  While PAC-Bayes is now an established learning framework for light-tailed
losses (\emph{e.g.}, subgaussian or subexponential), its extension to the case
of heavy-tailed losses remains largely uncharted and has attracted a growing
interest in recent years. We contribute PAC-Bayes generalisation bounds for
heavy-tailed losses under the sole assumption of bounded variance of the loss
function. Under that assumption, we extend previous results from
\citet{kuzborskij2019efron}. Our key technical contribution is exploiting an
extention of Markov's inequality for supermartingales. Our proof technique
unifies and extends different PAC-Bayesian frameworks by providing bounds for
unbounded martingales as well as bounds for batch and online learning with
heavy-tailed losses.
