Indicator fractional stable motions

Paul Jung

doi:10.1214/ecp.v16-1611

Abstract

Using the framework of random walks in random scenery, Cohen and Samorodnitsky (2006) introduced a family of symmetric $\alpha$-stable motions called local time fractional stable motions. When $\alpha=2$, these processes are precisely fractional Brownian motions with $1/2 < H < 1$. Motivated by random walks in alternating scenery, we find a complementary family of symmetric $\alpha$-stable motions which we call indicator fractional stable motions. These processes are complementary to local time fractional stable motions in that when $\alpha=2$, one gets fractional Brownian motions with $0 < H < 1/2$.

Highlights

There are a plethora of integral representations for Fractional Brownian motion (FBM) with Hurst parameter H ∈ (0, 1), and not surprisingly there are several generalizations of these integral representations to stable processes
LT-FSM is interesting because it is a subordinated process
Subordinated processes are processes constructed from integral representations with random kernels, or said another way, where the stable random measure has a control measure related in some way to a probability measure of some other stochastic process

Summary

Introduction

There are a plethora of integral representations for Fractional Brownian motion (FBM) with Hurst parameter H ∈ (0, 1), and not surprisingly there are several generalizations of these integral representations to stable processes. Using characterizations of the generating flows for the respective processes (see Section 3 below), [CS06] showed that the class of LT-FSMs is disjoint from the classes of RH-FSMs and L-FSMs. The first question one must ask is: are these new stable processes a legitimate new class of processes or are they just a different representation of L-FSMs and/or RH-FSMs? Using characterizations of the generating flows for the respective processes (see Section 3 below), [CS06] showed that the class of LT-FSMs is disjoint from the classes of RH-FSMs and L-FSMs Following their lead, we use the same characterizations to show that when the (discretized) subordinating process {An}n∈ is recurrent, the class of I-FSMs is disjoint from the two classes, RH-FSMs and L-FSMs. Since I-FSMs and LT-FSMs have disjoint self-similarity exponents when 1 < α < 2, these two classes of processes are disjoint when 1 < α < 2.

Indicator fractional stable motions

Background

Ergodic properties of indicator fractional stable noise