arXiv 1503.051522015-03-17math.PR

关于强无序下的规范化乘法级联

On Normalized Multiplicative Cascades under Strong Disorder

Partha S. Dey, Edward Waymire

备注：: 11 pages, 1 figure, submitted

摘要：

乘法级联，在弱或强无序下，是指正随机测度$μ_{n,β}, n = 1,2,\dots$的序列，由正无序参数$β$参数化，并定义在$\partial T = \{0,1,\dots b-1\}^\infty$的Borel $σ$ -域${\mathcal B}$上用于乘积拓扑。归一化级联由通过配分函数$Z_{n,β}$归一化为概率的随机概率测量$prob_{n,β}:= Z_{n,β}^{-1}μ_{n,β}, n = 1,2\dots,$的相应序列定义。在本文中，Madaule（2011）的一个最新结果被用来显式地构造强无序参数$β> β_c$的树索引概率测度$prob_{\infty,β}$族，几乎必然定义在公共概率空间上。此外，将$\{prob_{n,β}: β> β_c\}_{n=1}^\infty$看作概率测度值随机过程序列，导致了概率测度值过程$\{prob_{\infty,β}: β> β_c\}$在分布上的有限维弱收敛.极限过程是由树指标的导数鞅随机场和Brunet-Derrida-Madaule装饰Poisson过程构造的。提供了一些推论来说明这种构造的效用。

英文摘要：

Multiplicative cascades, under weak or strong disorder, refer to sequences of positive random measures $μ_{n,β}, n = 1,2,\dots$, parameterized by a positive disorder parameter $β$, and defined on the Borel $σ$-field ${\mathcal B}$ of $\partial T = \{0,1,\dots b-1\}^\infty$ for the product topology. The normalized cascade is defined by the corresponding sequence of random probability measures $prob_{n,β}:= Z_{n,β}^{-1}μ_{n,β}, n = 1,2\dots,$ normalized to a probability by the partition function $Z_{n,β}$. In this note, a recent result of Madaule (2011) is used to explicitly construct a family of tree indexed probability measures $prob_{\infty,β}$ for strong disorder parameters $β> β_c$, almost surely defined on a common probability space. Moreover, viewing $\{prob_{n,β}: β> β_c\}_{n=1}^\infty$ as a sequence of probability measure valued stochastic process leads to finite dimensional weak convergence in distribution to a probability measure valued process $\{prob_{\infty,β}: β> β_c\}$. The limit process is constructed from the tree-indexed random field of derivative martingales, and the Brunet-Derrida-Madaule decorated Poisson process. A number of corollaries are provided to illustrate the utility of this construction.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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