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Infinitely divisible laws associated with hyperbolic functions

机译：与双曲函数相关的无限可整定律

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摘要

The infinitely divisible distributions on R+ of random variables C-t, S-t and T-t with Laplace transforms (1/cosh root 2lambda)(t), (root2lambda/sinh root2lambda)(t), and (tanhroot2lambda/root2lambda)(t) respectively are characterized for various t > 0 in a number of different ways: by simple relations between their moments and cumulants, by corresponding relations between the distributions and their Levy measures, by recursions for their Mellin transforms, and by differential equations satisfied by their Laplace transforms. Some of these results are interpreted probabilistically via known appearances of these distributions for t = 1 or 2 in the description of the laws of various functionals of Brownian motion and Bessel processes, such as the heights and lengths of excursions of a one-dimensional Brownian motion. The distributions of C-1 and S-2 are also known to appear in the Mellin representations of two important functions in analytic number theory, the Riemann zeta function and the Dirichlet L-function associated with the quadratic character modulo 4. Related families of infinitely divisible laws, including the gamma, logistic and generalized hyperbolic secant distributions, are derived from S-t and C-t by operations such as Brownian subordination, exponential tilting, and weak limits, and characterized in various ways. [References: 76]
机译：利用Laplace变换（1 / cosh根2lambda）（t），（root2lambda / sinh root2lambda）（t）和（tanhroot2lambda / root2lambda）（t）分别表征随机变量Ct，St和Tt在R +上的无穷可分分布对于t> 0的各种变化，有多种不同的方式：通过它们的矩和累积量之间的简单关系，通过分布与Levy度量之间的对应关系，通过其Mellin变换的递归，以及通过Laplace变换满足的微分方程。通过描述布朗运动和贝塞尔过程的各种泛函的定律，例如一维布朗运动的高度和长度，可以通过t = 1或2的这些分布的已知出现概率地解释其中一些结果。 。已知C-1和S-2的分布也出现在解析数论中的两个重要函数的Mellin表示中，即与二次模4相关的黎曼zeta函数和Dirichlet L函数。包括伽玛，对数分布和广义双曲正割分布在内的可除法则是通过诸如布朗从属，指数倾斜和弱极限之类的操作从St和Ct派生而来的，并且以各种方式表征。 [参考：76]

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