Limit Theorems for Large Deviations

Inhaltsverzeichnis

• 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 randomvectors 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.