Fractional Tail-Event Timing Diagnostics: Extreme Value Theory Severity, Mittag-Leffler Recurrence, and Stress Episodes

Abstract

Extreme financial events are usually defined by loss magnitude alone, which quantifies severity but leaves open whether tail losses arrive in isolation or in temporally concentrated patterns. This paper develops a fractional tail-event timing diagnostic combining peaks-over-threshold (POT) generalized Pareto distribution (GPD) severity with Mittag-Leffler recurrence of interarrival times. We show that when exceedance times are driven by a stationary and ergodic latent intensity, the log-moment recurrence estimator has an explicit probability limit below the Poisson boundary that depends on the intensity only through its marginal log-variance, so volatility-driven dispersion of the arrival rate suffices to produce sub-Poisson estimates. In six daily series through 2026, the estimator is calibrated against a permutation null preserving the marginal loss distribution and the integer-gap discretization. That null sits near 1.05 rather than at unity, so the nominal Poisson boundary is not the correct reference for trading-day gaps. Against the calibrated null, the Deutscher Aktienindex (DAX), Nasdaq Composite, Nikkei 225, platinum, and the Standard & Poor’s 500 (S&P 500) display significant temporal concentration and coffee does not, whereas the monotone decline of the estimator across thresholds proves to be mechanical. Once goodness-of-fit tests are bootstrap-calibrated, no waiting-time law survives for four assets. The Mittag-Leffler parameter is therefore used only as an interpretable diagnostic index of temporal concentration; it is not a fitted-law claim and should not be interpreted as evidence of structural long memory or of a fractional data-generating mechanism.