Recurrence and Transience of Critical Branching Processes in Random Environment with Immigration and an Application to Excited Random Walks

We establish recurrence and transience criteria for critical branching processes in random environments with immigration. These results are then applied to the recurrence and transience of a recurrent random walk in a random environment on ℤ disturbed by cookies inducing a drift to the right of strength 1.

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Random walk in a random environment with correlated sites

Journal of Applied Probability ◽

10.1239/jap/1011994189 ◽

2001 ◽

Vol 38 (4) ◽

pp. 1018-1032 ◽

Cited By ~ 1

Author(s):

T. Komorowski ◽

G. Krupa

Keyword(s):

Random Walk ◽

Random Walks ◽

Random Field ◽

Random Environment ◽

Law Of Large Numbers ◽

Transition Probabilities ◽

Direct Application ◽

Random Environments ◽

Large Numbers ◽

Stationary Random Field

We prove the law of large numbers for random walks in random environments on the d-dimensional integer lattice Zd. The environment is described in terms of a stationary random field of transition probabilities on the lattice, possessing a certain drift property, modeled on the Kalikov condition. In contrast to the previously considered models, we admit possible correlation of transition probabilities at different sites, assuming however that they become independent at finite distances. The possible dependence of sites makes impossible a direct application of the renewal times technique of Sznitman and Zerner.

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Combining Losing Games into a Winning Game

Fluctuation and Noise Letters ◽

10.1142/s0219477519500032 ◽

2019 ◽

Vol 18 (01) ◽

pp. 1950003 ◽

Cited By ~ 3

Author(s):

Bruno Rémillard ◽

Jean Vaillancourt

Keyword(s):

Asymptotic Behavior ◽

Random Walk ◽

Random Walks ◽

Random Environment ◽

Regime Switching ◽

Random Environments ◽

Full Characterization ◽

Parrondo’S Paradox ◽

Random Subspaces

Parrondo’s paradox is extended to regime switching random walks in random environments. The paradoxical behavior of the resulting random walk is explained by the effect of the random environment. Full characterization of the asymptotic behavior is achieved in terms of the dimensions of some random subspaces occurring in Oseledec’s theorem. The regime switching mechanism gives our models a richer and more complex asymptotic behavior than the simple random walks in random environments appearing in the literature, in terms of transience and recurrence.

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Random walk in a random environment with correlated sites

Journal of Applied Probability ◽

10.1017/s0021900200019203 ◽

2001 ◽

Vol 38 (04) ◽

pp. 1018-1032 ◽

Cited By ~ 1

Author(s):

T. Komorowski ◽

G. Krupa

Keyword(s):

Random Walk ◽

Random Walks ◽

Random Field ◽

Random Environment ◽

Law Of Large Numbers ◽

Transition Probabilities ◽

Direct Application ◽

Random Environments ◽

Large Numbers ◽

Stationary Random Field

We prove the law of large numbers for random walks in random environments on the d-dimensional integer lattice Z d . The environment is described in terms of a stationary random field of transition probabilities on the lattice, possessing a certain drift property, modeled on the Kalikov condition. In contrast to the previously considered models, we admit possible correlation of transition probabilities at different sites, assuming however that they become independent at finite distances. The possible dependence of sites makes impossible a direct application of the renewal times technique of Sznitman and Zerner.

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A comparison of random walks in dependent random environments

Advances in Applied Probability ◽

10.1017/apr.2015.13 ◽

2016 ◽

Vol 48 (1) ◽

pp. 199-214

Author(s):

Werner R. W. Scheinhardt ◽

Dirk P. Kroese

Keyword(s):

Random Walks ◽

Random Environment ◽

Moving Average ◽

Random Environments ◽

Dependency Structure ◽

Interesting Behavior

Abstract We provide exact computations for the drift of random walks in dependent random environments, including k-dependent and moving average environments. We show how the drift can be characterized and evaluated using Perron–Frobenius theory. Comparing random walks in various dependent environments, we demonstrate that their drifts can exhibit interesting behavior that depends significantly on the dependency structure of the random environment.

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Empirical processes for recurrent and transient random walks in random scenery

ESAIM Probability and Statistics ◽

10.1051/ps/2019030 ◽

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}.

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A coupling proof of weak convergence

Journal of Applied Probability ◽

10.2307/3213788 ◽

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.

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Branching processes in random environment die slowly

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.3578 ◽

2008 ◽

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

Author(s):

Vladimir Vatutin ◽

Andreas Kyprianou

Keyword(s):

Random Walk ◽

Random Environment ◽

Generating Functions ◽

Limit Theorems ◽

Branching Process ◽

Branching Processes ◽

Limit Distributions ◽

Conditional Limit Theorems ◽

International Audience ◽

Conditional Limit

International audience Let $Z_n,n=0,1,\ldots,$ be a branching process evolving in the random environment generated by a sequence of iid generating functions $f_0(s),f_1(s),\ldots,$ and let $S_0=0$, $S_k=X_1+ \ldots +X_k,k \geq 1$, be the associated random walk with $X_i=\log f_{i-1}^{\prime}(1), \tau (m,n)$ be the left-most point of minimum of $\{S_k,k \geq 0 \}$ on the interval $[m,n]$, and $T=\min \{ k:Z_k=0\}$. Assuming that the associated random walk satisfies the Doney condition $P(S_n > 0) \to \rho \in (0,1), n \to \infty$, we prove (under the quenched approach) conditional limit theorems, as $n \to \infty$, for the distribution of $Z_{nt}, Z_{\tau (0,nt)}$, and $Z_{\tau (nt,n)}, t \in (0,1)$, given $T=n$. It is shown that the form of the limit distributions essentially depends on the location of $\tau (0,n)$ with respect to the point $nt$.

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