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Haar-Based Multiresolution Stochastic Processes

机译：基于Haar的多分辨率随机过程

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Modifying a Haar wavelet representation of Brownian motion yields a class of Haar-based multiresolution stochastic processes in the form of an infinite series $$X_t = sum_{n=0}^inftylambda_nvarDelta _n(t)epsilon_n,$$ where λ n Δ n (t) is the integral of the nth Haar wavelet from 0 to t, and ε n are i.i.d. random variables with mean 0 and variance 1. Two sufficient conditions are provided for X t to converge uniformly with probability one. Each stochastic process , the collection of all almost sure uniform limits, retains the second-moment properties and the same roughness of sample paths as Brownian motion, yet lacks some of the features of Brownian motion, e.g., does not have independent and/or stationary increments, is not Gaussian, is not self-similar, or is not a martingale. Two important tools are developed to analyze elements of , the nth-level self-similarity of the associated bridges and the tree structure of dyadic increments. These tools are essential in establishing sample path results such as Hölder continuity and fractional dimensions of graphs of the processes.
机译：修改布朗运动的Haar小波表示形式会产生一类基于Haar的多分辨率随机过程，形式为无穷级$$ X_t = sum_ {n = 0} ^ inftylambda_nvarDelta _n（t）epsilon_n，$$，其中λn Δn （t）是从0到t的第n个Haar小波的积分，而εn 是iid均值为0且方差为1的随机变量。X t 提供了两个充分条件，以1的概率均匀收敛。每个随机过程（几乎所有确定的统一限制的集合）都保留了第二矩属性，并且采样路径的粗糙度与布朗运动相同，但缺乏布朗运动的某些特征，例如，没有独立和/或固定的增量，不是高斯，不是自相似或不是mar。开发了两个重要的工具来分析的元素，相关桥的n级自相似性和二进增量的树结构。这些工具对于建立样品路径结果（例如Hölder连续性和过程图的分数维）至关重要。

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