Sparre-Andersen identity and the last passage time

It is shown that the celebrated result of Sparre Andersen for random walks and Lévy processes has intriguing consequences when the last time of the process in $(-\infty,0]$, say $σ$, is added to the picture. In the case of no positive jumps this leads to six random times, all of which have the same distribution - the uniform distribution on $[0,σ]$. Surprisingly, this result does not appear in the literature, even though it is based on some classical observations concerning exchangeable increments.

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