LARGE DEVIATION ASYMPTOTICS FOR RANDOM-WALK TYPE PERTURBATIONS

2007 ◽  
Vol 07 (01) ◽  
pp. 75-89
Author(s):  
ZHIHUI YANG
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
White Noise ◽  
Wiener Process ◽  
Stochastic Resonance ◽  
Large Deviation ◽  
Correction Term ◽  
Exit Problem ◽  
Wiener Processes ◽  
Noise Type

Symmetric random walks can be arranged to converge to a Wiener process in the area of normal deviation. However, random walks and Wiener processes have, in general, different asymptotics of the large deviation probabilities. The action functionals for random-walks and Wiener processes are compared in this paper. The correction term is calculated. Exit problem and stochastic resonance for random-walk-type perturbation are also considered and compared with the white-noise-type perturbation.

Embedding submartingales in wiener processes with drift, with applications to sequential analysis

1969 ◽  
Vol 6 (03) ◽  
pp. 612-632 ◽  
Author(s):  
W. J. Hall
Keyword(s):  
Lower Bound ◽  
Random Walks ◽  
Sample Size ◽  
Wiener Process ◽  
Likelihood Ratio ◽  
Sequential Analysis ◽  
Stopping Times ◽  
Wiener Processes ◽  
Size Function ◽  
Log Likelihood

Summary Skorokhod (1961) demonstrated how the study of martingale sequences (and zero-mean random walks) can be reduced to the study of the Wiener process (without drift) at a sequence of random stopping times. We show how the study of certain submartingale sequences, including certain random walks with drift and log likelihood ratio sequences, can be reduced to the study of the Wiener process with drift at a sequence of stopping times (Theorem 4.1). Applications to absorption problems are given. Specifically, we present new derivations of a number of the basic approximations and inequalities of classical sequential analysis, and some variations on them — including an improvement on Wald's lower bound for the expected sample size function (Corollary 7.5).

scholarly journals DYNAMIC QUANTUM BERNOULLI RANDOM WALKS

2008 ◽  
Vol 11 (02) ◽  
pp. 213-229
Author(s):  
NADINE GUILLOTIN-PLANTARD ◽  
RENÉ SCHOTT
Keyword(s):  
Random Walk ◽  
Limit Theorem ◽  
Random Walks ◽  
Limit Theorems ◽  
Law Of Large Numbers ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Local Limit Theorem ◽  
Local Limit ◽  
Large Numbers

Quantum Bernoulli random walks can be realized as random walks on the dual of SU(2). We use this realization in order to study a model of dynamic quantum Bernoulli random walk with time-dependent transitions. For the corresponding dynamic random walk on the dual of SU(2), we prove several limit theorems (local limit theorem, central limit theorem, law of large numbers, large deviation principle). In addition, we characterize a large class of transient dynamic random walks.

scholarly journals Gumbel and Fréchet convergence of the maxima of independent random walks

2020 ◽  
Vol 52 (1) ◽  
pp. 213-236 ◽  
Author(s):  
Thomas Mikosch ◽  
Jorge Yslas
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Point Process ◽  
Asymptotic Theory ◽  
Large Deviation ◽  
Moment Generating Function ◽  
Independent Random Variables ◽  
Finite Moment ◽  
Process Convergence ◽  
Precise Large Deviation

AbstractWe consider point process convergence for sequences of independent and identically distributed random walks. The objective is to derive asymptotic theory for the largest extremes of these random walks. We show convergence of the maximum random walk to the Gumbel or the Fréchet distributions. The proofs depend heavily on precise large deviation results for sums of independent random variables with a finite moment generating function or with a subexponential distribution.

A large deviation local limit theorem

1989 ◽  
Vol 105 (3) ◽  
pp. 575-577 ◽  
Author(s):  
R. A. Doney
Keyword(s):  
Random Walk ◽  
Limit Theorem ◽  
Random Walks ◽  
Large Deviation ◽  
Local Limit Theorem ◽  
Local Limit ◽  
Large Deviation Result

The following elegant one-sided large deviation result is given by S. V. Nagaev in [2].Theorem 0. Suppose that {Sn,n ≤ 0} is a random walk whose increments Xi are independent copies of X, where(X) = 0 andPr{X > x} ̃ x−αL(x) as x→ + ∞,and where 1 < α < ∞ and L is slowly varying at ∞. Then for any ε > 0 and uniformly in x ≥ εnPr{Sn > x} ̃ n Pr{X > x} as n→∞.It is the purpose of this note to point out that for lattice-valued random walks there is an analogous local limit theorem.

Embedding submartingales in wiener processes with drift, with applications to sequential analysis

1969 ◽  
Vol 6 (3) ◽  
pp. 612-632 ◽  
Author(s):  
W. J. Hall
Keyword(s):  
Lower Bound ◽  
Random Walks ◽  
Sample Size ◽  
Wiener Process ◽  
Likelihood Ratio ◽  
Sequential Analysis ◽  
Stopping Times ◽  
Wiener Processes ◽  
Size Function ◽  
Log Likelihood

SummarySkorokhod (1961) demonstrated how the study of martingale sequences (and zero-mean random walks) can be reduced to the study of the Wiener process (without drift) at a sequence of random stopping times. We show how the study of certain submartingale sequences, including certain random walks with drift and log likelihood ratio sequences, can be reduced to the study of the Wiener process with drift at a sequence of stopping times (Theorem 4.1). Applications to absorption problems are given. Specifically, we present new derivations of a number of the basic approximations and inequalities of classical sequential analysis, and some variations on them — including an improvement on Wald's lower bound for the expected sample size function (Corollary 7.5).

scholarly journals Large Deviations for Continuous Time Random Walks

Entropy ◽  
2020 ◽  
Vol 22 (6) ◽  
pp. 697 ◽  
Author(s):  
Wanli Wang ◽  
Eli Barkai ◽  
Stanislav Burov
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Exponential Decay ◽  
Continuous Time ◽  
Large Deviation ◽  
Waiting Times ◽  
Continuous Time Random Walk ◽  
Random Walk Model ◽  
Logarithmic Correction ◽  
Continuous Time Random Walks

Recently observation of random walks in complex environments like the cell and other glassy systems revealed that the spreading of particles, at its tails, follows a spatial exponential decay instead of the canonical Gaussian. We use the widely applicable continuous time random walk model and obtain the large deviation description of the propagator. Under mild conditions that the microscopic jump lengths distribution is decaying exponentially or faster i.e., Lévy like power law distributed jump lengths are excluded, and that the distribution of the waiting times is analytical for short waiting times, the spreading of particles follows an exponential decay at large distances, with a logarithmic correction. Here we show how anti-bunching of jump events reduces the effect, while bunching and intermittency enhances it. We employ exact solutions of the continuous time random walk model to test the large deviation theory.

scholarly journals Large Deviations for Radial Random Walks on Homogeneous Trees

2007 ◽  
Vol 187 ◽  
pp. 75-90
Author(s):  
Kanji Ichihara
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Large Deviations ◽  
Random Walks ◽  
Large Deviation ◽  
Rate Function ◽  
Homogeneous Tree ◽  
Homogeneous Trees ◽  
Radial Random Walks

AbstractDonsker-Varadhan’s type large deviation will be discussed for the pinned motion of a radial random walk on a homogeneous tree. We shall prove that the rate function corresponding to the large deviation is associated with a new Markov chain constructed from the above random walk through a harmonic transform based on a positive principal eigenfunction for the generator of the random walk.

scholarly journals Current Trends in Random Walks on Random Lattices

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1148
Author(s):  
Jewgeni H. Dshalalow ◽  
Ryan T. White
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Real Time ◽  
Real Space ◽  
Historical Background ◽  
Escape Time ◽  
The Real ◽  
Current Trends ◽  
One Step ◽  
Time Position

In a classical random walk model, a walker moves through a deterministic d-dimensional integer lattice in one step at a time, without drifting in any direction. In a more advanced setting, a walker randomly moves over a randomly configured (non equidistant) lattice jumping a random number of steps. In some further variants, there is a limited access walker’s moves. That is, the walker’s movements are not available in real time. Instead, the observations are limited to some random epochs resulting in a delayed information about the real-time position of the walker, its escape time, and location outside a bounded subset of the real space. In this case we target the virtual first passage (or escape) time. Thus, unlike standard random walk problems, rather than crossing the boundary, we deal with the walker’s escape location arbitrarily distant from the boundary. In this paper, we give a short historical background on random walk, discuss various directions in the development of random walk theory, and survey most of our results obtained in the last 25–30 years, including the very recent ones dated 2020–21. Among different applications of such random walks, we discuss stock markets, stochastic networks, games, and queueing.

scholarly journals Strong Local Survival of Branching Random Walks is Not Monotone

2014 ◽  
Vol 46 (02) ◽  
pp. 400-421 ◽  
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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