Let [Formula: see text] be a sequence of random variables with different distributions [Formula: see text]. The partial sums are denoted by [Formula: see text], [Formula: see text]. This paper mainly investigates the precise large deviations of [Formula: see text], for the widely orthant dependent random variables [Formula: see text]. Under some mild conditions, the lower and upper bounds of th...

The study of precise large deviations of random sums is an important topic in insurance and finance. In this paper, extended precise large deviations of random sums in the presence of END structure and consistent variation are investigated. The obtained results extend those of Chen and Zhang 2007 and Chen et al. 2011 . As an application, precise large deviations of the prospectiveloss process o...

The tail probabilities of two weighted sums of independent gamma random variables are compared when the first vector of weights majorizes the second vector of weights. The conjecture that the two cumulative distribution .functions cross exactly once is established in four special cases by means of the variation-diminishing property of totally positive kernels. Bounds are obtained for the locati...

Stein's method is used to prove approximations in total variation to the distributions of integer valued random variables by (possibly signed) compound Poisson measures. For sums of independent random variables, the results obtained are very explicit, and improve upon earlier work of Kruopis (1983) and Cekanavicius (1997); coupling methods are used to derive concrete expressions for the error b...

Operator stable laws are the limits of operator normed and centered sums of independent, identically distributed random vectors. The operator ν-stable laws are the analogous limit distributions for randomized sums. In this paper we characterize operator ν-stable laws and their domains of attraction. We also discuss several applications, including the scaling limits of continuous time random wal...