Lyapunov Exponents of Random Walks in Small Random Potential: The Lower Bound

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).

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Monotonicity of Avoidance Coupling on KN

Combinatorics Probability Computing ◽

10.1017/s0963548316000195 ◽

2016 ◽

Vol 26 (1) ◽

pp. 16-23 ◽

Cited By ~ 1

Author(s):

OHAD N. FELDHEIM

Keyword(s):

Lower Bound ◽

Random Walks ◽

Complete Graph ◽

Partially Ordered ◽

Monotonicity Result

Answering a question by Angel, Holroyd, Martin, Wilson and Winkler [1], we show that the maximal number of non-colliding coupled simple random walks on the complete graph KN, which take turns, moving one at a time, is monotone in N. We use this fact to couple [N/4] such walks on KN, improving the previous Ω(N/log N) lower bound of Angel et al. We also introduce a new generalization of simple avoidance coupling which we call partially ordered simple avoidance coupling, and provide a monotonicity result for this extension as well.

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On the sum of the non-negative Lyapunov exponents for some cocycles related to the Anderson model

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2015.59 ◽

2015 ◽

Vol 37 (2) ◽

pp. 369-388

Author(s):

ILIA BINDER ◽

MICHAEL GOLDSTEIN ◽

MIRCEA VODA

Keyword(s):

Lyapunov Exponent ◽

Lower Bound ◽

Lyapunov Exponents ◽

Anderson Model ◽

Explicit Lower Bound

We provide an explicit lower bound for the the sum of the non-negative Lyapunov exponents for some cocycles related to the Anderson model. In particular, for the Anderson model on a strip of width $W$, the lower bound is proportional to $W^{-\unicode[STIX]{x1D716}}$, for any $\unicode[STIX]{x1D716}>0$. This bound is consistent with the fact that the lowest non-negative Lyapunov exponent is conjectured to have a lower bound proportional to $W^{-1}$.

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