Pál Révész

Random walk in random and non-random environments

Simple Symmetric Random Walk in ℤ1: Introduction of Part I Distributions Recurrence and the Zero-One Law From the Strong Law of Large Numbers to the Law of Iterated Logarithm Levy Classes Wiener Process and Invariance Principle Increments Strassen Type Theorems Distribution of the Local Time Local Time and Invariance Principle Strong Theorems of the Local Time Excursions Frequently and Rarely Visited Sites An Embedding Theorem A Few Further Results Summary of Part I Simple Symmetric Random Walk in ℤd: The Recurrence Theorem Wiener Process and Invariance Principle The Law of Iterated Logarithm Local Time The Range Heavy Points and Heavy Balls Crossing and Self-crossing Large Covered Balls Long Excursions Speed of Escape A Few Further Problems Random Walk in Random Environment: Introduction of Part III In the First Six Days After the Sixth Day What Can a Physicist Say About the Local Time ξ(0,n)? On the Favourite Value of the RWIRE A Few Further Problems Random Walks in Graphs: Introduction of Part IV Random Walk in Comb Random Walk in a Comb and in a Brush with Crossings Random Walk on a Spider Random Walk in Half-Plane-Half-Comb.

A new method to prove strassen type laws of invariance principle. II

SummaryA new method is developed to produce strong laws of invariance principle without making use of the Skorohod representation. As an example, it will be proved that

The local time of iterated Brownian motion

{{\mathop {\lim }\limits_{n \to \infty } \left( {S_n - W(n)} \right)} \mathord{\left/ {\vphantom {{\mathop {\lim }\limits_{n \to \infty } \left( {S_n - W(n)} \right)} {n^{{1 \mathord{\left/ {\vphantom {1 {6 + \varepsilon }}} \right. \kern-\nulldelimiterspace} {6 + \varepsilon }}} }}} \right. \kern-\nulldelimiterspace} {n^{{1 \mathord{\left/ {\vphantom {1 {6 + \varepsilon }}} \right. \kern-\nulldelimiterspace} {6 + \varepsilon }}} }} = 0

Global Strassen-type theorems for iterated Brownian motions

with probability 1, for any g3>0, where Sn=X1 + ... +Xn, Xi is a sequence of i.i.d.r.v.s with P(Xi<t)=F(t), and F(t) is a distribution function obeying (i), (ii) and W(n) is a suitable Wiener-process. Strassen in [1], proved (under weaker conditions):

A generalization of Strassen's functional law of iterated logarithm

S_n - W\left( n \right) = O\left( {\sqrt[4]{{n{\text{ log log }}n}}\sqrt {{\text{log }}n} {\text{ }}} \right)

Simple random walk on the line in random environment

with probability one. He conjectured that if