The binomial, the negative binomial, the Poisson, the compound Poisson and the Erlang distribution do all admit integral representations with respect to its (continuous) parameter. We use the Margulis-Russo type formulas for Bernoulli and Poisson processes to derive these representations in a unified way and to provide a probabilistic interpretation for the derivatives. By similar variational methods, we obtain apparently new integro-differential identities which the density of a strictly $α$-stable multivariate density satisfies. Then, we extend Crofton's derivative formula known in integral geometry to the case of a Poisson process. Finally we use this extension to give a new probabilistic proof of a version of this formula for binomial point processes.
Clark--Ocone formula and variational representation for Poisson functionals
本文首先证明了Poisson空间上任意有界可测泛函的一个Clark-Ocone公式。然后利用这个公式，在Poisson随机测度的强度测度的一定条件下，证明了Dupuis和Ellis [A Weak Convergence Approach to the Theory of大偏差（1997）Wiley]，p. 122所证明的有界Poisson泛函的拉普拉斯变换的一个变分表示公式.
A subgeometric convergence formula for finite-level M/G/1-type Markov chains: via a block-decomposition-friendly solution for the Poisson equation of deviation matrix
On Poisson approximations for the Ewens sampling formula when the mutation parameter grows with the sample size
Ewens抽样公式最早是由Warren John Ewens于1972年在群体遗传学的背景下提出的，并已出现在许多其他科学领域。有丰富的近似结果与Ewens抽样公式，特别是当其中一个参数，样本量$n$或突变参数$θ$，表示缩放突变率，趋于无穷大，而另一个是固定的。相比之下，$θ$与$n$一起增长的情况在相对较少的工作中被考虑，尽管这种渐近设置也是自然的。本文从Poisson近似的观点出发，研究了当$θ$随$n$增长时，在Ewens抽样公式假设下，等位基因总数和等位基因划分中组分计数的渐近性质。
