A coupling proof of weak convergence

Weak convergence to reflected Brownian motion is deduced for certain upwardly drifting random walks by coupling them to a simple reflected random walk. The argument is quite elementary, and also gives the right conditions on the drift. A similar argument works for a corresponding continuous-time problem.

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Discrete Random Walks on One-Sided ``Periodic'' Graphs

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.3344 ◽

2003 ◽

Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽

Author(s):

Michael Drmota

Keyword(s):

Brownian Motion ◽

Random Walk ◽

Random Walks ◽

Generating Functions ◽

Connected Graph ◽

Infinite Graphs ◽

Reflected Brownian Motion ◽

Strongly Connected ◽

International Audience ◽

Null Recurrence

International audience In this paper we consider discrete random walks on infinite graphs that are generated by copying and shifting one finite (strongly connected) graph into one direction and connecting successive copies always in the same way. With help of generating functions it is shown that there are only three types for the asymptotic behaviour of the random walk. It either converges to the stationary distribution or it can be approximated in terms of a reflected Brownian motion or by a Brownian motion. In terms of Markov chains these cases correspond to positive recurrence, to null recurrence, and to non recurrence.

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Weak convergence of random walks, conditioned to stay away from small sets

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.50.2013.1.1232 ◽

2013 ◽

Vol 50 (1) ◽

pp. 122-128

Author(s):

Zsolt Pajor-Gyulai ◽

Domokos Szász

Keyword(s):

Brownian Motion ◽

Random Walk ◽

Weak Convergence ◽

Random Walks ◽

Continuous Process ◽

Random Variables ◽

Weak Limit ◽

Limit Laws ◽

Diffusive Limit

Let {Xn}n∈ℕ be a sequence of i.i.d. random variables in ℤd. Let Sk = X1 + … + Xk and Yn(t) be the continuous process on [0, 1] for which Yn(k/n) = Sk/n1/2 for k = 1, … n and which is linearly interpolated elsewhere. The paper gives a generalization of results of ([2]) on the weak limit laws of Yn(t) conditioned to stay away from some small sets. In particular, it is shown that the diffusive limit of the random walk meander on ℤd: d ≧ 2 is the Brownian motion.

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Weak Convergence Theorem of a Nonnegative Random Walk to Sticky Reflected Brownian Motion

Journal of Theoretical Probability ◽

10.1007/s10959-009-0244-4 ◽

2009 ◽

Vol 23 (4) ◽

pp. 1157-1181

Author(s):

Hirotaka Fushiya

Keyword(s):

Brownian Motion ◽

Random Walk ◽

Weak Convergence ◽

Convergence Theorem ◽

Reflected Brownian Motion ◽

Weak Convergence Theorem

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Moderate deviation principle for multivalued stochastic differential equations

Stochastics and Dynamics ◽

10.1142/s021949372050015x ◽

2019 ◽

Vol 20 (03) ◽

pp. 2050015 ◽

Cited By ~ 1

Author(s):

Hua Zhang

Keyword(s):

Brownian Motion ◽

Weak Convergence ◽

Differential Equations ◽

Stochastic Differential Equations ◽

Moderate Deviation ◽

Reflected Brownian Motion ◽

Moderate Deviation Principle ◽

Deviation Principle ◽

Weak Convergence Approach

In this paper, we prove a moderate deviation principle for the multivalued stochastic differential equations whose proof are based on recently well-developed weak convergence approach. As an application, we obtain the moderate deviation principle for reflected Brownian motion.

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Strong Local Survival of Branching Random Walks is Not Monotone

Advances in Applied Probability ◽

10.1017/s000186780000714x ◽

2014 ◽

Vol 46 (02) ◽

pp. 400-421 ◽

Cited By ~ 1

Author(s):

Daniela Bertacchi ◽

Fabio Zucca

Keyword(s):

Random Walk ◽

Random Walks ◽

Generating Function ◽

Continuous Time ◽

Local Property ◽

Branching Random Walk ◽

Infinite Dimensional ◽

Branching Random Walks ◽

Local Survival ◽

Branching Law

In this paper we study the strong local survival property for discrete-time and continuous-time branching random walks. We study this property by means of an infinite-dimensional generating functionGand a maximum principle which, we prove, is satisfied by every fixed point ofG. We give results for the existence of a strong local survival regime and we prove that, unlike local and global survival, in continuous time, strong local survival is not a monotone property in the general case (though it is monotone if the branching random walk is quasitransitive). We provide an example of an irreducible branching random walk where the strong local property depends on the starting site of the process. By means of other counterexamples, we show that the existence of a pure global phase is not equivalent to nonamenability of the process, and that even an irreducible branching random walk with the same branching law at each site may exhibit nonstrong local survival. Finally, we show that the generating function of an irreducible branching random walk can have more than two fixed points; this disproves a previously known result.

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Strong Local Survival of Branching Random Walks is Not Monotone

Advances in Applied Probability ◽

10.1239/aap/1401369700 ◽

2014 ◽

Vol 46 (2) ◽

pp. 400-421 ◽

Cited By ~ 5

Author(s):

Daniela Bertacchi ◽

Fabio Zucca

Keyword(s):

Random Walk ◽

Random Walks ◽

Generating Function ◽

Continuous Time ◽

Local Property ◽

Branching Random Walk ◽

Infinite Dimensional ◽

Branching Random Walks ◽

Local Survival ◽

Branching Law

In this paper we study the strong local survival property for discrete-time and continuous-time branching random walks. We study this property by means of an infinite-dimensional generating function G and a maximum principle which, we prove, is satisfied by every fixed point of G. We give results for the existence of a strong local survival regime and we prove that, unlike local and global survival, in continuous time, strong local survival is not a monotone property in the general case (though it is monotone if the branching random walk is quasitransitive). We provide an example of an irreducible branching random walk where the strong local property depends on the starting site of the process. By means of other counterexamples, we show that the existence of a pure global phase is not equivalent to nonamenability of the process, and that even an irreducible branching random walk with the same branching law at each site may exhibit nonstrong local survival. Finally, we show that the generating function of an irreducible branching random walk can have more than two fixed points; this disproves a previously known result.

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Limit theorems for some continuous-time random walks

Advances in Applied Probability ◽

10.1239/aap/1316792670 ◽

2011 ◽

Vol 43 (3) ◽

pp. 782-813 ◽

Cited By ~ 1

Author(s):

M. Jara ◽

T. Komorowski

Keyword(s):

Markov Chain ◽

Random Walk ◽

Random Walks ◽

Quantum Transport ◽

Continuous Time ◽

Limit Theorems ◽

Transport Theory ◽

Sufficient Conditions ◽

Continuous Time Random Walks ◽

Quantum Transport Theory

In this paper we consider the scaled limit of a continuous-time random walk (CTRW) based on a Markov chain {Xn,n≥ 0} and two observables, τ(∙) andV(∙), corresponding to the renewal times and jump sizes. Assuming that these observables belong to the domains of attraction of some stable laws, we give sufficient conditions on the chain that guarantee the existence of the scaled limits for CTRWs. An application of the results to a process that arises in quantum transport theory is provided. The results obtained in this paper generalize earlier results contained in Becker-Kern, Meerschaert and Scheffler (2004) and Meerschaert and Scheffler (2008), and the recent results of Henry and Straka (2011) and Jurlewicz, Kern, Meerschaert and Scheffler (2010), where {Xn,n≥ 0} is a sequence of independent and identically distributed random variables.

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Portfolio choice and the Bayesian Kelly criterion

Advances in Applied Probability ◽

10.1017/s0001867800027592 ◽

1996 ◽

Vol 28 (04) ◽

pp. 1145-1176 ◽

Cited By ~ 10

Author(s):

Sid Browne ◽

Ward Whitt

Keyword(s):

Stochastic Process ◽

Brownian Motion ◽

Portfolio Choice ◽

Random Environment ◽

Continuous Time ◽

Success Probability ◽

Simple Random Walk ◽

Diffusion Limit ◽

Kiefer Process ◽

Time Problem

We derive optimal gambling and investment policies for cases in which the underlying stochastic process has parameter values that are unobserved random variables. For the objective of maximizing logarithmic utility when the underlying stochastic process is a simple random walk in a random environment, we show that a state-dependent control is optimal, which is a generalization of the celebrated Kelly strategy: the optimal strategy is to bet a fraction of current wealth equal to a linear function of the posterior mean increment. To approximate more general stochastic processes, we consider a continuous-time analog involving Brownian motion. To analyze the continuous-time problem, we study the diffusion limit of random walks in a random environment. We prove that they converge weakly to a Kiefer process, or tied-down Brownian sheet. We then find conditions under which the discrete-time process converges to a diffusion, and analyze the resulting process. We analyze in detail the case of the natural conjugate prior, where the success probability has a beta distribution, and show that the resulting limit diffusion can be viewed as a rescaled Brownian motion. These results allow explicit computation of the optimal control policies for the continuous-time gambling and investment problems without resorting to continuous-time stochastic-control procedures. Moreover they also allow an explicit quantitative evaluation of the financial value of randomness, the financial gain of perfect information and the financial cost of learning in the Bayesian problem.

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Distributions of the Longest Excursions in a Tied Down Simple Random Walk and in a Brownian Bridge

Journal of Applied Probability ◽

10.1017/s0021900200003739 ◽

2007 ◽

Vol 44 (04) ◽

pp. 1056-1067 ◽

Cited By ~ 1

Author(s):

Andreas Lindell ◽

Lars Holst

Keyword(s):

Brownian Motion ◽

Random Walk ◽

Weak Convergence ◽

Joint Distribution ◽

Marginal Distribution ◽

Brownian Bridge ◽

Simple Random Walk ◽

Marginal Distributions

Expressions for the joint distribution of the longest and second longest excursions as well as the marginal distributions of the three longest excursions in the Brownian bridge are obtained. The method, which primarily makes use of the weak convergence of the random walk to the Brownian motion, principally gives the possibility to obtain any desired joint or marginal distribution. Numerical illustrations of the results are also given.

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