Professor Ryszard Zieliński's contribution to Monte Carlo methods and random number generators. Uniform asymptotics in statistics

Abstract

The aim of the paper is to summarize contributions of Ryszard Zielinski to two important areas of research. First, we discuss his work related to Monte Carlo methods. Ryszard Zielinski was particularly interested in Monte Carlo optimization. About 10 of his papers concerned stochastic algorithms for seeking extrema. He examined methods related to stochastic approximation, random search and global optimization. We stress that Zielinski often considered computational problems from a statistical perspective. In several articles he explicitly indicated that optimization can be reformulated as a statistical estimation problem. We also discuss relation between the family of Simulated Annealing algorithms on the one hand and some procedures examined earlier by Ryszard Zielinski on the other. Another topic belonging to Monte Carlo methods, in which Ryszard Zielinski has achieved interesting results, is construction of random number generators and examination of their statistical properties. Zielinski proposed an aperiodic generator based on Weil sequences and showed how it can be efficiently implemented. Later he constructed an algorithm which uses several such generators and produces pseudo-random sequences with better statistical properties. The second area of Zielinski’s work discussed here is related to uniform limit theorems of mathematical statistics. We stress the methodological motivation behind the research in this direction. In Zielinski’s view, asymptotic results should hold uniformly with respect to the family of probability distributions under consideration. In his opinion, this requirement comes from the very nature of statistical models and the needs of practical applications. Zielinski examined uniform versions the Weak Law of Large Numbers, Strong Law of Large Numbers and Central Limit Theorem in several statistical models. Some results were rather unexpected. He also gave a necessary and sufficient condition for uniform consistency of sample quantiles. Two papers of Ryszard Zielinski were devoted to uniform consistency of smoothed versions of empirical cumulative distribution function. In one of them he proved a version of Dvoretzky-Kiefer-Wolfowitz inequality.