When is the deepest drawdown also the longest?

Yinhao Wu studies iid increments with positive finite mean. When losses have a finite exponential moment, the deepest episode’s duration, relative to the longest among \(n\) episodes, converges in probability to an explicit \(\rho \in (0,1)\), and its duration rank is \(n^{1-\rho+o_{\mathbb{P}}(1)}\). In the Gaussian model \(\rho = 1/2\), independently of drift and variance.

For regularly varying losses a single large loss governs both depth and duration, so the duration ratio tends to one almost surely and the \(k\) deepest and \(k\) longest episodes share the same indices with probability tending to one. The split reflects the different exponential costs of reaching a large depth and of sustaining a long episode.