An invariance principle for sums and record times of regularly varying stationary sequences

TL;DR

This paper develops new limit theorems for sums and record times of stationary sequences with heavy tails, capturing temporal order and dependence structures often missed by classical approaches.

Contribution

It introduces a novel invariance principle and functional limit theorems for dependent, regularly varying sequences, extending existing asymptotic theory to broader models.

Findings

Limit distribution for extreme clusters derived.

New point process convergence preserving temporal order.

Record times converge to a scale-invariant Poisson process.

Abstract

We prove a sequence of limiting results about weakly dependent stationary and regularly varying stochastic processes in discrete time. After deducing the limiting distribution for individual clusters of extremes, we present a new type of point process convergence theorem. It is designed to preserve the entire information about the temporal ordering of observations which is typically lost in the limit after time scaling. By going beyond the existing asymptotic theory, we are able to prove a new functional limit theorem. Its assumptions are satisfied by a wide class of applied time series models, for which standard limiting theory in the space $D$ of \cadlag\ functions does not apply. To describe the limit of partial sums in this more general setting, we use the space~$E$ of so--called decorated \cadlag\ functions. We also study the running maximum of partial sums for which a corresponding functional theorem can be still expressed in the familiar setting of space $D$. We further apply our method to analyze record times in a sequence of dependent stationary observations, even when their marginal distribution is not necessarily regularly varying. Under certain restrictions on dependence among the observations, we show that the record times after scaling converge to a relatively simple compound scale invariant Poisson process.