Asymptotic entropy of transformed random walks

We consider general transformations of random walks on groups determined by Markov stopping times and prove that the asymptotic entropy (respectively, rate of escape) of the transformed random walks is equal to the asymptotic entropy (respectively, rate of escape) of the original random walk multiplied by the expectation of the corresponding stopping time. This is an analogue of the well-known Abramov formula from ergodic theory; its particular cases were established earlier by Kaimanovich [Differential entropy of the boundary of a random walk on a group. Uspekhi Mat. Nauk38(5(233)) (1983), 187–188] and Hartman et al [An Abramov formula for stationary spaces of discrete groups. Ergod. Th. & Dynam. Sys.34(3) (2014), 837–853].

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A General ‘Bang-Bang’ Principle for Predicting the Maximum of a Random Walk

Journal of Applied Probability ◽

10.1017/s0021900200007373 ◽

2010 ◽

Vol 47 (04) ◽

pp. 1072-1083 ◽

Cited By ~ 1

Author(s):

Pieter Allaart

Keyword(s):

Random Walk ◽

Optimal Stopping ◽

Stopping Time ◽

Stopping Times ◽

Optimal Prediction ◽

Prediction Problem ◽

Brownian Motion With Drift ◽

Natural Filtration ◽

Mathematical Justification ◽

Bernoulli Random Walk

Let (B t )0≤t≤T be either a Bernoulli random walk or a Brownian motion with drift, and let M t := max{B s: 0 ≤ s ≤ t}, 0 ≤ t ≤ T. In this paper we solve the general optimal prediction problem sup0≤τ≤T E[f(M T − B τ], where the supremum is over all stopping times τ adapted to the natural filtration of (B t ) and f is a nonincreasing convex function. The optimal stopping time τ* is shown to be of ‘bang-bang’ type: τ* ≡ 0 if the drift of the underlying process (B t ) is negative and τ* ≡ T if the drift is positive. This result generalizes recent findings of Toit and Peskir (2009) and Yam, Yung and Zhou (2009), and provides additional mathematical justification for the dictum in finance that one should sell bad stocks immediately, but keep good stocks as long as possible.

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Asymptotic entropy of the ranges of random walks on discrete groups

Science China Mathematics ◽

10.1007/s11425-017-9369-6 ◽

2020 ◽

Vol 63 (6) ◽

pp. 1153-1168

Author(s):

Xinxing Chen ◽

Jiansheng Xie ◽

Minzhi Zhao

Keyword(s):

Random Walks ◽

Discrete Groups ◽

Asymptotic Entropy

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A General ‘Bang-Bang’ Principle for Predicting the Maximum of a Random Walk

Journal of Applied Probability ◽

10.1239/jap/1294170520 ◽

2010 ◽

Vol 47 (4) ◽

pp. 1072-1083 ◽

Cited By ~ 7

Author(s):

Pieter Allaart

Keyword(s):

Random Walk ◽

Optimal Stopping ◽

Stopping Time ◽

Stopping Times ◽

Optimal Prediction ◽

Prediction Problem ◽

Brownian Motion With Drift ◽

Natural Filtration ◽

Mathematical Justification ◽

Bernoulli Random Walk

Let (Bt)0≤t≤T be either a Bernoulli random walk or a Brownian motion with drift, and let Mt := max{Bs: 0 ≤ s ≤ t}, 0 ≤ t ≤ T. In this paper we solve the general optimal prediction problem sup0≤τ≤TE[f(MT − Bτ], where the supremum is over all stopping times τ adapted to the natural filtration of (Bt) and f is a nonincreasing convex function. The optimal stopping time τ* is shown to be of ‘bang-bang’ type: τ* ≡ 0 if the drift of the underlying process (Bt) is negative and τ* ≡ T if the drift is positive. This result generalizes recent findings of Toit and Peskir (2009) and Yam, Yung and Zhou (2009), and provides additional mathematical justification for the dictum in finance that one should sell bad stocks immediately, but keep good stocks as long as possible.

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Hausdorff dimension of the harmonic measure on trees

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385798108180 ◽

1998 ◽

Vol 18 (3) ◽

pp. 631-660 ◽

Cited By ~ 15

Author(s):

VADIM A. KAIMANOVICH

Keyword(s):

Hausdorff Dimension ◽

Random Walks ◽

Large Class ◽

Harmonic Measure ◽

Random Environment ◽

Free Groups ◽

Equivalence Relations ◽

Markov Operators ◽

Rate Of Escape ◽

Asymptotic Entropy

For a large class of Markov operators on trees we prove the formula ${\bf HD}\,\nu=h/l$ connecting the Hausdorff dimension of the harmonic measure $\nu$ on the tree boundary, the rate of escape $l$ and the asymptotic entropy $h$. Applications of this formula include random walks on free groups, conditional random walks, random walks in random environment and random walks on treed equivalence relations.

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Ergodic theory on Galton—Watson trees: speed of random walk and dimension of harmonic measure

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700008543 ◽

1995 ◽

Vol 15 (3) ◽

pp. 593-619 ◽

Cited By ~ 47

Author(s):

Russell Lyons ◽

Robin Pemantle ◽

Yuval Peres

Keyword(s):

Random Walk ◽

Ergodic Theory ◽

Harmonic Measure ◽

Branching Process ◽

Main Tool ◽

Simple Random Walk ◽

Family Tree ◽

First Order ◽

The Family ◽

Rate Of Escape

AbstractWe consider simple random walk on the family treeTof a nondegenerate supercritical Galton—Watson branching process and show that the resulting harmonic measure has a.s. strictly smaller Hausdorff dimension than that of the whole boundary ofT. Concretely, this implies that an exponentially small fraction of thenth level ofTcarries most of the harmonic measure. First-order asymptotics for the rate of escape, Green function and the Avez entropy of the random walk are also determined. Ergodic theory of the shift on the space of random walk paths on trees is the main tool; the key observation is that iterating the transformation induced from this shift to the subset of ‘exit points’ yields a nonintersecting path sampled from harmonic measure.

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Rate of escape of random walks on free products

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700036375 ◽

2007 ◽

Vol 83 (1) ◽

pp. 31-54 ◽

Cited By ~ 6

Author(s):

Lorenz A. Gilch

Keyword(s):

Random Walk ◽

Random Walks ◽

Free Product ◽

Finite Family ◽

Block Length ◽

Probability Measures ◽

Free Products ◽

Rate Of Escape ◽

Transient Random Walk

AbstractSuppose we are given the free product V of a finite family of finite or countable sets (Vi)i∈∮ and probability measures on each Vi, which govern random walks on it. We consider a transient random walk on the free product arising naturally from the random walks on the Vi. We prove the existence of the rate of escape with respect to the block length, that is, the speed at which the random walk escapes to infinity, and furthermore we compute formulae for it. For this purpose, we present three different techniques providing three different, equivalent formulae.

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On the validity of Wald's equation

Journal of Applied Probability ◽

10.1017/s0021900200099484 ◽

1994 ◽

Vol 31 (04) ◽

pp. 949-957 ◽

Cited By ~ 2

Author(s):

Markus Roters

Keyword(s):

Random Walk ◽

Stopping Time ◽

Random Variables ◽

Stopping Times ◽

Stieltjes Transform ◽

Positive Part ◽

Negative Drift ◽

The Given ◽

Independent Identically Distributed

In this paper we review conditions under which Wald's equation holds, mainly if the expectation of the given stopping time is infinite. As a main result we obtain what is probably the weakest possible version of Wald's equation for the case of independent, identically distributed (i.i.d.) random variables. Moreover, we improve a result of Samuel (1967) concerning the existence of stopping times for which the expectation of the stopped sum of the underlying i.i.d. sequence of random variables does not exist. Finally, we show by counterexamples that it is impossible to generalize a theorem of Kiefer and Wolfowitz (1956) relating the moments of the supremum of a random walk with negative drift to moments of the positive part of X 1 to the case where the expectation of X 1 is —∞. Here, the Laplace–Stieltjes transform of the supremum of the considered random walk plays an important role.

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On the validity of Wald's equation

Journal of Applied Probability ◽

10.2307/3215320 ◽

1994 ◽

Vol 31 (4) ◽

pp. 949-957 ◽

Cited By ~ 7

Author(s):

Markus Roters

Keyword(s):

Random Walk ◽

Stopping Time ◽

Random Variables ◽

Stopping Times ◽

Stieltjes Transform ◽

Positive Part ◽

Negative Drift ◽

The Given ◽

Independent Identically Distributed

In this paper we review conditions under which Wald's equation holds, mainly if the expectation of the given stopping time is infinite. As a main result we obtain what is probably the weakest possible version of Wald's equation for the case of independent, identically distributed (i.i.d.) random variables. Moreover, we improve a result of Samuel (1967) concerning the existence of stopping times for which the expectation of the stopped sum of the underlying i.i.d. sequence of random variables does not exist. Finally, we show by counterexamples that it is impossible to generalize a theorem of Kiefer and Wolfowitz (1956) relating the moments of the supremum of a random walk with negative drift to moments of the positive part of X1 to the case where the expectation of X1 is —∞. Here, the Laplace–Stieltjes transform of the supremum of the considered random walk plays an important role.

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