A Mittag-Leffler mixed Poisson framework for lifetime flood exposure under non-stationary climate: Exact factorial moments and a dispersion diagnostic for memory

The probability that an engineered asset experiences design floods over its service life, the lifetime exposure risk, is conventionally computed under the assumption that annual exceedance events are independent, which yields the Poisson binomial distribution once climate drift is admitted through year-specific probabilities. Climate persistence modes additionally induce memory and clustering that no independent-event model can represent. We propose a mixed Poisson model in which all years of the design life share a single random rate multiplier, the Mittag-Leffler frailty, whose law is indexed by a memory parameter between zero and one; at the memoryless value one the classical model is recovered. The design-life count is shown to coincide in distribution, at the fixed design horizon, with the fractional Poisson process of Laskin, while differing from it at the process level, and the scope of the dependence it represents is stated explicitly. The probability mass function takes three-parameter Mittag-Leffler and Fox H-function form. The descending factorial moments admit a closed expression that is exact at every intensity, in contrast to ordinary fractional moments whose power-law scaling holds only asymptotically and underestimates the exact moment of order two and a half by a factor above two at the operational intensity of about one and a half expected events; the asymptotic scaling recovers the unified fractional moment theory of Chishtie (2025) in the large-intensity limit. The ratio of the second factorial moment to the squared mean depends on the memory index alone and gives a one-line diagnostic for memory from two sample moments. A Mittag-Leffler generalisation of the Hong inversion computes the distribution at the cost of the classical algorithm with aliasing error below ten to the minus thirty-four. Applied to a published SSP5 exposure trajectory under two calibration conventions, memory moves probability from the centre of the distribution to both the zero-exceedance class and the upper tail, raising the probability of five or more exceedances from 1.9 to between 6.3 and 8.5 percent as the memory index falls from one to one half, and the within-year Poisson approximation alters no probability by more than half a percentage point. A simulation study over a grid of intensities, memory indices and ensemble sizes shows that the two-moment estimator implied by the dispersion identity has lower root mean square error than four-moment matching in every configuration, with bootstrap coverage of 0.91 to 0.98 for ensembles of 500 or more, and provides detection-power tables for observational deployment. • Mittag-Leffler mixed Poisson model captures regime-scale climate memory in lifetime flood risk. • Exact factorial moments hold at all intensities, unlike asymptotic scaling laws. • Dispersion identity gives a one-line memory diagnostic from two sample moments. • Mittag-Leffler Hong inversion computes exposure at unchanged O(n log n) cost. • Two-moment estimator beats four-moment matching across a grid of 48 configurations.