Stable convergence of random sequences with random indices

In this paper, we give a complete solution to Rényi's conjecture about stable convergence under a general metric space setup, and provide new Anscombe-type theorems concerning the stable convergence of random sequences with random indices. In particular, we present a random version of the famous arcsine law. Supported by the Postdoctoral Foundation of China. Proceedings of the Seminar on Stability Problems for Stochastic Models, Vologda, Russia, 1998, Part II.     

Convergence of multidimensional random sequences with independent random indices

The paper present necessary and sufficient conditions for the weak convergence of R^m-valued random sequences with independent random indices under some additional assumptions. Operator normalization is considered. Supported by the Russian Foundation for Fundamental Research (grant No. 93-01-01446). Proceedings of the XVI Seminar on Stability Problems for Stochastic Models, Part I, Eger, Hungary, 1994.     

Asymptotic approximation of crossing probabilities of random sequences

Summary Let {X _i , i1} be a random sequence and {u _ni ,1in, n1} be an array of boundary values. We consider the asymptotic approximation of the probability P _n =P{X _i u _ni ,1in} by . We give sufficient conditions on X _i such that P _n–P _n ^* 0 as n. This generalizes the situation considered in extreme-value theory where the boundary is constant in i. The general theory is applied in particular to Gaussian cases.     

Asymptotic properties of Markov-modulated random sequences with fast and slow timescales

This work is concerned with asymptotic properties of Markov-modulated random processes having two timescales. The model contains a number of mixing sequences modulated by a switching process that is a discrete-time Markov chain. The motivation of our study stems from applications in manufacturing systems, communication networks and economic systems, in which regime-switching models are used. This paper focuses on asymptotic properties of the Markov-modulated processes under suitable scaling. Our main effort focuses on obtaining a strong approximation result.     

Asymptotic expansion for vector-valued sequences of random variables with focus on Wiener chaos

We develop the asymptotic expansion theory for vector-valued sequences ${\left(F_N\right)}_{N\ge 1}$ of random variables in terms of the convergence of the Stein–Malliavin matrix associated with the sequence $F_N$. Our approach combines the classical Fourier approach and the recent Stein–Malliavin theory. We find the second order term of the asymptotic expansion of the density of $F_N$ and we illustrate our results by several examples.     

The asymptotics of randomly indexed random sequences: Independent indices

Translated from:Problemy Ustoichivosti Stokhasticheskikh Modelei, Trudy Seminara, 1989, pp. 60–72.     

Asymptotics of randomly indexed infinite-dimensional random sequences: Independent indices

Translated from Problemy Ustoichivosti Stokhasticheskikh Modelei, Trudy Seminara, pp. 38–44, 1990.     

Asymptotic sine laws arising from alternating random permutations and sequences

In the last century, Désiré André obtained many remarkable properties of the numbers of alternating permutations, linking them to trigonometric functions among other things. By considering the probability that a random permutation is alternating and that a random sequence (from a uniform distribution) is alternating, and by conditioning on the first element of the sequence, his results are extended and illuminated. In particular, several “asymptotic sine laws” are obtained, some with exponential rates of convergence. © 1996 John Wiley & Sons, Inc.     

Asymptotic distribution of products of sums of independent random variables

In the paper we consider the asymptotic distribution of products of weighted sums of independent random variables.     

Asymptotic distribution of singular values of powers of random matrices

Let x be a complex random variable such that \( {\mathbf{E}}x = 0,\,{\mathbf{E}}{\left| x \right|^2} = 1 \), and \( {\mathbf{E}}{\left| x \right|^4} < \infty \). Let \( {x_{ij}},i,j \in \left\{ {1,2, \ldots } \right\} \), be independent copies of x. Let \( {\mathbf{X}} = \left( {{N^{ - 1/2}}{x_{ij}}} \right) \), 1≤i,j≤N, be a random matrix. Writing X ^? for the adjoint matrix of X, consider the product X ^m X ^?m with some m ∈{1,2,...}. The matrix X ^m X ^?m is Hermitian positive semidefinite. Let λ₁,λ₂,...,λ _N be eigenvalues of X ^m X ^?m (or squared singular values of the matrix X ^m ). In this paper, we find the asymptotic distribution function \( {G^{(m)}}(x) = {\lim_{N \to \infty }}{\mathbf{E}}F_N^{(m)}(x) \) of the empirical distribution function \( F_N^{(m)}(x) = {N^{ - 1}}\sum\nolimits_{k = 1}^N {\mathbb{I}\left\{ {{\lambda_k} \leqslant x} \right\}} \), where \( \mathbb{I}\left\{ A \right\} \) stands for the indicator function of an event A. With m=1, our result turns to a well-known result of Marchenko and Pastur [V. Marchenko and L. Pastur, The eigenvalue distribution in some ensembles of random matrices, Math. USSR Sb., 1:457–483, 1967].     

Asymptotic normal distribution of multidimensional statistics of dependent random variables

A central limit theorem for multidimensional processes in the sense of [9], [10] is proved. In particular the asymptotic normal distribution of a sum of dependent random functions of m variables defined on the positive part of the integral lattice is established by the method of moments. The results obtained can be used, for example, in proving the asymptotic normality of different statistics of n₀-dependent random variables as well as to determine the asymptotic behaviour of the resultant of reflected waves of telluric type.     

Asymptotic distribution of two-protected nodes in random binary search trees

We derive exact moments of the number of 2-protected nodes in binary search trees grown from random permutations. Furthermore, we show that a properly normalized version of this tree parameter converges to a Gaussian limit.     

Weak convergence with random indices

Suppose {X_nn?-0} are random variables such that for normalizing constants a_n>0, b_n, n?0 we have Y_n(·)=(X[_{n, ·}]-b_n/a_n ? Y(·) in D(0.∞) . Then a_n and b_n must in specific ways and the process Y possesses a scaling property. If {N_n} are positive integer valued random variables we discuss when Y_Nn → Y and Y'_n=(X[_Nn]-b_n)/a_n ? Y'. Results given subsume random index limit theorems for convergence to Brownian motion, stable processes and extremal processes.     

Asymptotic theorems for estimating the distribution function under random truncation

Representation theorem and local asymptotic minimax theorem are derived for nonparametric estimators of the distribution function on the basis of randomly truncated data. The convolution-type representation theorem asserts that the limiting process of any regular estimator of the distribution function is at least as dispersed as the limiting process of the product-limit estimator. The theorems are similar to those results for the complete data case due to Beran (1977, Ann. Statist., 5, 400–404) and for the censored data case due to Wellner (1982, Ann. Statist., 10, 595–602). Both likelihood and functional approaches are considered and the proofs rely on the method of Begun et al. (1983, Ann. Statist., 11, 432–452) with slight modifications.Division of Biostatistics, School of Public Health, Columbia Univ.     

On the uniform distribution of sequences of random variables

Summary This paper deals with the almost sure uniform distribution (modulo 1) of sequences of random variables. In the case where the law of the increments X _n+h –X _n of the sequence X ₀, X ₁, does not depend on n, sufficient conditions are given to assure the uniform distribution (modulo 1) with probability one. As an illustrative example the partial sums of a sequence of independent, identically distributed variables is considered.     

Asymptotic behavior for random walk in random environment with holding times

In this article, we mainly discuss the asymptotic behavior for multi-dimensional continuous-time random walk in random environment with holding times. By constructing a renewal structure and using the point “environment viewed from the particle”, under General Kalikow's Condition, we show the law of large numbers (LLN) and central limit theorem (CLT) for the escape speed of random walk.     

Convergence and relative compactness in distribution of sequences of random hypermeasures

Let Φ be a compact set in a vector space equipped with a convergence which is metrizable in Φ but not certainly in the whole space. We endow the space of continuous on Φ linear functionals on span Φ with the norm \( {\left\| u \right\|_\Phi } = \sup \varphi \in \Phi \left| {u\varphi } \right| \) and call the elements of the completion of Φ hypermeasures. We prove theorems on the convergence in probability or in distribution and relative compactness in distribution of a sequence of random hypermeasures.