M. Sreehari

Another characterization of multivariate normal distribution

We establish a characterization of the multivariate normal based on a maximal property relating Var[g([zeta])] and the gradient of g(·).

Characterization of a family of discrete distributions via a Chernoff type inequality

A characterization theorem is proved for a family of non-negative integer valued random variables. The result is based on an inequality of the type proved by Chernoff (1981). Applications of the result for various standard distributions are also discussed.

On the limiting behavior of randomly weighted partial sums

We study the almost sure limiting behavior and convergence in probability of weighted partial sums of the form where {Wnj, 1[less-than-or-equals, slant]j[less-than-or-equals, slant]n, n[less-than-or-equals, slant]1} and {Xnj, 1[less-than-or-equals, slant]j[less-than-or-equals, slant]n, n[greater-or-equal, slanted]1} are triangular arrays of random variables. The results obtain irrespective of the joint distributions of the random variables within each array. Applications concerning the Efron bootstrap and queueing theory are discussed.

Chernoff-type inequality and variance bounds

After a brief review of the work on Chernoff-type inequalities, bounds for the variance of functions g(X, Y) of a bivariate random vector (X, Y) are derived when the marginal distribution of X is normal, gamma, binomial, negative binomial or Poisson assuming that the variance of g(X, Y) is finite. These results follow as a consequence of Chernoff inequality, Stein-identity for the normal distribution and their analogues for other distributions as obtained by Cacoullos, Papathanasiou, Prakasa Rao, Sreehari among others. Some interesting inequalities in real analysis are derived as special cases.

Maximum stability and a generalization

Stability of the maximum of a random number N of independent identically distributed r.v.s Xn was discussed by Voorn (1987). We consider a generalized version of this problem and study the connection between the distributions of X1 and N.

Mean Convergence Theorems with or without Random Indices for Randomly Weighted Sums of Random Elements in Rademacher Type p Banach Spaces

Abstract Some mean convergence theorems are established for randomly weighted sums of the form ∑ j = 1 k n A nj V nj and ∑ j = 1 T n A nj V nj where {A nj , j ≥ 1, n ≥ 1} is an array of random variables, {V nj , j ≥ 1, n ≥ 1} is an array of mean 0 random elements in a separable real Rademacher type p (1 ≤ p ≤ 2) Banach space, and {k n , n ≥ 1} and {T n , n ≥ 1} are sequences of positive integers and positive integer‐valued random variables, respectively. The results take the form or where 1 ≤ r ≤ p. It is assumed that the array {A nj V nj , j ≥ 1, n ≥ 1} is comprised of rowwise independent random elements and that for all n ≥ 1, A nj and V nj are independent for all j ≥ 1 and T n and {A nj V nj , j ≥ 1} are independent. No conditions are imposed on the joint distributions of the random indices {T n , n ≥ 1}. The sharpness of the results is illustrated by examples.

A Weak Law with Random Indices for Randomly Weighted Sums of Rowwise Independent Random Elements in Rademacher Type p Banach Spaces

For randomly weighted and randomly indexed sums of the form ∑Tn j=1 Anj ( Vnj − E(VnjI(||Vnj|| ≤ cn)) ) where {Anj, j ≥ 1, n ≥ 1} is an array of rowwise independent random variables, {Vnj, j ≥ 1, n ≥ 1} is an array of rowwise independent random elements in a separable real Rademacher type p Banach space, {cn, n ≥ 1} is a sequence of positive constants, and {Tn, n ≥ 1} is a sequence of positive integer-valued random variables, we present conditions under which the general weak law of large numbers ∑Tn j=1 Anj ( Vnj − E(VnjI(||Vnj|| ≤ cn)) ) P → 0 holds. It is not assumed that the {Vnj, j ≥ 1, n ≥ 1} have expected values or absolute moments. The sequences {Anj, j ≥ 1} and {Vnj, j ≥ 1} are assumed to be independent for all n ≥ 1. However, no conditions are imposed on the joint distributions of the random indices {Tn, n ≥ 1} and no independence conditions are imposed between {Tn, n ≥ 1} and {Anj, Vnj, j ≥ 1, n ≥ 1}. The sharpness of the weak law is illustrated by examples.

Modelling some stationary Markov processes and related characterizations

Abstract In this paper we look at the problem of modelling some stationary Markov processes in a unified manner. This gives rise to some new models. We also give some interesting characterizations of Poisson, exponential and geometric distributions.