# The left tail of renewal measure

## TL;DR

This paper derives precise asymptotics for the left tail of the renewal measure in two-sided random walks, using a change of measure approach under broad conditions.

## Contribution

It provides a new, nearly equivalent formulation to Blackwell's Theorem for the left tail asymptotics of renewal measures with finite exponential moments.

## Key findings

- Exact asymptotics for the left tail of renewal measure
- Applicable to broad class of two-sided random walks
- Uses a simple change of measure approach

## Abstract

In the paper, we find exact asymptotics of the left tail of renewal measure for a broad class of two-sided random walks. We only require that an exponential moment of the left tail is finite. Through a simple change of measure approach, our result turns out to be almost equivalent to Blackwell's Theorem.