Non-standard limits for a family of autoregressive stochastic sequences

TL;DR

This paper investigates the limiting behavior of a family of autoregressive stochastic sequences with restart mechanisms, revealing non-standard limit distributions that include mixtures of atomic and continuous parts, with conditions for normality.

Contribution

It introduces a novel limit theorem for autoregressive sequences with restart, describing their distributional limits as parameters vary, including conditions for normality.

Findings

Derived a non-standard limit theorem for autoregressive sequences with restart.

Identified conditions under which the limit distribution is normal.

Provided multiple examples illustrating the limit behaviors.

Abstract

We consider a family of multivariate autoregressive stochastic sequences that restart when hit a neighbourhood of the origin, and study their distributional limits when the autoregressive coefficient tends to one, the noise scaling parameter tends to zero, and the neighbourhood size varies. We obtain a non-standard limit theorem where the limiting distribution is a mixture of an atomic distribution and an absolutely continuous distribution whose marginals, in turn, are mixtures of distributions of signed absolute values of normal random variables. In particular, we provide conditions for the limiting distribution to be normal, like in the case without restart mechanism. The main theorem is accompanied by a number of examples and auxiliary results of their own interest.