Limit Processes with Independent Increments for the Ewens Sampling Formula

Ewens sampling formula (ESF) is a one-parameter family of probability distributions with a number of intriguing combinatorial connections. This elegant closed-form formula first arose in biology as the stationary probability distribution of a sample configuration at one locus under the infinite-alleles model of mutation. Since its discovery in the early 1970s, the ESF has been used in various biological applications, and has sparked several interesting mathematical generalizations. In the population genetics community, extending the underlying random-mating model to include recombination has received much attention in the past, but no general closed-form sampling formula is currently known even for the simplest extension, that is, a model with two loci. In this paper, we show that it is possible to obtain useful closed-form results in the case the population-scaled recombination rate ρ is large but not necessarily infinite. Specifically, we consider an asymptotic expansion of the two-locus sampling formula in inverse powers of ρ and obtain closed-form expressions for the first few terms in the expansion. Our asymptotic sampling formula applies to arbitrary sample sizes and configurations.

Since the highlights of our professional life are what makes it interesting and rewarding, I have decided to describe in this account the events surrounding the derivation of the so-called “Ewens sampling formula” of population genetics, which has formed a high point in my career as an applied probabilist. Before doing this I will briefly describe my background and early training in population genetics theory.

Large deviation principles are established for some random\ncombinatorial structures including the Ewens sampling formula and the Pitman\nsampling formula. A path-level large deviation principle is established for the\nformer on the cadlag space D$(o, 1], R)$ equipped with the uniform convergence\ntopology, and the rate function is the same as for a Poisson process justifying\nthe Poisson process approximation for the Ewens sampling formula at the large\ndeviation level. A large deviation principle for the total number of parts in a\npartition is obtained for the Pitman formula; here the rate function depends\nonly on one of the two parameters which display the different roles of the two\nparameters at different scales. In addition to these large deviation results,\nwe also provide an embedding scheme which gives the Pitman sampling formula. A\nproduct of this embedding is an intuitive alternate proof of a result of Pitman\non the limiting total number of parts.

Previous article Next article On the Extension of the Class of Stable DistributionsV. M. KruglovV. M. Kruglovhttps://doi.org/10.1137/1117081PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] B. V. Gnedenko and , A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, Addison-Wesley, Cambridge, Mass., 1962 Google Scholar[2] Yu. V. Linnik, Decomposition of probability distributions, Edited by S. J. Taylor, Dover Publications Inc., New York, 1964xii+242 MR0189085 (32:6512) 0121.35502 Google Scholar[3] B. V. Gnedenko, Some theorems on distribution functions, Učenie Zapiski Moskovskogo Univ., 30 (1939), 61–82, (In Russian.) Google Scholar[4] V. M. Kruglov, A note on infinitely divisible distributions, Theory Prob. Applications, 15 (1970), 319–324 10.1137/1115038 0329.60011 LinkGoogle Scholar[5] I. A. Ibragimov and , Yu. V. Linnik, Independent and stationary sequences of random variables, Wolters-Noordhoff Publishing, Groningen, 1971, 443– MR0322926 (48:1287) 0219.60027 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Functional L.I.L. for the Upper Extremes of i.i.d. Random Variarbles from a Sub-class of Heavy Tailed Distributions31 October 2021 | Journal of the Indian Society for Probability and Statistics, Vol. 22, No. 2 Cross Ref Functional law of the iterated logarithm for partial sum processes of non-negative valued iid random variables belonging to the domain of partial attraction of a semi-stable lawStatistics & Probability Letters, Vol. 137 Cross Ref A Log Log Law for Subsequences of Delayed Random Sums27 July 2017 | Journal of the Indian Society for Probability and Statistics, Vol. 18, No. 2 Cross Ref Merging to Semistable ProcessesI. 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Recurrence equations for the number of types and the frequency of each type in a random sample drawn from a finite population undergoing discrete, nonoverlapping generations and reproducing according to the Cannings exchangeable model are deduced under the assumption of a mutation scheme with infinitely many types. The case of overlapping generations in discrete time is also considered. The equations are developed for the Wright-Fisher model and the Moran model, and extended to the case of the limit coalescent with nonrecurrent mutation as the population size goes to ∞ and the mutation rate to 0. Computations of the total variation distance for the distribution of the number of types in the sample suggest that the exact Moran model provides a better approximation for the sampling formula under the exact Wright-Fisher model than the Ewens sampling formula in the limit of the Kingman coalescent with nonrecurrent mutation. On the other hand, this model seems to provide a good approximation for a Λ-coalescent with nonrecurrent mutation as long as the probability of multiple mergers and the mutation rate are small enough.

This paper works with functions defined in metric spaces and takes values in complete paranormed vector spaces or in Banach spaces, and proves some necessary and sufficient conditions for weak convergence of probability measures. Our main result is as follows: Let X be a complete paranormed vector space and Ω an arbitrary metric space, then a sequence {μn} of probability measures is weakly convergent to a probability measure μ if and only if limn→∞∫Ωg(s)dμn=∫Ωg(s)dμ for every bounded continuous function g: Ω → X. A special case is as the following: if X is a Banach space, Ω an arbitrary metric space, then {μn} is weakly convergent to μ if and only if limn→∞∫Ωg(s)dμn=∫Ωg(s)dμ for every bounded continuous function g: Ω → X. Our theorems and corollaries in the article modified or generalized some recent results regarding the convergence of sequences of measures.

The paper studies convex stochastic optimization problems in a reproducing kernel Hilbert space (RKHS). The objective (risk) functional depends on functions from this RKHS and takes the form of a mathematical expectation (integral) of a nonnegative integrand (loss function) over a probability measure. The problem is generally ill-posed, a difficulty that in statistical learning is addressed through Tihonov regularization, with Monte Carlo approximation of integrals, which also makes it possible to solve the problem by finite dimensional (convex) quadratic optimization. The approximate solutions are referred to as kernel learning estimators and are expressed as a linear combination of kernels evaluated at the sample points. They are functional random variables that depend on the full sample. The paper studies a probabilistic convergence of these approximate solutions under a gradual elimination of the regularization parameter with rising number of observations. Its intended contribution is to derive novel nonasymptotic bounds on the minimization error and exponential bounds on the tail distribution of errors and to establish novel sufficient conditions for uniform convergence of kernel estimators to the true (normal) solution with probability one, jointly with a rule for downward adjustment of the regularization factor with increasing sample size. Applications to least squares, median, and quantile regression estimation, as well as to binary classification, are discussed.

On some inequalities in the probabilistic number theory

We consider linear iterated function systems with a random multiplicative error on the real line. Our system is $\{x\mapsto d_i + \lambda_i Y x\}_{i=1}^m$, where $d_i\in {\mathbb R}$ and $\lambda_i>0$ are fixed and $Y> 0$ is a random variable with an absolutely continuous distribution. The iterated maps are applied randomly according to a stationary ergodic process, with the sequence of independent and identically distributed errors $y_1,y_2,\dotsc$, distributed as $Y$, independent of everything else. Let $h$ be the entropy of the process, and let $\chi = {\mathbb E}[\log(\lambda Y)]$ be the Lyapunov exponent. Assuming that $\chi |\chi|$, then $\nu_{\bf y}$ is absolutely continuous with respect to the Lebesgue measure for almost every ${\bf y}$. We also prove that if $h < |\chi|$, then the measure $\nu_{\bf y}$ is singular and has dimension $h/|\chi|$ for almost every ${\bf y}$. These results are applied to a randomly perturbed iterated function system suggested by Sinai, and to a class of random sets considered by Arratia, motivated by probabilistic number theory.

On the class of labelled combinatorial structures called assemblies we define complex-valued multiplicative functions and examine their asymptotic mean values. The problem reduces to the investigation of quotients of the Taylor coefficients of exponential generating series having Euler products. Our approach, originating in probabilistic number theory, requires information on the generating functions only in the convergence disc and rather weak smoothness on the circumference. The results could be applied to studying the asymptotic value distribution of decomposable mappings defined on assemblies.

Sums of multiplicative functions and limit theorems in probabilistic number theory. I

ABSTRACTThe aim of this paper is to give some new characterizations of discrete compound Poisson distributions. Firstly, we give a characterization by the Lévy–Khintchine formula of infinitely divisible distributions under some conditions. The second characterization need to present by row sum of random triangular arrays converges in distribution. And we give an application in probabilistic number theory, the strongly additive function converging to a discrete compound Poisson in distribution. The next characterization, is an extension of Watanabe’s theorem of characterization of homogeneous Poisson process. The last characterization will be illustrated by waiting time distributions, especially the matrix-exponential representation.

Let y = {ii, ... ,4t} be a finite set of coprime integers and let {l, 2,...} denote the mutiplicative semigroup generated by y, and arranged in increasing order.Let Dn(w) denote the discrepancy of the sequence {nkco}%=i mod 1, w e [0, 1).In this paper we solve a problem posed by R.C. Baker [3], by proving that for all w except on a set of Lebesgue measure 0 l<limsuP /*(ft,) <C.Here the constant C only depends on the total number of primes involved in the prime factorization of q ,... , qx.The lower bound is obtained from a strong approximation theorem for the partial sums of the sequence {cos2nnk<}kx>=l by sums of independent standard normal random variables.P{co : t]k((o) < x} = x, 0<JC<1, *>1.If Fff denotes the empirical distribution function of the sequence {nk} at stage N, i.e., FN(s) = FN(s ; to) := N-l*{k < N : nk(w) <s}, 0 < s < 1, then DN(co):= sup \Fjf(s) -s\ 0<s<X denotes the discrepancy (or the Kolmogorov-Smirnov statistic) of the sequence {jt<y}f=1 mod 1. (For the basic definition and facts on uniform distribution

The paper presents an overview of main methods and results of the value distribution problem of additive arithmetical functions.

Size biased sampling from the Dickman subordinator