Convex Minorants of Random Walks and Brownian Motion

Abstract

CONVEX MINORANTS OF RANDOM WALKS AND BROWNIAN MOTIONПусть (S,-)JL 0 -процесс случайного блуждания, порожденный последовательностью независимых и одинаково распределенных ве-щественнозначных случайных величин (Х,)" =1 , имеющих плот ность. Изучаются вероятностные распределения, связанные с ас социированным процессом выпуклой миноранты. В частности, ис следуется длина самого длинного сегмента выпуклой миноранты. Используя теорию случайных перестановок, мы полностью харак теризуем распределение длины r-го по величине сегмента выпуклой миноранты броуновского движения на конечных интервалах; мы также указываем явный вид плотности совместного распределения г первых по длине сегментов. Кроме того, мы используем развитые здесь методы для доказательства формулы (Е. Sparre Andersen, [9]), позволяющей вычислить вероятность того, что выпуклая миноран та случайного блуждания длины N будет состоять из т сегментов. Приводятся аналогичные утверждения для случайных блужданий со случайными приращениями времени. Эти результаты недавно использованы автором для изучения динамики одномерных частиц с прилипанием.Ключевые слова и фразы: случайное блуждание, выпуклая ми норанта, броуновское движение, случайные перестановки. 1.Introduction. In this paper we study properties of convex minorants of random walks generated by real-valued independent identically dis tributed (i.i. (i) For a random walk of length TV, the probability that the associated convex minorant contains a segment of length n is n~l for n > N/2. The expected number of segments of length n is n~l.(ii) For s G [0,1] and any positive integer r, the probability that the length of the r-th longest segment, i^, of the convex minorant generated by Brownian motion on [0,1] is less than s is F r (s), where F r (s) has an explicit formula; the joint density for the lengths of the r longest segments,

Convex minorants of random walks and Lévy processes

“…almost surely sum to one and their law once arranged in decreasing order is referred to as the Poisson-Dirichlet distribution with parameter one which is the limiting distribution of the cycle structure of a permutation chosen uniformly at random (see [14]). We can now state the following result and note that a proof in the special case of Brownian motion was sketched in [19].…”

“…Since Brownian motion is a Lévy process (and stable with index 2), the results of the previous section apply to the convex minorant of Brownian motion, and some of these results were known (from [6,7,10,13,19]). However, Brownian motion offers extra analysis due to its special properties among Lévy processes (e.g.…”

“…The greatest convex minorant (or simply convex minorant for short) of a real-valued function (x u , u ∈ U ) with domain U contained in the real line is the maximal convex function (c u , u ∈ I) defined on a closed interval I containing U with c u ≤ x u for all u ∈ U . A number of authors have provided descriptions of certain features of the convex minorant for various stochastic processes such as random walks [, Brownian motion [6,7,10,13,19,3], Cauchy processes [4], Markov Processes [11], and Lévy processes (Chapter XI of [12]). Figure 1 illustrates an instance of the convex minorant for each of a random walk, a Brownian motion, and a Cauchy process on a finite interval.…”

The distribution of the maximal difference between a Brownian bridge and its concave majorant

“…Now let T (1) ≥ T (2) ≥ · · · denote the values of T i put in decreasing order. According to the result of Suidan [25], the distribution of the sequence (T (j) , j = 1, 2 . .…”

The greatest convex minorant of Brownian motion, meander, and bridge

“…The greatest convex minorant (or simply convex minorant for short) of a realvalued function (x u , u ∈ U ) with domain U contained in the real line is the maximal convex function (c u , u ∈ I) defined on a closed interval I containing U with c u ≤ x u for all u ∈ U . A number of authors have provided descriptions of certain features of the convex minorant for various stochastic processes such as random walks [17], Brownian motion [9,11,19,25,28], Cauchy processes [6], Markov Processes [4], and Lévy processes (Chapter XI of [23]).…”