基于伴随积分变换的Levy稳定分布
Levy stable distributions via associated integral transform
摘要：
本文给出了单侧重尾Levy稳定概率分布g_α（x），0 \leq x < \infty，0 < α< 1的精确显式的生成方法.我们证明，只要知道一个这样的分布g_α（x），就足以精确地得到g_{α^{p}}（x），p=2，3，.同样地，由已知的g_α（x）和g_β（x），0 < α，β< 1，我们得到了g_{αβ}（x）.该方法基于积分算子的构造，称为Levy变换，它实现了上述操作。对于α有理数，α= l/k，l < k，我们用这种方法重现了最近得到的关于g_{l/k}（x）的许多精确结果.这种方法也可以被改写为Efros定理在广义拉普拉斯卷积中的应用。它完全依赖于有效的定积分。
英文摘要：
We present a method of generation of exact and explicit forms of one-sided, heavy-tailed Levy stable probability distributions g_α(x), 0 \leq x < \infty, 0 < α< 1. We demonstrate that the knowledge of one such a distribution g_α(x) suffices to obtain exactly g_{α^{p}}(x), p=2, 3,... Similarly, from known g_α(x) and g_β(x), 0 < α, β< 1, we obtain g_{αβ}(x). The method is based on the construction of the integral operator, called Levy transform, which implements the above operations. For αrational, α= l/k with l < k, we reproduce in this manner many of the recently obtained exact results for g_{l/k}(x). This approach can be also recast as an application of the Efros theorem for generalized Laplace convolutions. It relies solely on efficient definite integration.
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