关于q-函数方程和偏移矩
On q-functional equations and excursion moments
摘要：
我们分析了q-函数方程所产生的树状组合结构，这是计数的大小，内部路径长度，以及某些概括。对应的计数参数由正整数k标记。当尺寸母函数以平方根为主导奇点时，我们证明了在无限尺寸极限下这些参数的联合极限分布的存在性。极限分布与标准布朗行程的k次幂积分的极限分布一致。我们的方法产生一个递归的时刻的极限分布。它可用于分析矩的渐近展开，并可推广到其它类型的奇点。
英文摘要：
We analyse q-functional equations arising from tree-like combinatorial structures, which are counted by size, internal path length, and certain generalisations thereof. The corresponding counting parameters are labelled by a positive integer k. We show the existence of a joint limit distribution for these parameters in the limit of infinite size, if the size generating function has a square root as dominant singularity. The limit distribution coincides with that of integrals of k-th powers of the standard Brownian excursion. Our approach yields a recursion for the moments of the limit distribution. It can be used to analyse asymptotic expansions of the moments, and it admits an extension to other types of singularity.
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