The Limit Distribution of the Maximum Increment of a Random Walk with Dependent and Regularly Varying Jump Size

• Activity Number: 181
• Type: Contributed
• Date/Time: Monday, August 1, 2011 : 10:30 AM to 12:20 PM
• Sponsor: IMS
• Author(s): Martin Moser*+ and Thomas Mikosch
• Companies: Technische Universität München and University of Copenhagen
• Keywords: change point analysis ; extreme value theory ; maximum increment of a random walk ; regular variation ; dependent random walk

Abstract

For a sequence of regularly varying random variables $X_i$ we consider the following statistical problem motivated by change point analysis: We test the null hypothesis of constant mean $E X_1 = \ldots = E X_n = \mu$ against the epidemic alternative of a change in the mean, i.e. $E X_1 = \ldots = E X_k = E X_{m+1} = \ldots = E X_n = \mu$ and $E X_{k+1} = \ldots = E_m = \nu$, where $\mu \not= \nu$ and $1 \leq k < m < n$. This leads to a test statistic $T_n$ given by the normalized maximum increment of the random walk $S_n = X_1 + \cdots + X_n$, $S_0 = 0$.\par For statistical reasons, it is very importand to understand the limit distribution of $T_n$. In this talk, we will derive this limit distribution for different linear and nonlinear dependence structures of the random variables $X_i$. It will turn out that $T_n$ will converge to a Fréchet distribution in every case.