Stable limits for sums of dependent infinite variance random variables

Abstract

The aim of this paper is to provide conditions which ensure that the affinely transformed partial sums of a strictly stationary process converge in distribution to an infinite variance stable distribution. Conditions for this convergence to hold are known in the literature. However, most of these results are qualitative in the sense that the parameters of the limit distribution are expressed in terms of some limiting point process. In this paper we will be able to determine the parameters of the limiting stable distribution in terms of some tail characteristics of the underlying stationary sequence. We will apply our results to some standard time series models, including the GARCH(1, 1) process and its squares, the stochastic volatility models and solutions to stochastic recurrence equations.

Heavy-Tailed Branching Process with Immigration

“…In either case, the limit is an -stable random variable with a Lévy triplet (0, , r ) identifiable as in Ref. [9] . Using results in Ref.…”

Estimation of and Inference About the Expected Shortfall for Time Series With Infinite Variance

“…Also see Embrechts et al (1997) for discussions on this topic. Recent work of Bartkiewicz, Jakubowski, Mikosch, and Wintenberger (2010), which clarifies and extends earlier work, for example Davis and Hsing (1995), provides conditions that determine the parameters of limiting distributions in terms of tail characteristics of the underlying stationary sequence. For simplicity, some conditions that we use are stronger than theirs, so as usual our conditions are sufficient but not necessary.…”

“…Assumptions A1-A4 are similar to Bartkiewicz et al (2010). Specifically, Assumption A1 is their condition RV (regular variation) restricted to the case where θ > 1 so that the expected shortfall exists.…”

On functional limits of short- and long-memory linear processes with GARCH(1,1) noises

“…As pointed out by a referee, using the same assumptions as ours and considering α ∈ (0, 2) or α ∈ (0, 2) ∪ (2, 4), Proposition 5 in [7] expressed the limiting results in terms of characteristic functions (contrast to Lemmas 4.1(a) and 4.3(a)). The major advantage of their classical blocking and mixing techniques over the point process approach is, by controlling clustering of big values, one may calculate the parameters of the stable limit in terms of quantities of the finitedimensional distributions of the underlying process.…”

“…in which for α > 4, with A 1 = a + bη 2 0 and defining π := a + b, 7 7 When α > 4 and π = 0, write γ (s) = ω 2  ∞ l=0 d l d l+s and observe that γ (s) = γ (−s), K…”