Modern Stochastics: Theory and Applications

In the Karlin infinite occupancy scheme, balls are thrown independently into an infinite array of boxes $1,2,\dots $ , with probability ${p_{k}}$ of hitting the box k. For $j,n\in \mathbb{N}$, denote by ${\mathcal{K}_{j}^{\ast }}(n)$ the number of boxes containing exactly j balls provided that n balls have been thrown. Small counts are the variables ${\mathcal{K}_{j}^{\ast }}(n)$, with j fixed. The main result is a law of the iterated logarithm (LIL) for the small counts as the number of balls thrown becomes large. Its proof exploits a Poissonization technique and is based on a new LIL for infinite sums of independent indicators ${\textstyle\sum _{k\ge 1}}{1_{{A_{k}}(t)}}$ as $t\to \infty $, where the family of events ${({A_{k}}(t))_{t\ge 0}}$ is not necessarily monotone in t. The latter LIL is an extension of a LIL obtained recently by Buraczewski, Iksanov and Kotelnikova (2023+) in the situation when ${({A_{k}}(t))_{t\ge 0}}$ forms a nondecreasing family of events.

A new formula for the ultimate ruin probability in the Cramér–Lundberg risk process is provided when the claims are assumed to follow a finite mixture of m Erlang distributions. Using the theory of recurrence sequences, the method proposed here shifts the problem of finding the ruin probability to the study of an associated characteristic polynomial and its roots. The found formula is given by a finite sum of terms, one for each root of the polynomial, and allows for yet another approximation of the ruin probability. No constraints are assumed on the multiplicity of the roots and that is illustrated via a couple of numerical examples.

In the present paper the change of measures technique for compound mixed renewal processes, developed in Tzaninis and Macheras [ArXiv:2007.05289 (2020) 1–25], is applied to the ruin problem in order to obtain an explicit formula for the probability of ruin in a mixed renewal risk model and to find upper and lower bounds for it.