Weighted maxima and sums of non-stationary random length sequences in heavy-tailed models

TL;DR

This paper extends the analysis of weighted sums and maxima of non-stationary sequences of heavy-tailed variables by relaxing previous constraints, allowing for more general conditions on the number of series, tail heaviness, and weights.

Contribution

It introduces new theoretical results for tail and extremal indices under broader assumptions on the structure of non-stationary heavy-tailed sequences.

Findings

Derived tail indices for sequences with random number of series

Established extremal indices under relaxed assumptions

Extended previous models to include real-valued weights

Abstract

The sums and maxima of weighted non-stationary random length sequences of regularly varying random variables may have the same tail and extremal indices, Markovich and Rodionov (2020). The main constraints are that there exists a unique series in a scheme of series with the minimum tail index, the tail of the term number is lighter than the tail of the terms and the weights are positive constants. These assumptions are changed here: a bounded random number of series is allowed to have the minimum tail index, the tail of the term number may be heavier than the tail of the terms and the weights may be real-valued. Then we derive the tail and extremal indices of the weighted non-stationary random length sequences under the new assumptions.