Limit theorems for one-dimensional diffusions and random walks in random environments

Summary The limiting behavior of one-dimensional diffusion process in an asymptotically self-similar random environment is investigated through the extension of Brox's method. Similar problems are then discussed for a random walk in a random environment with the aid of optional sampling from a diffusion model; an extension of the result of Sinai is given in the case of asymptotically self-similar random environments.     

Limit theorems for mergesort

Central and local limit theorems (including large deviations) are established for the number of comparisons used by the standard top-down recursive mergesort under the uniform permutation model. The method of proof utilizes Dirichlet series, Mellin transforms, and standard analytic methods in probability theory. © 1996 John Wiley & Sons, Inc.     

Limit theorems for the median deviation

Summary In this paper, we obtain a strong law and central limit theorem for the median deviation under only very mild smoothness conditions on the underlying distribution. Under an additional condition implied by symmetry, we derive a weak Bahadur representation for the median deviation and establish the asymptotic equivalence of the median deviation and the semi-interquartile range.     

Limit theorems for polynomial-type distributions

Distributions are found of independent nonnegative integer valued random variables under linear constraints. Limit theorems for these distributions are proved. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipolez, pp. 136–148, Perm, 1993.     

Limit theorems for polylinear forms

The limit theorems for polylinear forms are obtained. Conditions are found under which the distribution of the polylinear form of many random variables is essentially the same as if all the distributions of arguments were normal.     

Limit theorems for monochromatic stars

Let T(K_1,r,G_n) be the number of monochromatic copies of the r‐star K_1,r in a uniformly random coloring of the vertices of the graph G_n. In this paper we provide a complete characterization of the limiting distribution of T(K_1,r,G_n), in the regime where is bounded, for any growing sequence of graphs G_n. The asymptotic distribution is a sum of mutually independent components, each term of which is a polynomial of a single Poisson random variable of degree at most r. Conversely, any limiting distribution of T(K_1,r,G_n) has a representation of this form. Examples and connections to the birthday problem are discussed.     

Limit theorems for empirical processes

Summary The empirical measure P _n for iid sampling on a distribution P is formed by placing mass n ^–1 at each of the first n observations. Generalizations of the classical Glivenko-Cantelli theorem for empirical measures have been proved by Vapnik and ervonenkis using combinatorial methods. They found simple conditions on a class C to ensure that sup {|P _n (C) – P(C)|: C C} converges in probability to zero. They used a randomization device that reduced the problem to finding exponential bounds on the tails of a hypergeometric distribution. In this paper an alternative randomization is proposed. The role of the hypergeometric distribution is thereby taken over by the binomial distribution, for which the elementary Bernstein inequalities provide exponential boundson the tails. This leads to easier proofs of both the basic results of Vapnik-ervonenkis and the extensions due to Steele. A similar simplification is made in the proof of Dudley's central limit theorem forn ^1/2(P P _n –P)— a result that generalizes Donsker's functional central limit theorem for empirical distribution functions.This research was supported in part by the Air Force Office of Scientific Research, Contract No. F49620-79-C-0164     

Limit theorems for finite dams

Limit distributions are given for both the dam content at time n and the limiting dam content in the n^th dam, as n tends to infinity, for a sequence of finite dams in discrete time, under assumptions which correspond to the various cases of heavy traffic in queueing theory. The proof employed is an application of the theory of weak convergence of probability measures.     

Limit theorems for trimmed sums

This paper studies the heavily trimmed sums (*) _{[ns] + 1} ^[nt] X _j ⁽ⁿ⁾ , where {X _j ⁽ⁿ⁾ } _{j = 1} ⁿ are the order statistics from independent random variables {X ₁,...,X _n } having a common distributionF. The main theorem gives the limiting process of (*) as a process oft. More smoothly trimmed sums like _{j = 1} ^[nt] J(j/n)X _j ⁽ⁿ⁾ are also discussed.     

Limit theorems for random walks

We consider a random walk $S_{\tau }$ which is obtained from the simple random walk $S$ by a discrete time version of Bochner’s subordination. We prove that under certain conditions on the subordinator $\tau$ appropriately scaled random walk $S_{\tau }$ converges in the Skorohod space to the symmetric $\alpha$-stable process $B^{\alpha }$. We also prove asymptotic formula for the transition function of $S_{\tau }$ similar to the Pólya’s asymptotic formula for $B^{\alpha }$.     

Limit theorems for convex hulls

It is shown that the process of vertices of the convex hull of a uniform sample from the interior of a convex polygon converges locally, after rescaling, to a strongly mixing Markov process, as the sample size tends to infinity. The structure of the limiting Markov process is determined explicitly, and from this a central limit theorem for the number of vertices of the convex hull is derived. Similar results are given for uniform samples from the unit disk.     

Limit theorems for the multivariate binomial distribution

Nonsingular limit distributions are determined for sequences of affine transformations of random vectors whose distributions are multivariate binomial. Each of these limit distributions is that of an affine transformation of a random vector having a multivariate normal distribution or a multivariate Possion distribution or a joint distribution of two independent random vectors, one normal and the other Poisson.     

Limit behavior of two-time-scale diffusions revisited

This work is concerned with diffusions with two-time scales or singularly perturbed diffusions. Asymptotic expansions of the solution of the associated Cauchy problem for parabolic partial differential equation are obtained and the desired error bounds are derived. These asymptotic expansions are then used to analyze related limit distributions of normalized integral functionals.     

Limit theorems for induced extreme values

For a bivariate sample (X_i, Y_i) of size n, let (U(x), V(x)) denote the following pair of induced extreme values: U(x) is the maximum of those Y_i-values with corresponding X_i-value less than x and V(x) is the maximum of the remaining Y_i-values. In the paper, we study the asymptotic behavior of the (suitably normalized) random vector (U(x), V(x)), and we consider several cases. First, we consider nonrandom x and let x=x_n so that as n→∞, x_n tends to the endpoint of F_X(x), or so that x_n tends to x₀, a point in the support of F_X(x). The second important situation appears when x=X_k∶n, i.e., we select Y-values on the basis of the random variable X_k∶n, the k-th order-statistic of the X-sample. Here we also consider two cases: (i) k=n−j with fixed j, and (ii) k=[np], where 0<p<1. The paper generalizes the earlier results of David, Joshi, and Nagaraja, where it is assumed that (X, Y) is in the bivariate (max-) domain of attraction of a bivariate stable law with independent marginals. Proceedings of the XVI Seminar on Stability Problems for Stochastic Models, Part I, Eger, Hungary, 1994.     

Limit theorems for free multiplicative convolutions

We determine the distributional behavior for products of free random variables in a general infinitesimal triangular array. The main theorems in this paper extend a result for measures supported on the positive half-line, and provide a new limit theorem for measures on the unit circle with nonzero first moment.

     

Limit theorems for optimal mass transportation

The optimal mass transportation was introduced by Monge some 200?years ago and is, today, the source of large number of results in analysis, geometry and convexity. Here I investigate a new, surprising link between optimal transformations obtained by different Lagrangian actions on Riemannian manifolds. As a special case, for any pair of non-negative measures ??⁺, ??^? of equal mass $$W_1(\lambda^-, \lambda^+)= \lim_{\varepsilon\rightarrow 0} \varepsilon^{-1} \inf_{\mu} W_p(\mu+\varepsilon\lambda^-, \mu+\varepsilon\lambda^+) $$ where W _p , p??? 1 is the Wasserstein distance and the infimum is over the set of probability measures in the ambient space.