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

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.

A coupling proof of weak convergence

1985 ◽  
Vol 22 (2) ◽  
pp. 447-453
Author(s):  
Peter Guttorp ◽  
Reg Kulperger ◽  
Richard Lockhart
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Weak Convergence ◽  
Random Walks ◽  
Continuous Time ◽  
Similar Argument ◽  
Reflected Brownian Motion ◽  
Time Problem ◽  
The Right ◽  
Reflected Random Walk

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.

A coupling proof of weak convergence

1985 ◽  
Vol 22 (02) ◽  
pp. 447-453
Author(s):  
Peter Guttorp ◽  
Reg Kulperger ◽  
Richard Lockhart
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Weak Convergence ◽  
Random Walks ◽  
Continuous Time ◽  
Similar Argument ◽  
Reflected Brownian Motion ◽  
Time Problem ◽  
The Right ◽  
Reflected Random Walk

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.

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

2007 ◽  
Vol 44 (04) ◽  
pp. 1056-1067 ◽  
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.

The ruin problem and cover times of asymmetric random walks and Brownian motions

2000 ◽  
Vol 32 (01) ◽  
pp. 177-192 ◽  
Author(s):  
K. S. Chong ◽  
Richard Cowan ◽  
Lars Holst
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Random Walks ◽  
Cover Time ◽  
Brownian Motions ◽  
Brownian Motion With Drift ◽  
And Brownian Motion ◽  
Cover Times ◽  
Ruin Problem

A simple asymmetric random walk on the integers is stopped when its range is of a given length. When and where is it stopped? Analogous questions can be stated for a Brownian motion. Such problems are studied using results for the classical ruin problem, yielding results for the cover time and the range, both for asymmetric random walks and Brownian motion with drift.

Limiting diffusion for random walks with drift conditioned to stay positive

1978 ◽  
Vol 15 (02) ◽  
pp. 280-291 ◽  
Author(s):  
Peichuen Kao
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Random Function ◽  
Random Variables ◽  
Principal Result ◽  
Hitting Time ◽  
Brownian Excursion ◽  
Norming Constant ◽  
Finite Dimensional ◽  
Independent Identically Distributed

Let {ξ k : k ≧ 1} be a sequence of independent, identically distributed random variables with E{ξ 1} = μ ≠ 0. Form the random walk {S n : n ≧ 0} by setting S 0, S n = ξ 1 + ξ 2 + ··· + ξ n , n ≧ 1. Define the random function Xn by setting where α is a norming constant. Let N denote the hitting time of the set (–∞, 0] by the random walk. The principal result in this paper is to show (under appropriate conditions on the distribution of ξ 1) that the finite-dimensional distributions of Xn , conditioned on n < N < ∞ converge to those of the Brownian excursion process.

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

2007 ◽  
Vol 44 (4) ◽  
pp. 1056-1067
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.

Winding angle and maximum winding angle of the two-dimensional random walk

1991 ◽  
Vol 28 (4) ◽  
pp. 717-726 ◽  
Author(s):  
Claude Bélisle ◽  
Julian Faraway
Keyword(s):  
Asymptotic Behavior ◽  
Brownian Motion ◽  
Random Walk ◽  
Random Walks ◽  
Computer Simulations ◽  
Asymptotic Distributions ◽  
Two Dimensional ◽  
Exact Distributions ◽  
The Difference ◽  
Winding Angle

Recent results on the winding angle of the ordinary two-dimensional random walk on the integer lattice are reviewed. The difference between the Brownian motion winding angle and the random walk winding angle is discussed. Other functionals of the random walk, such as the maximum winding angle, are also considered and new results on their asymptotic behavior, as the number of steps increases, are presented. Results of computer simulations are presented, indicating how well the asymptotic distributions fit the exact distributions for random walks with 10m steps, for m = 2, 3, 4, 5, 6, 7.

scholarly journals Empirical processes for recurrent and transient random walks in random scenery

2020 ◽  
Vol 24 ◽  
pp. 127-137
Author(s):  
Nadine Guillotin-Plantard ◽  
Françoise Pène ◽  
Martin Wendler
Keyword(s):  
Random Walk ◽  
Asymptotic Behaviour ◽  
Random Walks ◽  
Stable Distribution ◽  
Random Variables ◽  
Empirical Processes ◽  
Independent Random Variables ◽  
Random Scenery ◽  
Recurrent Random Walk ◽  
Transient Random Walk

In this paper, we are interested in the asymptotic behaviour of the sequence of processes (Wn(s,t))s,t∈[0,1] with \begin{equation*} W_n(s,t):=\sum_{k=1}^{\lfloor nt\rfloor}\big(\mathds{1}_{\{\xi_{S_k}\leq s\}}-s\big) \end{equation*} where (ξx, x ∈ ℤd) is a sequence of independent random variables uniformly distributed on [0, 1] and (Sn)n ∈ ℕ is a random walk evolving in ℤd, independent of the ξ’s. In M. Wendler [Stoch. Process. Appl. 126 (2016) 2787–2799], the case where (Sn)n ∈ ℕ is a recurrent random walk in ℤ such that (n−1/αSn)n≥1 converges in distribution to a stable distribution of index α, with α ∈ (1, 2], has been investigated. Here, we consider the cases where (Sn)n ∈ ℕ is either: (a) a transient random walk in ℤd, (b) a recurrent random walk in ℤd such that (n−1/dSn)n≥1 converges in distribution to a stable distribution of index d ∈{1, 2}.

scholarly journals Adapted integral representations of random variables

2015 ◽  
Vol 36 ◽  
pp. 1560004 ◽  
Author(s):  
Georgiy Shevchenko ◽  
Lauri Viitasaari
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Stochastic Integral ◽  
Continuous Process ◽  
Random Variables ◽  
Random Variable ◽  
Integral Representations ◽  
Hölder Continuous ◽  
Continuous Processes ◽  
Mixed Fractional Brownian Motion

We study integral representations of random variables with respect to general Hölder continuous processes and with respect to two particular cases; fractional Brownian motion and mixed fractional Brownian motion. We prove that an arbitrary random variable can be represented as an improper integral, and that the stochastic integral can have any distribution. If in addition the random variable is a final value of an adapted Hölder continuous process, then it can be represented as a proper integral. It is also shown that in the particular case of mixed fractional Brownian motion, any adapted random variable can be represented as a proper integral.

scholarly journals Extensions and solutions for nonlinear diffusion equations and random walks

2019 ◽  
Vol 475 (2231) ◽  
pp. 20190432 ◽  
Author(s):  
E. K. Lenzi ◽  
M. K. Lenzi ◽  
H. V. Ribeiro ◽  
L. R. Evangelista
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Random Walks ◽  
Nonlinear Diffusion ◽  
Fractional Derivatives ◽  
Distribution Functions ◽  
Diffusion Equations ◽  
Nonlinear Diffusion Equations

We investigate a connection between random walks and nonlinear diffusion equations within the framework proposed by Einstein to explain the Brownian motion. We show here how to properly modify that framework in order to handle different physical scenarios. We obtain solutions for nonlinear diffusion equations that emerge from the random walk approach and analyse possible connections with a generalized thermostatistics formalism. Finally, we conclude that fractal and fractional derivatives may emerge in the context of nonlinear diffusion equations, depending on the choice of distribution functions related to the spreading of systems.

Sign in / Sign up

Export Citation Format

Share Document