In this paper, we investigate the large deviations of sums weighted random variables that are approximately independent, generalizing and improving some results Montgomery Odlyzko. We motivated by examples arising from number theory, including sequences pit, χ(p), χd(p), λf (p), Klq(a − n, b), where p ranges over primes, t varies in a interval, χ among all characters modulo q, χd quadratic atta...

We consider the zeros of sum independent random polynomials as their degrees tend to infinity. Namely, let p and q be two degree n, whose roots are chosen independently from probability measures μ ν in complex plane, respectively. compute limiting distribution for p+q n tends The can described by its logarithmic potential, which we show is pointwise maximum potentials ν. More generally, m when ...

The study of precise large deviations for random sums is an important topic in insurance and finance. In this paper, we extend recent results of Tang (2006) and Liu (2009) to random sums in various situations. In particular, we establish a precise large deviation result for a nonstandard renewal risk model in which innovations, modelled as real-valued random variables, are negatively dependent ...

Let {Xi} be a sequence of independent, identically distributed random variables with an intermediate regularly varying right tail F̄ . Let (N,C1, C2, . . .) be a nonnegative random vector independent of the {Xi} with N ∈ N ∪ {∞}. We study the weighted random sum SN = ∑Ni=1 CiXi , and its maximum, MN = sup1≤k<N+1 ∑ki=1 CiXi . This type of sum appears in the analysis of stochastic recursions, incl...

In this paper, we adapt the very effective Berry–Esseen theorems of Chen and Shao (2004), which apply to sums of locally dependent random variables, for use with randomly indexed sums. Our particular interest is in random variables resulting from integrating a random field with respect to a point process. We illustrate the use of our theorems in three examples; in a rather general model of the ...

We study a higher-dimensional ‘balls-into-bins’ problem. An infinite sequence of i.i.d. random vectors is revealed to us one vector at a time, and we are required to partition these vectors into a fixed number of bins in such a way as to keep the sums of the vectors in the different bins close together; how close can we keep these sums almost surely? This question, our primary focus in this pap...