Joint path properties of two of my most favourite friends in mathematics: a tribute to Endre Csáki and Pál Révész. (English) Zbl 1094.01008

Csörgő, Miklós

Period. Math. Hung. 50, No. 1-2, 1-27 (2005).

The paper is written in honour of the 70th birthday of Endre Csáki and Pál Révész. It is an overview of their most interesting results emphasizing their joint work. The historical background and the influence of the results are described. The paper consists of seven sections.
1. Brownian motion, random walks, local times and additive functionals. Here the first question is: how big are the increments of the standard Wiener process? The author lists the famous asymptotic theorems by Csáki and Révész on the increments. Then their results on the largest excursion of the random walk and on the location of the maximum of the standard Wiener process are mentioned. Then the importance of the strong invariance principle regarding the local time of the random walk and the local time of the standard Wiener process is explained. The approximation of the Brownian local time by a Wiener sheet is described. The strong approximation concerning additive functionals of a Wiener process is mentioned.
2. Iterated processes, and their local and occupation times. First the law of the iterated logarithm for the iterated Brownian motion is given. Then the local time is studied. Limit theorems are presented for the occupation time of iterated processes having no local time. The titles of the remaining short sections are the following:
3. Almost sure local and global central limit theorems. 4. Integral functions of geometric stochastic processes. 5. Favourite sites, favourite values and jump sizes for random walk and Brownian motion. 6. Random walking in random scenery. 7. Large void zones and occupation times for coalescing random walks.

Reviewer: István Fazekas (Debrecen)

MSC:

01A70	Biographies, obituaries, personalia, bibliographies
60-03	History of probability theory
60F15	Strong limit theorems
60F17	Functional limit theorems; invariance principles
60G15	Gaussian processes
60G50	Sums of independent random variables; random walks
60J55	Local time and additive functionals

Keywords:

Brownian motion; random walks; path properties, excursions; local times and additive functionals; almost sure limit theorems; integral functionals of geometric processes; random walks in random scenery; coalescing random walks