Large deviations for longest runs in Markov chains

2020 ◽  
Vol 26 (2) ◽  
pp. 309-314
Author(s):  
Zhenxia Liu ◽  
Yurong Zhu
Keyword(s):  
Distribution Function ◽  
Markov Chain ◽  
Markov Chains ◽  
Present Note ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Large Deviation Principles ◽  
Success Run ◽  
Longest Success Run ◽  
Math Statist

AbstractWe continue our investigation on general large deviation principles (LDPs) for longest runs. Previously, a general LDP for the longest success run in a sequence of independent Bernoulli trails was derived in [Z. Liu and X. Yang, A general large deviation principle for longest runs, Statist. Probab. Lett. 110 2016, 128–132]. In the present note, we establish a general LDP for the longest success run in a two-state (success or failure) Markov chain which recovers the previous result in the aforementioned paper. The main new ingredient is to implement suitable estimates of the distribution function of the longest success run recently established in [Z. Liu and X. Yang, On the longest runs in Markov chains, Probab. Math. Statist. 38 2018, 2, 407–428].

LARGE DEVIATIONS FOR SMALL PERTURBATIONS OF SDES WITH NON-MARKOVIAN COEFFICIENTS AND THEIR APPLICATIONS

2006 ◽  
Vol 06 (04) ◽  
pp. 487-520 ◽  
Author(s):  
FUQING GAO ◽  
JICHENG LIU
Keyword(s):  
Large Deviations ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Wiener Space ◽  
Small Perturbations ◽  
Deviation Principle ◽  
Large Deviation Principles ◽  
Sobolev Norms

We prove large deviation principles for solutions of small perturbations of SDEs in Hölder norms and Sobolev norms, where the SDEs have non-Markovian coefficients. As an application, we obtain a large deviation principle for solutions of anticipating SDEs in terms of (r, p) capacities on the Wiener space.

Stochastic scrabble: large deviations for sequences with scores

1988 ◽  
Vol 25 (1) ◽  
pp. 106-119 ◽  
Author(s):  
Richard Arratia ◽  
Pricilla Morris ◽  
Michael S. Waterman
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Law Of Large Numbers ◽  
Large Deviation ◽  
Additive Functional ◽  
Finite Alphabet ◽  
Real Numbers ◽  
Similar Treatment ◽  
Large Numbers ◽  
Large Deviation Result

A derivation of a law of large numbers for the highest-scoring matching subsequence is given. Let Xk, Yk be i.i.d. q=(q(i))i∊S letters from a finite alphabet S and v=(v(i))i∊S be a sequence of non-negative real numbers assigned to the letters of S. Using a scoring system similar to that of the game Scrabble, the score of a word w=i1 · ·· im is defined to be V(w)=v(i1) + · ·· + v(im). Let Vn denote the value of the highest-scoring matching contiguous subsequence between X1X2 · ·· Xn and Y1Y2· ·· Yn. In this paper, we show that Vn/K log(n) → 1 a.s. where K ≡ K(q,v). The method employed here involves ‘stuttering’ the letters to construct a Markov chain and applying previous results for the length of the longest matching subsequence. An explicit form for β ∊Pr(S), where β (i) denotes the proportion of letter i found in the highest-scoring word, is given. A similar treatment for Markov chains is also included.Implicit in these results is a large-deviation result for the additive functional, H ≡ Σn < τv(Xn), for a Markov chain stopped at the hitting time τ of some state. We give this large deviation result explicitly, for Markov chains in discrete time and in continuous time.

scholarly journals A Markov Chain Analysis of Genetic Algorithms: Large Deviation Principle Approach

2010 ◽  
Vol 47 (04) ◽  
pp. 967-975
Author(s):  
Joe Suzuki
Keyword(s):  
Genetic Algorithms ◽  
Markov Chain ◽  
Large Deviation Principle ◽  
Selective Pressure ◽  
Large Deviation ◽  
Sufficient Condition ◽  
Deviation Principle ◽  
Chain Analysis ◽  
Fitness Value ◽  
Uniform Population

In this paper we prove that the stationary distribution of populations in genetic algorithms focuses on the uniform population with the highest fitness value as the selective pressure goes to ∞ and the mutation probability goes to 0. The obtained sufficient condition is based on the work of Albuquerque and Mazza (2000), who, following Cerf (1998), applied the large deviation principle approach (Freidlin-Wentzell theory) to the Markov chain of genetic algorithms. The sufficient condition is more general than that of Albuquerque and Mazza, and covers a set of parameters which were not found by Cerf.

Monte Carlo simulation and large deviations theory for uniformly recurrent Markov chains

1990 ◽  
Vol 27 (1) ◽  
pp. 44-59 ◽  
Author(s):  
James A Bucklew ◽  
Peter Ney ◽  
John S. Sadowsky
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chains ◽  
Large Deviations ◽  
Large Deviation ◽  
Homogeneous Markov Chain ◽  
Underlying Distribution ◽  
Monte Carlo Simulation Technique ◽  
Large Deviations Theory

Importance sampling is a Monte Carlo simulation technique in which the simulation distribution is different from the true underlying distribution. In order to obtain an unbiased Monte Carlo estimate of the desired parameter, simulated events are weighted to reflect their true relative frequency. In this paper, we consider the estimation via simulation of certain large deviations probabilities for time-homogeneous Markov chains. We first demonstrate that when the simulation distribution is also a homogeneous Markov chain, the estimator variance will vanish exponentially as the sample size n tends to∞. We then prove that the estimator variance is asymptotically minimized by the same exponentially twisted Markov chain which arises in large deviation theory, and furthermore, this optimization is unique among uniformly recurrent homogeneous Markov chain simulation distributions.

scholarly journals Large deviation principles of one-dimensional maps for Hölder continuous potentials

2014 ◽  
Vol 36 (1) ◽  
pp. 127-141 ◽  
Author(s):  
HUAIBIN LI
Keyword(s):  
Large Deviation Principle ◽  
Large Deviation ◽  
Weak Form ◽  
Periodic Points ◽  
Hölder Continuous ◽  
One Dimensional ◽  
Deviation Principle ◽  
Large Deviation Principles ◽  
One Dimensional Maps ◽  
Level 2

We show some level-2 large deviation principles for real and complex one-dimensional maps satisfying a weak form of hyperbolicity. More precisely, we prove a large deviation principle for the distribution of iterated preimages, periodic points, and Birkhoff averages.

scholarly journals Extended Laplace principle for empirical measures of a Markov chain

2019 ◽  
Vol 51 (01) ◽  
pp. 136-167 ◽  
Author(s):  
Stephan Eckstein
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Large Deviations ◽  
Large Deviation ◽  
Dual Pair ◽  
Wasserstein Distance ◽  
Large Deviations Principle ◽  
Empirical Measures ◽  
Leibler Divergence ◽  
Laplace Principle

AbstractWe consider discrete-time Markov chains with Polish state space. The large deviations principle for empirical measures of a Markov chain can equivalently be stated in Laplace principle form, which builds on the convex dual pair of relative entropy (or Kullback– Leibler divergence) and cumulant generating functional f ↦ ln ʃ exp (f). Following the approach by Lacker (2016) in the independent and identically distributed case, we generalize the Laplace principle to a greater class of convex dual pairs. We present in depth one application arising from this extension, which includes large deviation results and a weak law of large numbers for certain robust Markov chains—similar to Markov set chains—where we model robustness via the first Wasserstein distance. The setting and proof of the extended Laplace principle are based on the weak convergence approach to large deviations by Dupuis and Ellis (2011).

scholarly journals Regeneration and general Markov chains

1994 ◽  
Vol 7 (3) ◽  
pp. 357-371 ◽  
Author(s):  
Vladimir V. Kalashnikov
Keyword(s):  
Distribution Function ◽  
Markov Chain ◽  
Quantitative Analysis ◽  
Markov Chains ◽  
Rates Of Convergence ◽  
Regenerative Process ◽  
Wide Sense ◽  
Visit Time ◽  
Finite Approximations

Ergodicity, continuity, finite approximations and rare visits of general Markov chains are investigated. The obtained results permit further quantitative analysis of characteristics, such as, rates of convergence, continuity (measured as a distance between perturbed and non-perturbed characteristics), deviations between Markov chains, accuracy of approximations and bounds on the distribution function of the first visit time to a chosen subset, etc. The underlying techniques use the embedding of the general Markov chain into a wide sense regenerative process with the help of splitting construction.

On exact and large deviation approximation for the distribution of the longest run in a sequence of two-state Markov dependent trials

2003 ◽  
Vol 40 (02) ◽  
pp. 346-360 ◽  
Author(s):  
James C. Fu ◽  
Liqun Wang ◽  
W. Y. Wendy Lou
Keyword(s):  
Large Deviation ◽  
Finite Markov Chain ◽  
Finite Markov Chain Imbedding ◽  
Markov Chain Imbedding ◽  
Longest Run ◽  
Exact Distributions ◽  
Markov Dependent Trials ◽  
Success Run ◽  
Longest Success Run ◽  
Theoretical Results

Consider a sequence of outcomes from Markov dependent two-state (success-failure) trials. In this paper, the exact distributions are derived for three longest-run statistics: the longest failure run, longest success run, and the maximum of the two. The method of finite Markov chain imbedding is used to obtain these exact distributions, and their bounds and large deviation approximation are also studied. Numerical comparisons among the exact distributions, bounds, and approximations are provided to illustrate the theoretical results. With some modifications, we show that the results can be easily extended to Markov dependent multistate trials.

A SUPPORT THEOREM AND A LARGE DEVIATION PRINCIPLE FOR KUNITA FLOWS

2012 ◽  
Vol 12 (03) ◽  
pp. 1150022 ◽  
Author(s):  
STEFFEN DEREICH ◽  
GEORGI DIMITROFF
Keyword(s):  
Large Deviation Principle ◽  
Large Deviation ◽  
Second Step ◽  
Stochastic Flow ◽  
Stochastic Flows ◽  
Global Flow ◽  
Rough Paths ◽  
Support Theorem ◽  
Large Deviation Principles ◽  
Support Theorems

In this paper, stochastic flows driven by Kunita-type stochastic differential equations are studied, focusing on support theorems (ST) and large deviation principles (LDP). We establish a new ST and LDP for Brownian flows with respect to a fine Hölder topology. Our approach is based on recent advances in rough paths theory, which is the natural framework for proving ST and LDP. Nevertheless, while rigorous, our presentation stays rather clear from the rough paths technicalities and is accessible for readers not familiar with them. We view the localized Brownian stochastic flow as a projection of the solution of a rough path differential equation implying the ST and LDP. In a second step the results are generalized for the global flow.

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