The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

Stable non-Gaussian random processes , by G. Samorodnitsky and M. S. Taqqu. Pp. 632. £49.50. 1994. ISBN 0-412-05171-0 (Chapman and Hall).

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Principles Of Random Walk

This article deals with the asymptotic behavior as $t \rightarrow +\infty $ of the survival function $\mathbb{P}[T > t]$, where T is the first passage time above a non negative level of a random process starting from zero. In many cases of physical significance, the behavior is of the type $\mathbb{P}[T > t] = t^{-\theta +o(1)}$ for a known or unknown positive parameter $\theta$ which is called the persistence exponent. The problem is well understood for random walks or Levy processes but becomes more difficult for integrals of such processes, which are more related to physics. We survey recent results and open problems in this field.

/pdf/persistence-probabilities-and-exponents-3bf0bsge4p.pdf

Persistence Probabilities and Exponents

On graded irreducible representations of Leavitt path algebras

The speed of a branching system of random walks in random environment

We consider a transient diffusion in a $(-\kappa/2)$-drifted Brownian potential $W_{\kappa}$ with $0<\kappa<1$.
 We prove its localization at time $t$ in the neighborhood of some random points depending only on the environment, which are
 the positive $h_t$-minima of the environment, for $h_t$ a bit smaller than $\log t$.
 We also prove an Aging phenomenon for the diffusion, a renewal theorem for the hitting time of the farthest visited valley, and provide a central limit theorem for the number of valleys visited up to time $t$.
The proof relies on a decomposition of the trajectory of $W_{\kappa}$ in the neighborhood of $h_t$-minima, with the help of results of A. Faggionato, and on a precise analysis of exponential functionals of $W_{\kappa}$ and of $W_{\kappa}$ Doob-conditioned to stay positive.

/pdf/localization-and-number-of-visited-valleys-for-a-transient-2wpp5jcmra.pdf

Localization and number of visited valleys for a transient diffusion in random environment

Nous nous interessons a la marche de Sinai $(S_{n})_{n\in\mathbb{N}}$. Nous prouvons que la probabilite annealed que $\sum_{k=0}^{n}f(S_{k})$ soit strictement positive pour tout $n\in[1,N]$ est egale a $1/(\log N)^{(3-\sqrt{5})/2+o(1)}$, pour une large classe de fonctions $f$, et en particulier pour $f(x)=x$. L’exposant de persistance $\frac{3-\sqrt{5}}{2}$ est d’abord apparu dans un article non rigoureux de Le Doussal, Monthus et Fischer, avec des motivations venant de la physique. La preuve est basee sur des techniques de localisation pour la marche de Sinai et utilise des resultats de Cheliotis sur les changements de signe des fonds de vallees d’un mouvement Brownien indexe par $\mathbb{R}$.

/pdf/persistence-of-some-additive-functionals-of-sinai-s-walk-3ud43ool01.pdf

Persistence of some additive functionals of Sinai’s walk

We study a one-dimensional diffusion process in a drifted Brownian potential. We characterize the upper functions of its hitting times in the sense of Paul Levy, and determine the lower limits in terms of an iterated logarithm law.

https://hal.archives-ouvertes.fr/hal-00013040/document

Almost sure asymptotics for a diffusion process in a drifted Brownian potential

We consider a one-dimensional diffusion process X in a (−κ∕2)-drifted Brownian potential for κ ≠ 0. We are interested in the maximum of its local time, and study its almost sure asymptotic behaviour, which is proved to be different from the behaviour of the maximum local time of the transient random walk in random environment. We also obtain the convergence in law of the maximum local time of X under the annealed law after suitable renormalization when κ ≥ 1. Moreover, we characterize all the upper and lower classes for the hitting times of X, in the sense of Paul Levy, and provide laws of the iterated logarithm for the diffusion X itself. To this aim, we use annealed technics.

Alexis Devulder

Papers

The speed of a branching system of random walks in random environment

Localization and number of visited valleys for a transient diffusion in random environment

Persistence of some additive functionals of Sinai’s walk

Almost sure asymptotics for a diffusion process in a drifted Brownian potential

The Maximum of the Local Time of a Diffusion Process in a Drifted Brownian Potential