# Tempered fractional Brownian and stable motions of second kind

## TL;DR

This paper introduces a new version of tempered fractional Brownian and stable motions of second kind, distinguishes them from existing models, and analyzes their large-time properties, revealing significant differences.

## Contribution

It proposes the tempered fractional Brownian/stable motion of second kind (TFBM II/TFSM II) and demonstrates they are distinct from the original tempered models, with unique large-time behaviors.

## Key findings

- TFBM II/TFSM II are different processes from TFBM/TFSM.
- Large time properties of TFBM II/TFSM II resemble FBM/FSM.
- Large time properties of TFBM/TFSM differ from TFBM/TFSM.

## Abstract

Meerschaert and Sabzikar [12], [13] introduced tempered fractional Brownian/stable motion (TFBM/TFSM) by including an exponential tempering factor in the moving average representation of FBM/FSM. The present paper discusses another tempered version of FBM/FSM, termed tempered fractional Brownian/stable motion of second kind (TFBM II/TFSM II).We prove that TFBM/TFSM and TFBM II/TFSM II are different processes. Particularly, large time properties of TFBM II/TFSM II are similar to those of FBM/FSM and are in deep contrast to large time properties of TFBM/TFSM.