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Limit Theorems for Large Deviations / by L. Saulis, V.A. Statulevičius.

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"Et moi ... - si j'avait su comment en revenir. One service mathematics has rendered the je n'y serais poin t aile.' human race. It has put common sense back Jules Verne where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded non- The series is...

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Bibliographic Details
Online Access: Full Text (via Springer)
Main Author: Saulis, L.
Other Authors: Statulevičius, V. A.
Format: eBook
Language:English
Published: Dordrecht : Springer Netherlands, 1991.
Series:Mathematics and its applications (Kluwer Academic Publishers). Soviet series ; 73.
Subjects:
Mathematics.
Distribution (Probability theory)
Statistics.
Table of Contents:
  • 1. The main notions
  • 2. The main lemmas
  • 2.1. General lemmas on the approximation of distribution of an arbitrary random variable by the normal distribution
  • 2.2. Proof of lemmas 2.1
  • 2.4
  • 3. Theorems on large deviations for the distributions of sums of independent random variables
  • 3.1. Theorems on large deviations under Bernstein's condition
  • 3.2. A theorem of large deviations in terms of Lyapunov's fractions
  • 4. Theorems of large deviations for sums of dependent random variables
  • 4.1. Estimates of the kth order centered moments of random processes with mixing
  • 4.2. Estimates of mixed cumulants of random processes with mixing
  • 4.3. Estimates of cumulants of sums of dependent random variables
  • 4.4. Theorems and inequalities of large deviations for sums of dependent random variables
  • 5. Theorems of large deviations for polynomial forms, multiple stochastic integrals and statistical estimates
  • 5.1. Estimates of cumulants and theorems of large deviations for polynomial forms, polynomial Pitman estimates and U-statistics
  • 5.2. Cumulants of multiple stochastic integrals and theorems of large deviations
  • 5.3. Large deviations for estimates of the spectrum of a stationary sequence
  • 6. Asymptotic expansions in the zones of large deviations
  • 6.1. Asymptotic expansion for distribution density of an arbitrary random variable
  • 6.2. Estimates for characteristic functions
  • 6.3. Asymptotic expansion in the Cramer zone for distribution density of sums of independent random variables
  • 6.4. Asymptotic expansions in integral theorems with large deviations
  • 7. Probabilities of large deviations for random vectors
  • 7.1. General lemmas on large deviations for a random vector with regular behaviour of cumulants
  • 7.2. Theorems on large deviations for sums of random vectors and quadratic forms
  • Appendices
  • Appendix 1. Proof of inequalities for moments and Lyapunov's fractions
  • Appendix 2. Proof of the lemma on the representation of cumulants
  • Appendix 3. Leonov
  • Shiryaev's formula
  • References.

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