Invariant measures under random integral mappings and marginal distributions of fractional L\'evy processes

TL;DR

The paper investigates invariant measures under specific random integral mappings and explores their relation to fractional Lévy processes, providing new insights into the structure of these distributions and their applications.

Contribution

It characterizes invariant convolution semigroups under certain random integral mappings and applies findings to fractional Lévy processes.

Findings

Invariant measures are preserved under specific integral mappings.

Converse implications are established for certain infinitely divisible distributions.

Applications to moving average fractional Lévy processes are demonstrated.

Abstract

It is shown that some convolution semigroups of infinitely divisible measures are invariant under the random integral mappings $I^{h,r}_{(a,b]}$ defined in $(\star)$ below. The converse implication is specified for the semigroups of generalized s-selfdecomposable and selfdecomposable distributions. Some application are given to the moving average fractional L\'evy process (MAFLP).