A note on $\alpha$-IDT processes

TL;DR

This paper introduces $\alpha$-IDT processes, a modified version of IDT processes, characterizes Gaussian examples including fractional Brownian motion, and explores their connections with selfdecomposability, stability, and self-similarity.

Contribution

It defines the new class of $\alpha$-IDT processes, provides examples, characterizes Gaussian cases, and links these processes to key properties like selfdecomposability and self-similarity.

Findings

Gaussian $\alpha$-IDT processes include fractional Brownian motion.

$\alpha$-IDT processes relate to selfdecomposability and stability.

Several examples of $\alpha$-IDT processes are provided.

Abstract

In this note, we introduce the notion of $\alpha$-IDT processes which is obtained from a slight and fundamental modification of the IDT property. Several examples of $\alpha$-IDT processes are given and Gaussian processes which are $\alpha$-IDT are characterized. A kind example of this Gaussian $\alpha$-IDT is the standard fractional Brownian motion. Also, we invest some links between the $\alpha$-IDT property, with selfdecomposability, temporal selfdecomposability, stability and self similarity.