## Limit Theorems of Probability TheoryYu.V. Prokhorov, V. Statulevicius 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

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 |

267 | |

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### Common terms and phrases

Akad Appl asymptotic expansions Banach spaces bounds Bulinskii central limit theorem coefficients condition convergence rate cumulants defined denotes distribution function distributions of sums Dokl English transl estimate exists finite finite-dimensional functions f Gaussian Götze Hilbert space Hölder's inequality Ibragimov identically distributed independent random variables inequality integral invariance principle iterated logarithm large deviation probabilities large deviations large numbers law of large Lemma Liet Lith Markov chain Math method metric mixing sequences Nagaev Nauk Nauka nonuniform obtain Paulauskas Petrov polynomial Primen Prokhorov proved random fields rate of convergence remainder term Rink Rudzkis Russian satisfies Saulis Sazonov sequence of independent spectral SSSR stationary Statist Statulevičius sums of independent sums of weakly Sunklodas Suppose Teor theorem for m-dependent theorems for sums Theory Probab Tikhomirov U-statistics Veroyatn Vilnius Wahrscheinlichkeitstheorie Verw weakly dependent random Zalesskii Zitikis