Limit Theorems of Probability Theory

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Yu.V. Prokhorov, V. Statulevicius
Springer Science & Business Media, Mar 14, 2013 - Mathematics - 273 pages
This book consists of five parts written by different authors devoted to various problems dealing with probability limit theorems. The first part, "Classical-Type Limit Theorems for Sums ofIndependent Random Variables" (V.v. Petrov), presents a number of classical limit theorems for sums of independent random variables as well as newer related results. The presentation dwells on three basic topics: the central limit theorem, laws of large numbers and the law of the iterated logarithm for sequences of real-valued random variables. The second part, "The Accuracy of Gaussian Approximation in Banach Spaces" (V. Bentkus, F. G6tze, V. Paulauskas and A. Rackauskas), reviews various results and methods used to estimate the convergence rate in the central limit theorem and to construct asymptotic expansions in infinite-dimensional spaces. The authors con fine themselves to independent and identically distributed random variables. They do not strive to be exhaustive or to obtain the most general results; their aim is merely to point out the differences from the finite-dimensional case and to explain certain new phenomena related to the more complex structure of Banach spaces. Also reflected here is the growing tendency in recent years to apply results obtained for Banach spaces to asymptotic problems of statistics.
 

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Contents

of Independent Random Variables
2
Laws of Large Numbers
10
The Law of the Iterated Logarithm
16
The Accuracy of Gaussian Approximation in Banach Spaces
25
Approximation of Distributions of Sums
113
Refinements of the Central Limit Theorem
166
Limit Theorems on Large Deviations
185
Name Index
267
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