The longest stretch L(n) of consecutive heads in n independent and identically distributed coin tosses is seen from the prism of large deviations. We first establish precise asymptotics for the moment generating function of L(n) and then show that there are precisely two large deviation principles, one concerning the behavior of the distribution of L(n) near its nominal value log(1/p) n and one away from it. We discuss applications to inference and to logarithmic asymptotics of functionals of L(n).

Let (X-1,. . ., X-n) be any n-dimensional centered Gaussian random vector, in this note the following expectation product inequality is proved: E Pi (n)(j=1) f(j)( X-j) amp;gt;= Pi (n)(j=1) Ef(j)(X-j) for functionsh, 1 amp;lt;= j amp;lt;= n, taking the forms f(j)(x) = integral(infinity)(0) where mu(j), 1 amp;lt;= j amp;lt;= n, are finite positive measures. The motivation of studying such an inequality comes from the Gaussian correlation conjecture (which was recently proved) and the Gaussian moment product conjecture (which is still unsolved). Several explicit examples of such functions f(j) are given. The proof is built on characteristic functions of Gaussian random variables. (C) 2017 Elsevier B.V. All rights reserved.

In this note we prove a general large deviation principle (LDP) for the longest success run in a sequence of independent Bernoulli trails. This study not only recovers several recently derived LDPs, but also gives new LDPs for the longest success run. The method is based on the Bryc’s inverse Varadhan lemma, which can be intuitively generalized to the longest success run in a two-state (success and failure) Markov chain.

Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.

Yang, Xiangfeng

Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.

ON THE LONGEST RUNS IN MARKOV CHAINS2018In: Probability and Mathematical Statistics, ISSN 0208-4147, Vol. 38, no 2, p. 407-428Article in journal (Refereed)

Abstract [en]

In the first n steps of a two-state (success and failure) Markov chain, the longest success run L(n) has been attracting considerable attention due to its various applications. In this paper, we study L(n) in terms of its two closely connected properties: moment generating function and large deviations. This study generalizes several existing results in the literature, and also finds an application in statistical inference. Our method on the moment generating function is based on a global estimate of the cumulative distribution function of L(n) proposed in this paper, and the proofs of the large deviations include the Gartner-Ellis theorem and the moment generating function.