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Cumulative Parisian ruin probability for two-dimensional Brownian risk model

Krystecki Konrad

Probability and Mathematical Statistics

2023

Vol. 43, Fasc. 1

63--81

Let (W1(s), W2(t)), s, t ≤ 0, be a bivariate Brownian motion with standard Brownian motion marginals and constant correlation ρ ∈ (−1, 1). We derive the exact asymptotics as u → ∞ for the cumulative Parisian ruin probability [Formula] for c1, c2 ∈ R, a ∈ (0, 1] and suitably adjusted H1(u), H2(u).

Extremes of moving averages and moving maxima on a regular lattice

Basrak B., Tafro A.

Probability and Mathematical Statistics

2014

Vol. 34, Fasc. 1

61--79

We study the extremal behaviour of spatial moving averages and moving maxima on a regular discrete grid. Our main assumption is that these random fields are stationary and regularly varying with the tail index α > 0. Using the asymptotic theory for point processes we characterise the limiting behaviour of their extremes over an increasing grid. Our approach builds on the results of Davis and Resnick concerning linear processes. By analogy to the analysis of time series data, an appropriate Hill estimator of the tail index can be defined.We exhibit a sufficient condition for the consistency of this estimator in a certain class of spatial lattice models. Finally, we show that this condition holds for the models in our title.