 1)!)^{r} 𝔼_{0}(Z_{m,θ}^{r}),  (38)

we get

μ_{r,θ} = 𝔼((1 − V)^{mr}) μ_{r,θ} + ∑_{k=1}^{r} \binom{r}{k}𝔼 (a_{m}^{k}(V)(1 − V)^{m(r−k)}) μ_{r−k,θ}.

This implies

μ_{r,θ} (1 − \frac{θ}{mr + θ} ) = ∑_{k=1}^{r} \binom{r}{k}𝔼 (a_{m}^{k}(V)(1 − V)^{m(r−k)}) μ_{r−k,θ},

and therefore we have the recursion

μ_{r,θ} = \frac{mr + θ}{mr} ∑_{k=1}^{r} \binom{r}{k}𝔼 (a_{m}^{k}(V)(1 − V)^{m(r−k)}) μ_{r−k,θ}.  (39)

So the lemma will follow if we can establish

sup_{θ>1} θ |𝔼 (a_{m}^{k}(V)(1 − V)^{m(r−k)})| < ∞,  (40)

for k = 1, …, r, where V ∼ Beta (1, θ). The cases k = 1 and k ≥ 2 must be treated separately.

We begin with k = 1.

θ |𝔼 (a_{m}(V)(1 − V)^{m(r−1)})|

= \sqrt{θ} |𝔼 \{θ^{m}V^{m}(1 − V)^{m(r−1)} − (m − 1)! θ [1 − (1 − V)^{m}](1 − V)^{m(r−1)}\}|

= \sqrt{θ} |\frac{m! θ^{m+1}}{(θ + m(r − 1)) ⋯ (θ + m(r − 1) + m)} − (m − 1)! θ (\frac{θ}{θ + m(r − 1)} − \frac{θ}{θ + mr} )|

= \sqrt{θ} |m!(1 − ϕ(θ)) − m!(1 − ψ(θ))| ,

where |ϕ(θ)| ∼ O(θ^{−1}) and |ψ(θ)| ∼ O(θ^{−1}). Thus,

θ |𝔼 (a_{m}(V)(1 − V)^{m(r−1)})| ≤ \sqrt{θ} m! (|ϕ(θ)| + |ψ(θ)|) ∼ O(1/\sqrt{θ})  (41)

and (40) is verified for k = 1.

To handle the case 2 ≤ k ≤ r, use (32) to see

sup_{θ>1} θ |𝔼 (a_{m}^{k}(V)(1 − V)^{m(r−k)})| ≤ sup_{θ>1} θ𝔼|a_{m}(V)|^{k} ≤ sup_{θ>1} 2^{k} θ^{1−k/2} [(mk)! + (m!2^{m})^{k}k!] < ∞.

Thus, (40) is satisfied when k ≥ 2, and this completes the proof.

The next three lemmas provide estimates that are used in the proof of the crucial uniform integrability result in Lemma 9. To obtain the desired uniform integrability, we will need to bound (uniformly in θ) an alternating series related to exp{−tZ_{m,θ}}. The following three lemmas (essentially calculus exercises which were not so obvious that we could omit the proofs) will aid us in the calculations involving the afore-mentioned series.

Lemma 6 Let ∑_{j} A_{j}x^{j} and ∑_{j} B_{j}x^{j} be two absolutely convergent series. Then, for any 1 ≤ k ≤ l,

|∑_{i=l}^{∞} ∑_{j=k}^{i} A_{j}B_{i−j}x^{i}| ≤ |∑_{j=k}^{∞} A_{j}x^{j} ∑_{i=0}^{∞} B_{i}x^{i}| + |x|^{k}C_{kl}(|x|),

where

C_{kl}(u) ≡ ∑_{j=k}^{l−1} |A_{j}| u^{j−k} ∑_{i=0}^{l−j−1} |B_{i}| u^{i}  (42)

is an increasing function of u ∈ [0, ∞).

Proof. Because each series is absolutely convergent, Fubini’s theorem implies, for k ≤ l,

\begin{aligned}∑_{i=l}^{∞} ∑_{j=k}^{i} A_{j}B_{i−j}x^{i} & = ∑_{j=k}^{l−1} ∑_{i=l}^{∞} A_{j}x^{j}B_{i−j}x^{i−j} + ∑_{j=l}^{∞} ∑_{i=j}^{∞} A_{j}x^{j}B_{i−j}x^{i−j} \\ & = ∑_{j=k}^{l−1} A_{j}x^{j} ∑_{q=l−j}^{∞} B_{q}x^{q} + ∑_{j=l}^{∞} A_{j}x^{j} ∑_{q=0}^{∞} B_{q}x^{q} \\ & = ∑_{j=k}^{∞} A_{j}x^{j} ∑_{q=0}^{∞} B_{q}x^{q} − ∑_{j=k}^{l−1} A_{j}x^{j} ∑_{q=0}^{l−j−1} B_{q}x^{q}. \end{aligned}  (43)

An application of the triangle inequality now yields the desired result.

Lemma 7 For k = 1, 2 and positive integers l > k, define h_{l,k} : ℝ → ℝ by h_{l,k}(z) ≡ ∑_{r=l−k}^{∞} \frac{(−z)^{r}}{(r + k)r!}.

Then, for l > k + 1,

(i) h′_{l,k}(z) = −h_{l,k+1}(z).

Furthermore, if x lies between z and 0, then

(ii) |h_{l,k}(x)| ≤ \frac{e^{−z}}{l} + 2 ∑_{r=0}^{l−1} \frac{|z|^{r}}{r!} , and

(iii) |z||h_{l,k}(x)| ≤ e^{−z} + 2 ∑_{r=0}^{l−1} \frac{|z|^{r}}{r!} .

Proof. The equality in (i) follows immediately from an interchange of summation and differentiation. The conditions that justify this interchange are easily verified in this case.

To prove (ii), begin by defining, for k = 1, 2, the continuous functions

h_{k}(z) ≡ \{ \begin{matrix}∑_{r=0}^{∞} \frac{(−z)^{r}}{(r + k)r!}, & if z ≠ 0 \\ \frac{1}{k} , & if z = 0. \end{matrix}  (44)

It is easy to see that, when z ≠ 0,

h_{1}(z) = \frac{1 − e^{−z}}{z}  and  h_{2}(z) = −\frac{d}{dz} h_{1}(z) = \frac{1 − e^{−z} − ze^{−z}}{z^{2}} .

It is also easy to check that h_{1}(z) and h_{2}(z) are nonnegative, decreasing functions of z ∈ ℝ. We now proceed to prove (ii) by considering separately the cases 0 ≤ x ≤ z and z ≤ x ≤ 0.

In the first case, h_{k}(z) ≤ h_{k}(0) and so, for k = 1, 2 and l > k,

|h_{l,k}(x)| = |h_{k}(x) − ∑_{r=0}^{l−k−1} \frac{(−x)^{r}}{(r + k)r!}| ≤ h_{k}(x) + ∑_{r=0}^{l−2} \frac{|x|^{r}}{(r + 1)!} ≤ h_{k}(0) + ∑_{r=0}^{l−2} \frac{|z|^{r}}{(r + 1)!} ≤ 2 ∑_{r=0}^{l−1} \frac{|z|^{r}}{r!} . (45)

Now suppose z ≤ x ≤ 0. Then 0 ≤ −x ≤ −z, and hence

|h_{l,k}(x)| = ∑_{r=l−k}^{∞} \frac{(−x)^{r}}{(r + k)r!} ≤ ∑_{r=l−k}^{∞} \frac{(−z)^{r}}{(r + k)r!} < \frac{1}{l} ∑_{r=l−k}^{∞} \frac{(−z)^{r}}{r!} ≤ \frac{e^{−z}}{l} .  (46)

So (ii) follows from (45) and (46).

To prove (iii), we begin by assuming 0 ≤ x ≤ z as before. Arguing as in (45), we obtain

|z||h_{l,k}(x)| ≤ |z|h_{k}(0) + ∑_{r=0}^{l−2} \frac{|z|^{r+1}}{(r + 1)!} ≤ 2 ∑_{r=0}^{l−1} \frac{|z|^{r}}{r!} .  (47)

In the case z ≤ x ≤ 0, we have

|z||h_{l,k}(x)| = −z ∑_{r=l−k}^{∞} \frac{(−x)^{r}}{(r + k)r!} ≤ ∑_{r=l−k}^{∞} \frac{(−z)^{r+1}}{(r + 1)r!} < ∑_{r=0}^{∞} \frac{(−z)^{r}}{r!} = e^{−z}.  (48)

Therefore, (iii) follows from (47) and (48).

Lemma 8 For m ≥ 2, let a_{m}(V) be defined as in (27), with V ∼ Beta (1, θ) and θ > 1. Then, for each j ≥ 1, there is a positive, increasing, deterministic function γ_{m,j}(t) such that, for all t > 0,

|∑_{k=j}^{∞} \frac{(−ta_{m}(V))^{k}}{k!} | ≤ t^{j}γ_{m,j}(t) |a_{m}(V)|^{j} .  (49)

Proof. First note that

|∑_{k=j}^{∞} \frac{(−ta_{m}(V))^{k}}{k!} | = |ta_{m}(V)|^{j} |∑_{k=j}^{∞} \frac{1}{k!} (−ta_{m}(V))^{k−j}| = t^{j}|a_{m}(V)|^{j} g_{j}(ta_{m}(V)) ,

where

0 < g_{j}(x) ≡ ∑_{k=j}^{∞} \frac{(−x)^{k−j}}{k!} = \{ \begin{matrix}\frac{e^{−x} − ∑_{k=0}^{j−1}(−x)^{k}/k!}{(−x)^{j}}, & x ≠ 0 \\ \frac{1}{j!}, & x = 0. \end{matrix}

To prove the desired inequality, it suffices to show that g_{j}(ta_{m}(V)) is a.s. bounded above as a function of θ > 1. A simple calculus exercise shows that g_{j}(x) is a decreasing function of x ∈ (−∞, ∞). Now define f(u) = u^{m} − m!2^{m}u. Note that f is bounded below on [0, ∞), and let u^{∗} = ((m − 1)!2^{m})^{1/(m−1)} be the point in [0, ∞) at which f achieves this minimum. Using (29) and the fact that θV > 0, we have

a_{m}(V) ≥ \frac{f(θV)}{\sqrt{θ}} ≥ \frac{f(u^{∗})}{\sqrt{θ}}.

Also, it is easy to see that f(u^{∗}) < 0, and hence f(u^{∗})/\sqrt{θ} > f(u^{∗}), for θ > 1. Therefore, for all t > 0,

g_{j}(ta_{m}(V)) ≤ g_{j}(\frac{tf(u^{∗})}{\sqrt{θ}}) ≤ g_{j}(tf(u^{∗})) ≡ γ_{m,j}(t) < ∞.

This complete the proof.

Lemma 9 Let Z_{m,θ} be defined by equation (13) and suppose m ≥ 2 and t > 0 are fixed. Then there exists θ_{0} = θ_{0}(t) < ∞ such that

sup_{θ>θ_{0}} 𝔼_{0}(e^{−tZ_{m,θ}}) < ∞.

Proof. Since m is fixed throughout, to simplify notation we will write

Z̄_{θ} ≡ (m − 1)!Z̃_{m,θ},  (50)

where Z̄_{θ} is independent of V as in (26). As before, let μ_{r,θ} = ((m − 1)!)^{r}𝔼_{0}(Z_{m,θ}^{r}) = 𝔼_{0}(Z̄_{θ}^{r}). By (34), it is enough to show, for t > 0, that there are finite values l = l(t) and θ_{0} = θ_{0}(l, t) such that

sup_{θ>θ_{0}} |∑_{r=l}^{∞} \frac{(−t)^{r}}{r!} μ_{r,θ}| < ∞.  (51)

(This is not true if −t is replaced with t, and it is this reliance on the alternating nature of the sum that makes the following argument rather delicate.)

First use (39) to write

|∑_{r=l}^{∞} \frac{(−t)^{r}}{r!} μ_{r,θ}| = |∑_{r=l}^{∞} \frac{mr + θ}{mr} \frac{(−t)^{r}}{r!} ∑_{k=1}^{r} \binom{r}{k}μ_{r−k,θ}𝔼 (a_{m}^{k}(V)(1 − V)^{m(r−k)})| ,  (52)

where a_{m}(V) is defined in (27). Motivated by Lemma 6, we define the following random variables:

A_{k} ≡ \frac{(a_{m}(V))^{k}}{k!} , B_{q} ≡ \frac{(Z̄_{θ}(1 − V)^{m})^{q}}{q!} .  (53)

Next, rewrite (52) as

\begin{aligned}|∑_{r=l}^{∞} \frac{(−t)^{r}}{r!} μ_{r,θ}| & = |∑_{r=l}^{∞} ∑_{k=1}^{r} (1 + \frac{θ}{mr} ) 𝔼 (A_{k}B_{r−k}) (−t)^{r}| \\ & = |𝔼 (∑_{r=l}^{∞} ∑_{k=1}^{r} (1 + \frac{θ}{mr} ) A_{k}B_{r−k} (−t)^{r})| \\ & ≤ |𝔼 (∑_{r=l}^{∞} ∑_{k=1}^{r} A_{k}B_{r−k} (−t)^{r})| \\ & + |𝔼 (∑_{r=l}^{∞} ∑_{k=1}^{r} \frac{θ}{mr} A_{k}B_{r−k} (−t)^{r})| \\ & ≡ |𝔼(I_{1})| + |𝔼(I_{2})|. \end{aligned}  (54)

Note that, for fixed θ, the random variables V, a_{m}(V), and Z̄_{θ} are bounded. Thus, there are positive constants, c_{θ} and d_{θ}, such that |A_{k}| ≤ c_{θ}^{k}/k! and |B_{q}| ≤ d_{θ}^{q}/q! hold almost surely. Therefore, the interchange of expectation with summation in (54) is justified by the dominated convergence theorem.

Our strategy is now to bound each of the two terms in the last line of (54). Indeed, we will demonstrate that for θ_{0} and l chosen sufficiently large, and all θ > θ_{0},

|𝔼(I_{1})| ≤ \frac{1}{4} ∑_{r=0}^{∞} \frac{(−t)^{r}}{r!} μ_{r,θ} + α_{1}(t),  (55)

|𝔼(I_{2})| ≤ \frac{1}{4} ∑_{r=0}^{∞} \frac{(−t)^{r}}{r!} μ_{r,θ} + α_{2}(t),  (56)

where α_{1}(t) and α_{2}(t) are suitably chosen deterministic functions of t. Once the bounds in equations (55) and (56) have been established, we will have

|∑_{r=l}^{∞} \frac{(−t)^{r}}{r!} μ_{r,θ}| ≤ \frac{1}{2} ∑_{r=0}^{∞} \frac{(−t)^{r}}{r!} μ_{r,θ} + α_{1}(t) + α_{2}(t).

Now collect all terms involving the common infinite series and put them on the left side to get

|∑_{r=l}^{∞} \frac{(−t)^{r}}{r!} μ_{r,θ}| − \frac{1}{2} ∑_{r=l}^{∞} \frac{(−t)^{r}}{r!} μ_{r,θ} ≤ \frac{1}{2} ∑_{r=0}^{l−1} \frac{t^{r}}{r!} |μ_{r,θ}| + α_{1}(t) + α_{2}(t).

Considering separately the cases where ∑_{r=l}^{∞}(−t)^{r} μ_{r,θ}/r! is positive and negative, it is easy to see that

\frac{1}{2} |∑_{r=l}^{∞} \frac{(−t)^{r}}{r!} μ_{r,θ}| ≤ \frac{1}{2} ∑_{r=0}^{l−1} \frac{t^{r}}{r!} |μ_{r,θ}| + α_{1}(t) + α_{2}(t).

Since the right-hand side is a sum of a finite number of terms, Lemma 5 can be used to conclude that

sup_{θ>θ_{0}} |∑_{r=l}^{∞} \frac{(−t)^{r}}{r!} μ_{r,θ}| < ∞.

Thus, the proof of Lemma 9 reduces to establishing the inequalities in (55) and (56).

(I_{1}): Recall that A_{k} and B_{k}, defined by (53), are bounded random variables. Therefore, we can apply Lemma 6 to I_{1}, with k = 1 and x = −t, to get

|I_{1}| ≤ |∑_{j=1}^{∞}(−t)^{j}A_{j} ∑_{q=0}^{∞}(−t)^{q}B_{q}| + tC_{1l}(t).  (57)

By Lemma 8, we have |∑_{j=1}^{∞}(−t)^{j}A_{j}| = |∑_{j=1}^{∞}(−ta_{m}(V))^{j}/j!| ≤ t γ_{m,1}(t)|a_{m}(V)|. In addition,

|∑_{q=0}^{∞} B_{q}(−t)^{q}| = exp{−tZ̄_{θ}(1 − V)^{m}} ≤ max {1, exp{−tZ̄_{θ}}} ≤ 1 + exp{−tZ̄_{θ}},  (58)

and so

|I_{1}| ≤ t γ_{m,1}(t) |a_{m}(V)| (1 + e^{−tZ̄_{θ}}) + tC_{1l}(t).  (59)

Therefore,

|𝔼(I_{1})| ≤ 𝔼|I_{1}| ≤ t γ_{m,1}(t) 𝔼|a_{m}(V)| · 𝔼(e^{−tZ̄_{θ}}) + t γ_{m,1}(t) 𝔼|a_{m}(V)| + t 𝔼(C_{1l}(t)).

Since C_{1l}(t) is a sum of a finite number of random variables with means uniformly bounded in θ, it follows that sup_{θ>1} 𝔼(C_{1l}(t)) < ∞. Since 𝔼|a_{m}(V)| ∼ O(θ^{−1/2}), there exists θ_{0} = θ_{0}(t) < ∞ such that tγ_{m,1}(t)𝔼|a_{m}(V)| < 1/4, for all θ > θ_{0}(t). Define α_{1}(t) ≡ sup_{θ>1} 𝔼(C_{1l}(t)) + 1/4. Then, for θ > θ_{0}(t),

|𝔼(I_{1})| ≤ \frac{1}{4}𝔼(e^{−tZ̄_{θ}}) + α_{1}(t) = \frac{1}{4} ∑_{r=0}^{∞} \frac{(−t)^{r}}{r!} μ_{r,θ} + α_{1}(t),

and this completes the proof of (55).

(I_{2}): Note that the factor θ/(mr) in I_{2} does not appear in I_{1}. Because of this, the proof of (56) requires a more delicate argument. We will need to consider separately the cases k = 1, k = 2, and k ≥ 3. Begin by writing

I_{2} = ∑_{r=l}^{∞} ∑_{k=1}^{r} \frac{θ}{mr} A_{k}B_{r−k} (−t)^{r}

= ∑_{r=l}^{∞} \frac{θ}{mr} A_{1}B_{r−1} (−t)^{r} + ∑_{r=l}^{∞} \frac{θ}{mr} A_{2}B_{r−2} (−t)^{r} + ∑_{r=l}^{∞} ∑_{k=3}^{r} \frac{θ}{mr} A_{k}B_{r−k} (−t)^{r}

≡ J_{1} + J_{2} + J_{3}.

To prove (56), we now proceed to bound the means of the last three terms.

(J_{1}): Defining X̄_{θ} ≡ (1 − V)^{m}Z̄_{θ} and using the notation in Lemma 7, we can rewrite J_{1} as

J_{1} = −\frac{tθ}{m} a_{m}(V) ∑_{r=l}^{∞} \frac{(−tX̄_{θ})^{r−1}}{r!} = −\frac{tθ}{m} a_{m}(V)h_{l,1}(tX̄_{θ}).

Now recall from Lemma 7 that h′_{l,1} = −h_{l,2} and apply the mean value theorem to h_{l,1} to get

\begin{aligned}J_{1} & = −\frac{tθ}{m} a_{m}(V) [h_{l,1}(tZ̄_{θ}) + t(X̄_{θ} − Z̄_{θ})h′_{l,1}(ξ)] \\ & = −\frac{tθ}{m} a_{m}(V) [h_{l,1}(tZ̄_{θ}) + t(Z̄_{θ} − X̄_{θ})h_{l,2}(ξ)] , \end{aligned}  (60)

where ξ lies between tX̄_{θ} and tZ̄_{θ}. Since Z̄_{θ} − X̄_{θ} = [1 − (1 − V)^{m}] Z̄_{θ}, it follows from (30) that

|Z̄_{θ} − X̄_{θ}| ≤ 2^{m}V |Z̄_{θ}|.  (61)

Also, X̄_{θ} and Z̄_{θ} have the same sign, so we can apply Lemma 7 to get

\begin{aligned}|tZ̄_{θ}||h_{l,2}(ξ)| & ≤ e^{−tZ̄_{θ}} + 2 ∑_{r=0}^{l−1} |tZ̄_{θ}|^{r} /r!, \\ and |h_{l,1}(tZ̄_{θ})| & ≤ e^{−tZ̄_{θ}} + 2 ∑_{r=0}^{l−1} |tZ̄_{θ}|^{r} /r!. \end{aligned}  (62)

Taking expectations in (60) and applying the bounds in (61) and (62), we get

\begin{aligned}|𝔼(J_{1})| & ≤ \frac{tθ}{m} |𝔼(a_{m}(V))| · 𝔼 |h_{l,1}(tZ̄_{θ})| + \frac{t^{2}θ}{m} 𝔼 |a_{m}(V)(Z̄_{θ} − X̄_{θ})h_{l,2}(ξ)| \\ & ≤ (\frac{tθ}{m} |𝔼(a_{m}(V))| + \frac{2^{m}t^{2}θ}{m} 𝔼 |V a_{m}(V)|) (𝔼(e^{−tZ̄_{θ}}) + 2 ∑_{r=0}^{l} 𝔼 |tZ̄_{θ}|^{r} /r!) , \end{aligned}

for each t > 0. It follows from (27), (28), and (37) that θ |𝔼(a_{m}(V))| ∼ m!O(1/\sqrt{θ}), and from (31) and (33) we have θ𝔼 |V a_{m}(V)| ∼ O(1/\sqrt{θ}). Therefore, we can choose θ large enough so that

|𝔼(J_{1})| ≤ \frac{1}{12}𝔼(e^{−tZ̄_{θ}}) + β_{1}(t) = \frac{1}{12} ∑_{r=0}^{∞} \frac{(−t)^{r}μ_{r,θ}}{r!} + β_{1}(t),  (63)

where β_{1}(t) = (1/6) sup_{θ>1} ∑_{r=0}^{l} 𝔼 |tZ̄_{θ}|^{r} /r!.

(J_{2}): Start with

|J_{2}| = |∑_{r=l}^{∞} \frac{θ}{mr} A_{2}B_{r−2}(−t)^{r}| = θa_{m}^{2}(V)\frac{t^{2}}{2m} |∑_{r=l}^{∞} \frac{(−tX̄_{θ})^{r−2}}{r(r − 2)!} | = θa_{m}^{2}(V)\frac{t^{2}}{2m} |h_{l,2}(tX̄_{θ})| .

Then, from part (ii) of Lemma 7,

|J_{2}| ≤ \frac{θt^{2}}{2m} a_{m}^{2}(V) (\frac{e^{−tZ̄_{θ}}}{l} + 2 ∑_{r=0}^{l} \frac{|tZ̄_{θ}|^{r}}{r!} ) .

Define λ(t) ≡ sup_{θ>1} \frac{θt^{2}}{2m} 𝔼 (a_{m}^{2}(V)). It follows from (32) that λ(t) < ∞, so choosing l > 12λ(t) leads to

𝔼|J_{2}| ≤ \frac{1}{12}𝔼(e^{−tZ̄_{θ}}) + 12λ(t)β_{1}(t) = \frac{1}{12} ∑_{r=0}^{∞} \frac{(−t)^{r}}{r!} μ_{r,θ} + β_{2}(t),  (64)

where β_{2}(t) ≡ 12λ(t)β_{1}(t).

(J_{3}): We now rewrite J_{3} and again apply Lemma 6 to get

|J_{3}| = |∫_{0}^{t} \frac{1}{s} ∑_{r=l}^{∞} ∑_{k=3}^{r} \frac{θ}{m}A_{k}B_{r−k}(−s)^{r}ds|

≤ ∫_{0}^{t} (\frac{1}{s} |∑_{k=3}^{∞} \frac{θ}{m}A_{k}(−s)^{k} ∑_{q=0}^{∞} B_{q}(−s)^{q}| + s^{3}C_{3l}(s)) ds

≤ ∫_{0}^{t} \frac{1}{s} |∑_{k=3}^{∞} \frac{θ}{m}A_{k}(−s)^{k}| (∑_{q=0}^{∞} B_{q}(−s)^{q}) ds + t^{2}C_{3l}(t).

It follows from Lemma 8 that |∑_{k=3}^{∞} A_{k}(−s)^{k}| = |∑_{k=3}^{∞}(−sa_{m}(V))^{k}/k!| ≤ s^{3}γ_{m,3}(t) |a_{m}(V)|^{3}, for 0 ≤ s ≤ t, and by the same argument used in (58), we have 0 < ∑_{q=0}^{∞} B_{q}(−s)^{q} ≤ 1 + exp{−tZ̄_{θ}}. Therefore,

|J_{3}| ≤ (1 + e^{−tZ̄_{θ}}) \frac{γ_{m,3}(t)}{m} θ |a_{m}(V)|^{3} ∫_{0}^{t} s^{2}ds + t^{2}C_{3l}(t),

and this implies, by independence of V and Z̄_{θ},

𝔼|J_{3}| ≤ t^{3} \frac{γ_{m,3}(t)}{m} θ 𝔼|a_{m}(V)|^{3} · 𝔼(e^{−tZ̄_{θ}}) + t^{3} \frac{γ_{m,3}(t)}{m} θ 𝔼|a_{m}(V)|^{3} + t^{2}𝔼 (C_{3l}(t)) .

By (32), we can choose θ so large that t^{3} \frac{γ_{m,3}(t)}{m} θ 𝔼|a_{m}(V)|^{3} < 1/12. Therefore, setting β_{3}(t) ≡ 1/12 + t^{2} sup_{θ>1} 𝔼 (C_{3l}(t)), we arrive at

𝔼|J_{3}| ≤ \frac{1}{12}𝔼(e^{−tZ̄_{θ}}) + β_{3}(t) = \frac{1}{12} ∑_{r=0}^{∞} \frac{(−t)^{r}μ_{r,θ}}{r!} + β_{3}(t),  (65)

when θ is sufficiently large. So (56) now follows from equations (63),(64), and (65), with α_{2}(t) = β_{1}(t) + β_{2}(t) + β_{3}(t).

§5. Rescaled Likelihood Ratios and Sampling Distributions

In this section we extend our uniform integrability and limit results to rescaled versions of the likelihood ratios and sampling distributions. These results will be used in the proofs of Theroems 3, 4, and 5.

Lemma 10 Suppose σ = cθ^{3/2−α}, where α > 0 and c > 0 are constants.

(i) If X_{θ} = (X_{1,θ}, X_{2,θ}, …) ∼ μ_{0}, then

θ^{α} (\frac{dμ_{σ}}{dμ_{0}} (X_{θ}) − 1) = −cZ_{2,θ} + L(Z_{2,θ}),

where lim_{θ→∞} 𝔼_{0}|L(Z_{2,θ})|^{r} = 0 for all r > 0.

(ii) If r > 0, then

sup_{θ>1} 𝔼_{0} |θ^{α} (\frac{dμ_{σ}}{dμ_{0}} (X_{θ}) − 1)|^{r} < ∞.

(iii) If Y_{θ}^{(−α)} = (Y_{1,θ}^{(−α)}, Y_{2,θ}^{(−α)}, …) ∼ μ_{σ}, then

θ^{α} (\frac{dμ_{σ}}{dμ_{0}} (Y_{θ}^{(−α)}) − 1) = −cZ_{2,θ}^{(−α)} + L_{α}(Z_{2,θ}^{(−α)}),

where lim_{θ→∞} 𝔼_{σ}|L_{α}(Z_{2,θ}^{(−α)})|^{r} = 0 for all r > 0.

Remark. Lemma 10, when combined with Lemma 1, will provide a quick proof of Theorem 3. Part (ii) yields a uniform integrability condition that will be needed in the proofs of Theorems 4 and 5.

Proof. Beginning with (i), recall that, for σ = cθ^{3/2−α}, \frac{dμ_{σ}}{dμ_{0}} (X_{θ}) = e^{−cθ^{−α}Z_{2,θ}}/𝔼_{0}(e^{−cθ^{−α}Z_{2,θ}}). Using the Taylor decomposition, e^{−x} = 1−x+R(x), with 0 < R(x) < max{x^{2}/2, e^{−x}x^{2}/2} ≤ (x^{2}/2)(1+e^{−x}), we can write

e^{−cθ^{−α}Z_{2,θ}} = 1 − cθ^{−α}Z_{2,θ} + R(cθ^{−α}Z_{2,θ}).

For notational convenience let us denote R(cθ^{−α}Z_{2,θ}) by R_{θ}.

As a consequence of the classical result 𝔼_{0}(H(X_{θ})) = 1/(1 + θ), we have

𝔼_{0}(e^{−cθ^{−α}Z_{2,θ}}) = 1 − 𝔼_{0}(cθ^{−α}Z_{2,θ}) + 𝔼_{0}(R_{θ})

= 1 + cθ^{1/2−α} \frac{1}{1 + θ} + 𝔼_{0}(R_{θ}).

Hence, the quantity of interest can be written as

θ^{α} (\frac{e^{−cθ^{−α}Z_{2,θ}}}{𝔼_{0}(e^{−cθ^{−α}Z_{2,θ}})} − 1) = \frac{θ^{α}}{𝔼_{0}(e^{−cθ^{−α}Z_{2,θ}})} (1 − cθ^{−α}Z_{2,θ} + R_{θ} − 𝔼_{0}(e^{−cθ^{−α}Z_{2,θ}}))

= \frac{θ^{α}}{𝔼_{0}(e^{−cθ^{−α}Z_{2,θ}})} (−cθ^{−α}Z_{2,θ} + R_{θ} − c \frac{θ^{1/2−α}}{1 + θ} − 𝔼_{0}(R_{θ}))

= \frac{1}{𝔼_{0}(e^{−cθ^{−α}Z_{2,θ}})} (−cZ_{2,θ} − c \frac{\sqrt{θ}}{1 + θ} + θ^{α}R_{θ} − 𝔼_{0}(θ^{α}R_{θ}))

= −cZ_{2,θ} + L(Z_{2,θ}),

where we define

L(Z_{2,θ}) ≡ \frac{1}{𝔼_{0}(e^{−cθ^{−α}Z_{2,θ}})} (−c \frac{\sqrt{θ}}{1 + θ} + θ^{α}R_{θ} − θ^{α}𝔼_{0}(R_{θ})) − cZ_{2,θ} (\frac{1}{𝔼_{0}(e^{−cθ^{−α}Z_{2,θ}})} − 1) .

Since Lemma 9 implies, for any constant δ > 1,

sup_{θ>θ_{0}(δ)} 𝔼_{0}(e^{−δθ^{−α}Z_{2,θ}}) ≤ 1 + sup_{θ>θ_{0}(δ)} 𝔼_{0}(e^{−δZ_{2,θ}}) < ∞,  (66)

we see that 𝔼_{0}(e^{−cθ^{−α}Z_{2,θ}}) → 1. Furthermore, using the Cauchy–Schwarz inequality, it is clear that

\begin{aligned}0 < θ^{rα} 𝔼_{0}|R_{θ}^{r}| & < θ^{rα} 𝔼_{0} ((cθ^{−α}Z_{2,θ})^{2r} (1 + exp\{−cθ^{−α}Z_{2,θ}\})^{r}) \\ & < c^{2r}θ^{−rα}\sqrt{𝔼_{0}(Z_{2,θ}^{4r}) 𝔼_{0}(1 + exp\{−cθ^{−α}Z_{2,θ}\})^{2r}} \end{aligned}

holds for all r > 0. Thus we can use Lemmas 5 and 9 to get θ^{rα} 𝔼_{0}|R_{θ}^{r}| → 0, as θ → ∞. A simple argument, together with Lemmas 5 and 9, now produces

lim_{θ→∞} 𝔼_{0}|L(Z_{2,θ})|^{r} = 0,

finishing the proof of (i).

Part (ii) follows immediately from part (i) and Lemma 5.

To prove part (iii), one argues as in the proof of (i), making the obvious changes. Replacing L(Z_{2,θ}) will be

L_{α}(Z_{2,θ}^{(−α)}) ≡ \frac{1}{𝔼_{0}(e^{−cθ^{−α}Z_{2,θ}})} (−c \frac{\sqrt{θ}}{1 + θ} + θ^{α}R_{θ}^{(−α)} − θ^{α}𝔼_{0}(R_{θ})) − cZ_{2,θ}^{(−α)} (\frac{1}{𝔼(e^{−cθ^{−α}Z_{2,θ}})} − 1) ,

where R_{θ}^{(−α)} ≡ R(cθ^{−α}Z_{2,θ}^{(−α)}). The last part of (iii) requires uniform integrability of Z_{2,θ}^{(−α)} and e^{−Z_{2,θ}^{(−α)}}. These follow from Lemmas 5 and 9 together with part (ii) above.

Sampling distributions and homozygosities. Recall that Theorems 4 and 5 involve limits for sampling distributions. Our next goal is to establish moment bounds for certain scalings of the conditional sampling functions, ϕ_{a}(X_{θ}). These will lead to the uniform integrability we need to prove the aforementioned theorems. For example, we wish to show that if the partition a satisfies a_{1} ≠ n (i.e., not all alleles in the sample are distinct), then

sup_{θ>1} 𝔼_{0} |\sqrt{θ} (\frac{ϕ_{a}(X_{θ}) − ESF(θ, a)}{ESF(θ, a)})|^{r} < ∞.  (67)

To aid us in our derivation of (67), we will exploit a relationship between the population moments, H_{m}(X_{θ}), and the conditional sampling function, ϕ_{a}(X_{θ}).

This is given by the following general identity, which holds for any partition a of a sample of size n:

∏_{m=2}^{n} (\frac{θ^{m−1}}{(m − 1)!} H_{m}(X_{θ}))^{a_{m}} = \frac{θ^{n}}{θ_{(n)}} \frac{ϕ_{a}(X_{θ})}{𝔼_{0}(ϕ_{a}(X_{θ}))} + \frac{θ^{n−k}}{∏_{m=2}^{n}((m − 1)!)^{a_{m}}} ∑_{b∈C_{a}} l_{b}ϕ_{b}(X_{θ}),  (68)

where C_{a} is defined to be the collection of all partitions formed by coalescing two or more of the classes associated with a, and l_{b} is a nonnegative combinatorial factor whose exact form is not important here. A derivation appears in Joyce, Krone, and Kurtz (2001). To understand why this identity will be useful, note that the term on the left involves population homozygosities and the last term is small when θ is large, as will be seen below. So the large-θ asymptotics of ϕ_{a}(X_{θ}) will be determined by those of the various population homozygosities. This is helpful because the population homozygosities are especially simple examples of sampling distributions.

Since 𝔼_{0}(ϕ_{a}(X_{θ})) = ESF(θ, a), the above identity can be used to obtain

|\frac{ϕ_{a}(X_{θ}) − ESF(θ, a)}{ESF(θ, a)}| = |\frac{ϕ_{a}(X_{θ})}{𝔼_{0}(ϕ_{a}(X_{θ}))} − 1|

≤ |\frac{θ_{(n)}}{θ^{n}} ∏_{m=2}^{n} (\frac{θ^{m−1}H_{m}(X_{θ})}{(m − 1)!})^{a_{m}} − 1| + \frac{θ_{(n)}}{θ^{n}} \frac{θ^{n−k}}{∏_{m=2}^{n}((m − 1)!)^{a_{m}}} ∑_{b∈C_{a}} l_{b}ϕ_{b}(X_{θ}).

Next, notice that \sqrt{θ} (\frac{θ_{(n)}}{θ^{n}} − 1) ∼ O(1/\sqrt{θ}), and set B ≡ sup_{θ>1} \sqrt{θ} |\frac{θ_{(n)}}{θ^{n}} − 1|. Thus the quantity whose rth moment appears in (67) has absolute value

\sqrt{θ} |\frac{ϕ_{a}(X_{θ})}{𝔼_{0}(ϕ_{a}(X_{θ}))} − 1|

≤ \sqrt{θ} | ∏_{m=2}^{n} (\frac{θ^{m−1}H_{m}(X_{θ})}{(m − 1)!})^{a_{m}} − 1| + \sqrt{θ} |\frac{θ_{(n)}}{θ^{n}} − 1| ∏_{m=2}^{n} (\frac{θ^{m−1}H_{m}(X_{θ})}{(m − 1)!})^{a_{m}}

+\frac{θ^{n−k}}{∏_{m=2}^{n}((m − 1)!)^{a_{m}}} ∑_{b∈C_{a}} l_{b}ϕ_{b}(X_{θ})  (69)

≤ \sqrt{θ} | ∏_{m=2}^{n} (\frac{θ^{m−1}H_{m}(X_{θ})}{(m − 1)!})^{a_{m}} − 1| + B ∏_{m=2}^{n} |\frac{Z_{m,θ}}{\sqrt{θ}} + 1|^{a_{m}} + \frac{θ^{n−k}}{∏_{m=2}^{n}((m − 1)!)^{a_{m}}} ∑_{b∈C_{a}} l_{b}ϕ_{b}(X_{θ}).

The first term in the last line of (69) is the most important, and we isolate it in the first part of the next lemma. The uniform integrability condition given in (67) will then be given, followed by the last part of the lemma treating the special case when a_{1} = n.

Lemma 11 Suppose X_{θ} ∼ μ_{0} and let a be a partition corresponding to a sample of size n drawn from a population with ordered frequencies X_{θ}.

(i) If a_{1} ≠ n, then

sup_{θ>1} θ^{r/2}𝔼_{0} | ∏_{m=2}^{n} (\frac{θ^{m−1}H_{m}(X_{θ})}{(m − 1)!})^{a_{m}} − 1|^{r} < ∞, for each r > 0.  (70)

(ii) If a_{1} ≠ n, then

sup_{θ>1} θ^{r/2}𝔼_{0} |\frac{ϕ_{a}(X_{θ}) − ESF(θ, a)}{ESF(θ, a)}|^{r} < ∞, for each r > 0.

(iii) If a_{1} = (n, 0, …, 0) and a_{2} = (n − 2, 1, 0, …, 0) are the allelic partitions with n singletons and n − 2 singletons, respectively, then

sup_{θ>1} θ^{r/2}𝔼_{0} |\frac{ϕ_{a_{1}}(X_{θ}) − ESF(θ, a_{1})}{ESF(θ, a_{2})}|^{r} < ∞, for each r > 0.

Proof. We prove (i) by induction on the sample size n. If n = 2, then the assumption, a_{1} ≠ n, implies a_{2} must be 1. In this case, equation (70) reduces to the uniform integrability result in Lemma 5:

sup_{θ>1} θ^{r/2}𝔼_{0} |θH_{2}(X_{θ}) − 1|^{r} = sup_{θ>1} 𝔼_{0}|Z_{2,θ}|^{r} < ∞.

Now assume that (70) holds for all partitions corresponding to samples of size less than or equal to n, and let a = (a_{1}, a_{2}, …, a_{n+1}) be a partition corresponding to a sample of size n + 1. For partitions with a_{n+1} = 1 and with a_{n} = 1, a_{1} = 1, the quantity of interest,

θ^{r/2} | ∏_{m=2}^{n+1} (\frac{θ^{m−1}H_{m}(X_{θ})}{(m − 1)!})^{a_{m}} − 1|^{r} ,

reduces to |Z_{n+1,θ}|^{r} and |Z_{n,θ}|^{r}, respectively, and hence can be handled by Lemma 5. (Notice that terms associated with a_{1} never appear in the product.) Since the hypothesis in (i) says a_{1} ≠ n + 1, we are left to treat the situation in which a_{j} ≥ 1 for some 1 < j ≤ n − 1. For such a j, define a_{j} = (a_{j1}, …, a_{jn}) to be a partition corresponding to a sample of size n + 1 − j formed by removing one allele with j representatives. That is, a_{jj} = a_{j} − 1, and a_{ji} = a_{i} for all i ≠ j. Then

θ^{r/2} | ∏_{m=2}^{n+1} (\frac{θ^{m−1}H_{m}(X_{θ})}{(m − 1)!})^{a_{m}} − 1|^{r}

= θ^{r/2} |\frac{θ^{j−1}H_{j}(X_{θ})}{(j − 1)!} ( ∏_{m=2}^{n+1−j} (\frac{θ^{m−1}H_{m}(X_{θ})}{(m − 1)!})^{a_{mj}} − 1) + \frac{θ^{j−1}H_{j}(X_{θ})}{(j − 1)!} − 1|^{r}  (71)

≤ 2^{r}θ^{r/2} |\frac{θ^{j−1}H_{j}(X_{θ})}{(j − 1)!} ( ∏_{m=2}^{n+1−j} (\frac{θ^{m−1}H_{m}(X_{θ})}{(m − 1)!})^{a_{mj}} − 1)|^{r} + 2^{r}|Z_{j,θ}|^{r}.

Since Lemma 5 can be used to handle the mean of the last term, it remains to treat the first term in the last line of (71). An appeal to the Cauchy–Schwarz inequality allows us to write

𝔼_{0} |\frac{θ^{j−1}H_{j}(X_{θ})}{(j − 1)!} ( ∏_{m=2}^{n+1−j} (\frac{θ^{m−1}H_{m}(X_{θ})}{(m − 1)!})^{a_{mj}} − 1)|^{r}

≤ \sqrt{𝔼_{0} |\frac{θ^{j−1}H_{j}(X_{θ})}{(j − 1)!}|^{2r} 𝔼_{0} | ∏_{m=2}^{n+1−j} (\frac{θ^{m−1}H_{m}(X_{θ})}{(m − 1)!})^{a_{mj}} − 1|^{2r}}.  (72)

Clearly, the induction hypothesis implies

sup_{θ>0} θ^{r/2}𝔼_{0} | ∏_{m=2}^{n+1−j} (\frac{θ^{m−1}H_{m}(X_{θ})}{(m − 1)!})^{a_{mj}} − 1|^{2r} < ∞,

and the first term under the square root is handled with Lemma 5. Combining all this with (71) and (72) completes the proof of (i).

To prove (ii), we proceed by induction on the number, k, of different alleles represented in the sample. If k = 1, then all the sampled individuals have the same allele; i.e., a_{n} = 1. Thus ϕ_{a}(X_{θ}) = ∑ X_{i,θ}^{n}, and (ii) follows as a special case of Lemma 5, after a simple argument to account for the fact that the scaling terms are only asymptotically the same. Next, assume (ii) holds for all partitions of n with fewer than k distinct alleles. In other words, if b is such a partition and we define

D_{b}(X_{θ}) ≡ \sqrt{θ} (\frac{ϕ_{b}(X_{θ})}{𝔼_{0}(ϕ_{b}(X_{θ}))} − 1) ,

then the induction hypothesis implies sup_{θ>1} 𝔼_{0}|D_{b}(X_{θ})|^{r} < ∞, for all r > 0. However, from the definition of D_{b}(X_{θ}), we can write

(ϕ_{b}(X_{θ}))^{r} = (𝔼_{0}(ϕ_{b}(X_{θ})))^{r} (\frac{D_{b}(X_{θ})}{\sqrt{θ}} + 1)^{r} .  (73)

To apply this to a partition a with k distinct alleles, recall the inequality in (69). Clearly, to complete the induction argument, it suffices to show that the rth moment of each of the terms in the last line of (69) is uniformly bounded in θ > 1. For the first term in (69), just apply part (i) above. For the last term in (69), using the fact that the number of distinct alleles represented in any partition b ∈ C_{a} is less than or equal to k − 1, we can infer from the Ewens sampling formula that 𝔼_{0}(ϕ_{b}(X_{θ})) ∼ O(θ^{k−1−n}), and so, taking expectations on both sides of (73), we get

sup_{θ>1} θ^{r(n−k)}𝔼_{0}(ϕ_{b}(X_{θ}))^{r} < ∞.  (74)

From this, it follows immediately that the rth moment of the last term in (69) is bounded in θ > 1. Finally, the middle term in (69) is easy to handle; just use Lemma 5 and Hölder’s inequality to see that

sup_{θ>1} 𝔼_{0} ( ∏_{m=2}^{n} |\frac{Z_{m,θ}}{\sqrt{θ}} + 1|^{ra_{m}}) < ∞.  (75)

To prove (iii), we note that ϕ_{a_{1}}(X_{θ}) = 1 − ∑_{a≠a_{1}} ϕ_{a}(X_{θ}). Therefore,

\frac{ϕ_{a_{1}}(X_{θ}) − 𝔼_{0} (ϕ_{a_{1}}(X_{θ}))}{𝔼_{0} (ϕ_{a_{2}}(X_{θ}))} = \frac{𝔼_{0} (∑_{a≠a_{1}} ϕ_{a}(X_{θ})) − ∑_{a≠a_{1}} ϕ_{a}(X_{θ})}{𝔼_{0} (ϕ_{a_{2}}(X_{θ}))}

= \frac{𝔼_{0}(ϕ_{a_{2}}(X_{θ})) − ϕ_{a_{2}}(X_{θ})}{𝔼_{0} (ϕ_{a_{2}}(X_{θ}))} + ∑_{a≠a_{1},a_{2}} \frac{𝔼_{0}(ϕ_{a}(X_{θ})) − ϕ_{a}(X_{θ})}{𝔼_{0} (ϕ_{a}(X_{θ}))} \frac{𝔼_{0} (ϕ_{a}(X_{θ}))}{𝔼_{0} (ϕ_{a_{2}}(X_{θ}))}.  (76)

It follows from direct calculation using the Ewens sampling formula (8) that lim_{θ→∞} \frac{𝔼_{0} (ϕ_{a}(X_{θ}))}{𝔼_{0} (ϕ_{a_{2}}(X_{θ}))} = 0, if a ≠ a_{1}, a_{2}, and thus

sup_{θ>1} \frac{𝔼_{0} (ϕ_{a}(X_{θ}))}{𝔼_{0} (ϕ_{a_{2}}(X_{θ}))} < ∞.  (77)

Therefore

sup_{θ>1} θ^{r/2}𝔼_{0} |\frac{ϕ_{a_{1}}(X_{θ}) − ESF(θ, a_{1})}{ESF(θ, a_{2})}|^{r} < ∞

follows from part (ii) and equations (76) and (77).

§6. Proofs of the Main Results

Proof of Theorem 1. It follows from (18) that cZ_{2,θ} ⇒ cZ_{2}, where Z_{2} ∼ N(0, 2). So, in the case σ = cθ^{3/2}, we can use (15), Lemma 4, and Lemma 9, with Y_{n} replaced by cZ_{2,θ} and n with θ, to get

\frac{dμ_{σ}}{dμ_{0}} (X_{θ}) = \frac{exp\{−cZ_{2,θ}\}}{𝔼_{0}(exp\{−cZ_{2,θ}\})} ⇒ \frac{e^{cZ_{2}}}{𝔼(e^{−cZ_{2}})} = exp\{cZ_{2} − c^{2}\}.  (78)

Here we have used 𝔼(e^{−cZ_{2}}) = e^{c^{2}} and the fact that Z_{2} is equal in distribution to −Z_{2}. (Note that the t in Lemma 9 can be chosen so ct > 1.) The cases σ = cθ^{3/2+γ} with γ < 0 and γ > 0 of Theorem 1 also follow by applying Lemma 4.

Proof of Theorem 2. Suppose Y_{θ}^{(γ)} ∼ μ_{σ}, where σ = cθ^{3/2+γ}. First consider the case γ ≤ 0. By (16), we have

\frac{dμ_{σ}}{dμ_{0}} (Y_{θ}^{(γ)}) = \frac{exp\{−cθ^{γ}Z_{2,θ}^{(γ)}\}}{𝔼_{0}(exp\{−cθ^{γ}Z_{2,θ}\})},  (79)

and so the desired result follows from the continuous mapping theorem for weak limits and the uniform integrability obtained from (66) and Lemma 9.

Now assume that σ = cθ^{3/2+γ}, with γ > 0, and let g ≡ \frac{dμ_{σ}}{dμ_{0}} . Then

𝔼_{σ} (\frac{1}{\sqrt{g(Y_{θ}^{(γ)})}}) = 𝔼_{0} (\frac{g(X_{θ})}{\sqrt{g(X_{θ})}}) = 𝔼_{0} (\sqrt{g(X_{θ})}) .

It follows from Lemma 3 and equation (15) that lim_{θ→∞} 𝔼_{0}(\sqrt{g(X_{θ})}) = 0. Thus, we conclude that lim_{θ→∞} 𝔼_{σ} (1/\sqrt{g(Y_{θ}^{(γ)})}) = 0, implying g(Y_{θ}^{(γ)}) ⇒ ∞.

Proof of Theorem 3. The result follows immediately from Lemmas 1 and 10.

Proof of Theorem 4. Suppose σ = cθ^{3/2−α} and a ≠ (n, 0, …, 0) is a partition for a sample of size n. Treating first the case α > 0, we have

θ^{α+1/2} |\frac{ℙ_{σ}(A_{n} = a)}{ℙ_{0}(A_{n} = a)} − 1| = θ^{α+1/2} |\frac{ℙ_{σ}(A_{n} = a) − ℙ_{0}(A_{n} = a)}{ℙ_{0}(A_{n} = a)}|

= θ^{α+1/2} |∫ \frac{dμ_{σ}}{dμ_{0}} (x) (\frac{ϕ_{a}(X_{θ}) − ESF(θ, a)}{ESF(θ, a)}) μ_{0}(dx)|

= |𝔼_{0} (\sqrt{θ} (\frac{ϕ_{a}(X_{θ}) − ESF(θ, a)}{ESF(θ, a)}) θ^{α} \frac{dμ_{σ}}{dμ_{0}} (X_{θ}))|

= |𝔼_{0} (\sqrt{θ} (\frac{ϕ_{a}(X_{θ}) − ESF(θ, a)}{ESF(θ, a)}) θ^{α} \frac{dμ_{σ}}{dμ_{0}} (X_{θ}) − θ^{α} + θ^{α})|

= |𝔼_{0} (\sqrt{θ} (\frac{ϕ_{a}(X_{θ}) − ESF(θ, a)}{ESF(θ, a)}) θ^{α} (\frac{dμ_{σ}}{dμ_{0}} (X_{θ}) − 1))| .

Note that Lemma 11 and part (ii) of Lemma 10, together with the Cauchy–Schwarz inequality, gives uniform integrability of the last expression. Therefore, we can bring limits inside the expectation and use part (i) of Lemma 2, together with Theorem 3, to get

lim_{θ→∞} θ^{α+1/2} |\frac{ℙ_{σ}(A_{n} = a)}{ℙ_{0}(A_{n} = a)} − 1| = c |∑_{i=2}^{n} a_{i}Cov (Z_{i}, Z_{2})| = 2c ∑_{i=2}^{n} a_{i}\binom{i}{2}.

In the case α = 0, applying Lemma 2 and noting that uniform integrability follows by Lemmas 9 and 11, we have

lim_{θ→∞} θ^{1/2} |\frac{ℙ_{σ}(A_{n} = a)}{ℙ_{0}(A_{n} = a)} − 1| = lim_{θ→∞} |𝔼_{0} (\sqrt{θ} (\frac{ϕ_{a}(X_{θ}) − ESF(θ, a)}{ESF(θ, a)}) \frac{e^{−cZ_{2,θ}}}{𝔼_{0}(e^{−cZ_{2,θ}})})|

= ∑_{i=2}^{n} a_{i} \frac{|𝔼 (Z_{i}e^{−cZ_{2}}) |}{𝔼(e^{−cZ_{2}})}.

It follows from standard properties of bivariate normal distributions that

𝔼(e^{−cZ_{2}}Z_{i}) = 𝔼 (e^{−cZ_{2}}𝔼(Z_{i}|Z_{2})) = \frac{Cov(Z_{2}, Z_{i})}{VarZ_{2}}𝔼 (e^{−cZ_{2}}Z_{2}) = \frac{Cov(Z_{2}, Z_{i})}{2}𝔼 (e^{−cZ_{2}}Z_{2})  (80)

and

𝔼 (e^{−cZ_{2}}Z_{2}) = ∫_{−∞}^{∞} \frac{1}{2\sqrt{π}} ze^{−z^{2}/4}e^{−cz}dz = e^{c^{2}} ∫_{−∞}^{∞} \frac{1}{2\sqrt{π}} ze^{−(z+2c)^{2}/4}dz = −2ce^{c^{2}}.  (81)

Therefore,

lim_{θ→∞} θ^{1/2} |\frac{ℙ_{σ}(A_{n} = a)}{ℙ_{0}(A_{n} = a)} − 1| = e^{−c^{2}} ∑_{i=2}^{n} a_{i} |𝔼 (Z_{i}e^{−cZ_{2}})| = e^{−c^{2}} ∑_{i=2}^{n} a_{i} \frac{Cov(Z_{2}, Z_{i})}{2}(2ce^{c^{2}})

= c ∑_{i=2}^{n} a_{i}Cov(Z_{2}, Z_{i}) = 2c ∑_{i=2}^{n} a_{i}\binom{i}{2}.

Proof of Theorem 5. Suppose a_{1} = (n, 0, …, 0) and a_{2} = (n − 2, 1, 0, …, 0) are the allelic partitions with n singletons and n − 2 singletons, respectively. Then, for α > 0,

θ^{α+3/2} |\frac{ℙ_{σ}(A_{n} = a_{1})}{ℙ_{0}(A_{n} = a_{1})} − 1|

= \frac{θ^{α+1}}{𝔼_{0}(e^{−cθ^{−α}Z_{2,θ}})} |𝔼_{0} (\sqrt{θ} (\frac{ϕ_{a_{1}}(X_{θ}) − ESF(θ, a_{1})}{ESF(θ, a_{1})}) \frac{dμ_{σ}}{dμ_{0}} (X_{θ}))|

= \frac{θ^{α+1}}{𝔼_{0}(e^{−cθ^{−α}Z_{2,θ}})} \frac{ESF(θ, a_{2})}{ESF(θ, a_{1})} |𝔼_{0} (\sqrt{θ} (\frac{ϕ_{a_{1}}(X_{θ}) − ESF(θ, a_{1})}{ESF(θ, a_{2})}) (\frac{dμ_{σ}}{dμ_{0}} (X_{θ}) − 1))|

= \frac{θ}{𝔼_{0}(e^{−cθ^{−α}Z_{2,θ}})} \frac{ESF(θ, a_{2})}{ESF(θ, a_{1})} |𝔼_{0} (\sqrt{θ} (\frac{ϕ_{a_{1}}(X_{θ}) − ESF(θ, a_{1})}{ESF(θ, a_{2})}) θ^{α} (\frac{dμ_{σ}}{dμ_{0}} (X_{θ}) − 1))| .

Note that θ ESF(θ, a_{2})/ESF(θ, a_{1}) → 1, and 𝔼_{0}(e^{−cθ^{−α}Z_{2,θ}}) → 1, as θ → ∞. Once again, the uniform integrability established in Lemmas 10 and 11 allows us to bring the limit inside the last expected value. It then follows from part (ii) of Lemma 2, and Theorem 3, that

lim_{θ→∞} θ^{α+3/2} |\frac{ℙ_{σ}(A_{n} = a_{1})}{ℙ_{0}(A_{n} = a_{1})} − 1| = c |𝔼_{0} (−Z_{2}Z_{2})| = cVar(Z_{2}) = 2c.

Acknowledgments: We thank John Gillespie for bringing this problem to our attention. We would also like to thank Bob Griffiths for suggesting the recursion satisfied by Z_{m,θ}.

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Paul Joyce  Stephen M. Krone
Department of Mathematics  Department of Mathematics
University of Idaho  University of Idaho
Moscow, ID 83844  Moscow, ID 83844

   joyce@uidaho.edu                         krone@uidaho.edu

Thomas G. Kurtz

Departments of Mathematics and Statistics

University of Wisconsin Madison, WI 53706-1388

   kurtz@math.wisc.edu