Weak Tail Conditions for Local Martingales

TL;DR

This paper characterizes the precise conditions under which a cadlag local martingale becomes a uniformly integrable martingale, focusing on tail behavior, jump convergence, and limit integrability.

Contribution

It establishes necessary and sufficient weak tail conditions for local martingales to be uniformly integrable, advancing the theoretical understanding of martingale convergence.

Findings

Weak tail of the supremum must be zero

Jumps at first-exit times must converge to zero in L^1

Almost sure limit must be integrable

Abstract

The following conditions are necessary and sufficient for an arbitrary c\`adl\`ag local martingale to be a uniformly integrable martingale: (i) The weak tail of the supremum of its modulus is zero; (ii) its jumps at the first-exit times from compact intervals converge to zero in $L^1$, on the events that those times are finite; and (iii) its almost sure limit is an integrable random variable.