Modern Stochastics: Theory and Applications

Multisets are like sets, except that they can contain multiple copies of their elements. If there are ${n_{i}}$ copies of i, $1\le i\le t$, in multiset ${M_{t}}$, then there are $\left(\genfrac{}{}{0.0pt}{}{{n_{1}}+\cdots +{n_{t}}}{{n_{1}},\dots ,{n_{t}}}\right)$ possible permutations of ${M_{t}}$. Knuth showed how to factor any multiset permutation into cycles. For fixed ${n_{i}}$, $i\ge 1$, we show how to adapt the Chinese restaurant process, which generates random permutations on n elements with weighting ${\theta ^{\# \hspace{0.1667em}\mathrm{cycles}}}$, $\theta \gt 0$, sequentially for $n=1,2,\dots $, to the multiset case, where we fix the ${n_{i}}$ and build permutations on ${M_{t}}$ sequentially for $t=1,2,\dots $. The number of cycles of a multiset permutation chosen uniformly at random, i.e. $\theta =1$, has distribution given by the sum of independent negative hypergeometric distributed random variables. For all $\theta \gt 0$, and under the assumption that ${n_{i}}=O(1)$, we show a central limit theorem as $t\to \infty $ for the number of cycles.

This paper presents some extensions of recent noncentral moderate deviation results. In the first part, the results in [Statist. Probab. Lett. 185, Paper No. 109424, 8 pp. (2022)] are generalized by considering a general Lévy process $\{S(t):t\ge 0\}$ instead of a compound Poisson process. In the second part, it is assumed that $\{S(t):t\ge 0\}$ has bounded variation and is not a subordinator; thus $\{S(t):t\ge 0\}$ can be seen as the difference of two independent nonnull subordinators. In this way, the results in [Mod. Stoch. Theory Appl. 11, 43–61] for Skellam processes are generalized.