# Prokhorov and Contemporary Probability Theory: In Honor of Yuri V. Prokhorov (Springer Proceedings in Mathematics & Statistics)

Language: English

Pages: 446

ISBN: 3642431682

Format: PDF / Kindle (mobi) / ePub

The role of Yuri Vasilyevich Prokhorov as a prominent mathematician and leading expert in the theory of probability is well known. Even early in his career he obtained substantial results on the validity of the strong law of large numbers and on the estimates (bounds) of the rates of convergence, some of which are the best possible. His findings on limit theorems in metric spaces and particularly functional limit theorems are of exceptional importance. Y.V. Prokhorov developed an original approach to the proof of functional limit theorems, based on the weak convergence of finite dimensional distributions and the condition of tightness of probability measures.

The present volume commemorates the 80^{th }birthday of Yuri Vasilyevich Prokhorov. It includes scientific contributions written by his colleagues, friends and pupils, who would like to express their deep respect and sincerest admiration for him and his scientific work.

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Statistics, 43. Moscow: “Nauka”, 1991, 445 pp. IV Works as Editor 1. D.A. Rodionov. Statistical methods of geological objects identification by set of characteristics. (Russian) Ed. by Yu.V. Prokhorov, -M. : “Nedra”, 1968, 158 pp. 2. Proc. of the Second Japan-USSR Symposium on Probability Theory. Ed. by G. Maruyama and Yu.V. Prokhorov. Lect. Notes Math., 330. Berlin etc.: Springer-Verlag, 1973, 550 pp. 3. Proceedings of the Third Japan-USSR Symposium on Probability Theory Held in Tashkent, Aug.

Probability Theory and Mathematical Statistics. Selected Works. (Russian) Ed. by Yu.V. Prokhorov, Compiled by A.N. Shiryaev. With commentaries. Moscow: “Nauka”, 1986, 535 pp. 10. First World Congress of the Bernoulli Society for Mathematical Statistics and Probability Theory. Vol. I. Sections 1–19. Abstracts from the Congress held in Tashkent, 1986. (Russian) Ed. by Yu.V. Prokhorov. Moscow: “Nauka”, 1986. 488 pp. 11. First World Congress of the Bernoulli Society for Mathematical Statistics and

allowed strategy for an agent i is to place calls, at the times of a Poisson (rate Â) process, to a random agent. 2.1 Finite Number of Rewards Before deriving the result (5) in our general framework, let us step outside that framework to derive a very easy variant result. Suppose that only the first two recipients of an item of information receive a reward, of amount wn say. Agent 10 D.J. Aldous strategy cannot affect the first recipient, only the second. Suppose ego uses rate and other

."/ D 2 n where n2 D E.j ni Gj2 / (see [8]). Under the Assumptions 1 and 2 the denominators . nÞ /2 and n2 have the same order, but the nominator of Þ n ."/ is much smaller than the nominator of n ."/. This has an important Limit Theorems for Functionals of Higher Order Differences 79 consequence: the central limit theorems for the multipower variation of the increments of X hold only for ˇ 2 . 12 ; 0/ while the corresponding results for the second order differences hold for all ˇ 2 . 12 ;

i.e. with f verifying f 00 .x/ D ax , a > 0. Their properties are given in Theorem 6. Levy Preservation and Properties for Minimal Equivalent Martingale Measures 167 2 Some Facts About Exponential Levy Models Let us describe our model in more details. We assume the financial market consists of a bank account B whose value at time t is Bt D B0 e rt ; where r 0 is the interest rate which we assume to be constant. We also assume that there are d 1 risky assets whose prices are described by a d