HARCH processes are heavy tailed
type: Item
instance of: scholarly article
title: HARCH processes are heavy tailed
zbMATH Open document ID: 0947.60049
DOI: 10.1023/A:1009972311258
author: Paul Embrechts
author: Rudolf Grübel
published in: Extremes
publication date: 2000-05-24T00:00:00Z
review text: A \(\text{HARCH}(k)\) process \(R_n\) is defined by the recursive relations \[ R_n=\sigma_n\varepsilon_n, \qquad \sigma_n^2=c_0+\sum_{j=1}^kc_j \left( \sum_{i=1}^j R_{n-i} \right)^2 \] where the \(c_j\) are some nonrandom constants, \(\varepsilon_n\) are i.i.d., \(E\varepsilon=0\), \(E\varepsilon^2<\infty\). Such models are used to describe the log-return process in financial asset market price modeling. Denote by \(R_\infty\) the distributional limit of \(R_n\) (if it exists). It is shown that if \(c_1>0\) and \(\Pr(\varepsilon^2>c_1^{-1})>0\), then \[ \liminf_{r\to\infty} { \log\Pr\{|R_\infty|>r\} \over \log r } >-\infty, \] i.e. \(R_\infty\) is a heavy tailed distribution.
Mathematics Subject Classification ID: 60G42
Mathematics Subject Classification ID: 62P05
zbMATH DE Number: 1452424
zbMATH Keywords: nonlinear time series
zbMATH Keywords: random recursion
zbMATH Keywords: heavy tailed distribution
MaRDI profile type: Publication
full work available at URL: a:1009972311258
OpenAlex ID: W2106447440
description: scientific article; zbMATH DE number 1452424