# Occupation times of discrete-time fractional Brownian motion

## TL;DR

This paper establishes a local limit theorem for discrete-time fractional Brownian motion with high Hurst parameter and demonstrates that its scaled occupation time converges to a Mittag-Leffler distribution, linking stochastic processes with ergodic theory.

## Contribution

It provides the first conditional local limit theorem for dfBm with Hurst parameter between 3/4 and 1 and connects occupation times to Mittag-Leffler distributions using ergodic theory.

## Key findings

- Proves a conditional local limit theorem for dfBm.
- Shows scaled occupation times converge to Mittag-Leffler distribution.
- Links stochastic process behavior with infinite ergodic theory.

## Abstract

We prove a conditional local limit theorem for discrete-time fractional Brownian motions (dfBm) with Hurst parameter 3/4<H<1. Using results from infinite ergodic theory it is then shown that the properly scaled occupation time of dfBm converges to a Mittag-Leffler distribution.