On the laws of large numbers for double arrays of independent random elements in Banach spaces

For a double array of independent random elements {Vmn,m ≥ 1,n ≥ 1} in a real separable Banach space,conditions are provided under which the weak and strong laws of large numbers for the double sums mi=1 nj=1Vij,m ≥ 1,n ≥ 1 are equivalent.Both the identically distributed and the nonidentically distributed cases are treated.In the main theorems,no assumptions are made concerning the geometry of the underlying Banach space.These theorems are applied to obtain Kolmogorov,Brunk–Chung,and Marcinkiewicz–Zygmund type strong laws of large numbers for double sums in Rademacher type p(1 ≤ p ≤ 2) Banach spaces.     

Asymptotic behavior of the stationary distribution in a finite QBD process with zero mean drift

We consider a finite QBD process with m levels. Assuming that the mean drift is 0, we obtain an asymptotic behavior as m→∞ in the stationary distribution , by finding an explicit expression for vector c. This solves the problem that was conjectured by Miyazawa et al. [Asymptotic behaviors of the loss probability for a finite buffer queue with QBD structure, 23 (2007) 79-95].     

A class of bivariate negative binomial distributions with different index parameters in the marginals

In this paper, we consider a new class of bivariate negative binomial distributions having marginal distributions with different index parameters. This feature is useful in statistical modelling and simulation studies, where different marginal distributions and a specified correlation are required. This feature also makes it more flexible than the existing bivariate generalizations of the negative binomial distribution, which have a common index parameter in the marginal distributions. Various interesting properties, such as canonical expansions and quadrant dependence, are obtained. Potential application of the proposed class of bivariate negative binomial distributions, as a bivariate mixed Poisson distribution, and computer generation of samples are examined. Numerical examples as well as goodness-of-fit to simulated and real data are also given here in order to illustrate the application of this family of bivariate negative binomial distributions.