Triangular Array Scheme Research Articles

Asymptotic behaviour of the survival probability of almost critical branching processes in a random environment

A generalization of the well-known result concerning the survival probability of a critical branching process in random environment $Z_k$ is considered. The triangular array scheme of branching processes in random environment $Z_{k,n}$ that are close to $Z_k$ for large $n$ is studied. The equivalence of the survival probabilities for the processes $Z_{n,n}$ and $Z_n$ is obtained under rather natural assumptions on the closeness of $Z_{k,n}$ and $Z_k$. Bibliography: 7 titles.

Generalization and Refinement of the Integro-Local Stone Theorem for Sums of Random Vectors

The integro-local Stone theorem on the asymptotics of the probability that a sum of random vectors enters a small cube is (a) refined under additional moment and structural conditions; (b) generalized to a case of nonidentically distributed summands in the triangular array scheme; (c) the results of item (b) are refined under additional moment and structural conditions.

Limit Theorem for the Maximum of Random Variables Connected by IT-Copulas of Student's $t$-Distribution

It is shown that the asymptotic of the maximum in a triangular array scheme of random variables connected by IT-copulas of Student's $t$-distribution is the same as for random variables with Gaussian copulas. Given this, assuming appropriate choice of basis, and corresponding projection system, generating the above triangular scheme, a convergence to max-stable Gumbel law takes place.

Tempered stable laws as random walk limits

Stable laws can be tempered by modifying the Lévy measure to cool the probability of large jumps. Tempered stable laws retain their signature power law behavior at infinity, and infinite divisibility. This paper develops random walk models that converge to a tempered stable law under a triangular array scheme. Since tempered stable laws and processes are useful in statistical physics, these random walk models can provide a basic physical model for the underlying physical phenomena.

Integro-Local and Local Theorems on Normal and Large Deviations of the Sums of Nonidentically Distributed Random Variables in the Triangular Array Scheme

Gnedenko's local theorem and Stone–Shepp's integro-local theorem (see [L. A. Shepp, Ann. Math. Statist., 35 (1964), pp. 419–423], [C. Stone, Ann. Math. Statist., 36 (1965), pp. 546–551], [B. V. Gnedenko and A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, Addison-Wesley, Cambridge, MA, 1954]) for sums on independent identically distributed random variables are extended to the case when the summands are nonidentically distributed in a triangular array scheme. Under Cramer's condition on the random variables, we also obtain integro-local and local theorems that are valid in both normal and large deviations zones.

Transient phenomena for random walks in the absence of the expected value of jumps

Let ξ, ξ1, ξ2, ... be independent identically distributed random variables, and Sn:=Σj=1n,ξj, \( \bar S \):= supn≥0Sn. If Eξ = −a < 0 then we call transient those phenomena that happen to the distribution \( \bar S \) as a → 0 and \( \bar S \) tends to infinity in probability. We consider the case when Eξ fails to exist and study transient phenomena as a → 0 for the following two random walk models: 1. The first model assumes that ξj can be represented as ξj = ζj + αηj, where ζ1, ζ2, ... and η1, η2, ... are two independent sequences of independent random variables, identically distributed in each sequence, such that supn≥0Σj=1n ζj = ∞, supn≥0Σj=1nηj < ∞, and \( \bar S \) < ∞ almost surely. 2. In the second model we consider a triangular array scheme with parameter a and assume that the right tail distribution P(ξj ≥ t) ∼ V (t) as t→∞ depends weakly on a, while the left tail distribution is P(ξj 0.

Transient Phenomena for Random Walks with Nonidentically Distributed Jumps with Infinite Variances

Let $\zeta_1,\zeta_2,\ldots$ be independent random variables, $$ Z_n=\sum_{i=1}^n\zeta_i,\qquad \overline{Z}_n=\max_{k\le n}Z_k,\qquad Z=\overline{Z}_\infty. $$ It is well known that if $\zeta_i$ are identically distributed, then Z is a proper random variable when ${{\bf E}\zeta_i=-a < 0}$, and $Z=\infty$ a.s. when $a=0$. The limiting distribution for $\overline{Z}_n$ as $n\to\infty$, $a\to 0$ (in the triangular array scheme) when ${\bf E}\zeta_i^2 < \infty$ is well studied (see, e.g., [J. F. C. Kingman, J. Roy. Statist. Soc. Ser. B, 24 (1962), pp. 383-392], [Yu. V. Prokhorov, Litov. Math. Sb., 3 (1963), pp. 199-204 (in Russian)], and [A. A. Borovkov, Stochastic Processes in Queueing Theory, Springer-Verlag, New York, 1976]). In the present paper, we study the limiting distribution for $\overline{Z}_n$ as ${\bf E \zeta_i\to 0}$, $n\to\infty$, in the case when $\bE \zeta_i^2=\infty$ and the summands $\zeta_i$ are nonidentically distributed.

The Strong Law of Large Numbers for a Triangular Array Scheme of Conditional Distributions of Stable Elliptically Contoured Measures

This paper deals with conditional distributions of stable elliptically contoured measures on real separable Hilbert space. We consider projections of these measures on an increasing sequence of finite-dimensional linear subspaces spanned by initial elements of orthonormal basis. It is shown that the asymptotic properties of corresponding conditional distributions depend on a choice of a basis of Hilbert space. We give sufficient conditions of a choice of a basis when triangular array schemes of conditional distributions (in a certain sense) obey the strong law of large numbers.

Asymptotic Analysis for Random Walks with Nonidentically Distributed Jumps Having Finite Variance

Let ξ1,ξ2,... be independent random variables with distributions F1F2,... in a triangular array scheme (Fi may depend on some parameter). Assume that Eξi = 0, Eξi2 x) and \(P(\bar S_n > x)\). We also study the asymptotics of these probabilities and of the probabilities that a trajectory {Sk} crosses the remote boundary {g(k)}; that is, the asymptotics of P(maxk≤n(Sk − g(k)) > 0). The case n = ∞ is not excluded. We also estimate the distribution of the first crossing time.

Asymptotic Expansion of the Distribution Density Function for the Sum of Random Variables in the Series Scheme in Large Deviation Zones

The work is designated for obtaining asymptotic expansions and determination of structures of the remainder terms that take into consideration large deviations both in the Cramer zone and Linnik power zones for the distribution density function of sums of independent random variables in a triangular array scheme. The result was obtained using general Lemma 6.1 of Saulis and Statulevicius in Limit Theorems for Large Deviations (Kluwer, 1991) and joining the methods of characteristic functions and cumulants. The work extends the theory of sums of random variables and in a special case, improves S. A.Book's results on sums of random variables with weights.

Limit distributions of the number of collections of H -equivalent segments in the triangular array scheme of equiprobable polynomial trials

AbstractIn this paper we study random variables which characterise collections of segments in an equiprobable polynomial scheme related by the H-equivalence. We give an upper bound for the variation distance between the distribution of the random variable ξ

Asymptotic properties of conditional quantiles of the Cauchy distribution in Hilbert space

In this paper, we consider the conditional distributions that are induced by finite-dimensional projections of a σ-additive Cauchy measure defined in a real Hilbert space. In the case where the dimension of projections tends to infinity, we establish the almost sure convergence of “conforming” sequences of finite-dimensional conditional quantiles and prove the strong law of large numbers for the triangular array scheme applied to a family of conditional distributions.

Testing the hypothesis concerning the type of distribution and normed order statistics in a triangular array scheme

Testing the hypothesis concerning the type of distribution and normed order statistics in a triangular array scheme