Explosive Poisson shot noise processes with applications to risk reserves (Q1903607)

A shot-noise process is defined as \(S(t)= \sum X_n (t- T_n)\) where the \(T_n\) are the epochs of a Poisson process and the \(X_n\) are i.i.d. non-negative random functions. In contrast to most of the literature where the \(X_n\) are assumed to be non-increasing with limit 0, this paper considers the non-decreasing case. Laws of large numbers and central limit theorems for \(S(t)\) are derived. Assuming regular variation of \(\text{Cov} (X (s), X(t))\), it is shown that the shot-noise process when properly scaled and normalised has a self-similar Gaussian limit with continuous sample paths. Conditions for the limit to be Brownian motion are investigated in the case \(\lim_{t\to \infty} X_n (t)< \infty\) a.s. As a corollary, ruin probability approximations for an insurance risk model are derived for the case where \(X_n (t)\) is the part of the \(n\)th claim which has been settled before \(t\) time units after its occurrence.