The Mittag-Leffler process and a scaling limit for the block counting process of the Bolthausen-Sznitman coalescent

The Mittag-Leffler process $X=(X_t{)}_{t\ge 0}$ is introduced. This Markov process has the property that its marginal random variables $X_t$ are Mittag-Leffler distributed with parameter $e^{-t}$, $t\in [0,\mathrm{\infty })$, and the semigroup $(T_t{)}_{t\ge 0}$ of $X$ satisfies $T_tf(x)={\mathbb E}(f(x^{e^{-t}}X_t))$forall$x\ge 0$andallboundedmeasurablefunctions$f:[0,\infty)\to{\mathbb R}$\mathrm{.}Furthercharacteristicsoftheprocess$X$ are derived, for example an explicit formula for the joint moments of its finite dimensional distributions. The main result states that the block counting process of the Bolthausen-Sznitman $n$-coalescent, properly scaled, converges in the Skorohod topology to the Mittag-Leffler process $X$ as the sample size $n$ tends to infinity.