"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 divergent; therefore we may be sense'. able to do something with it. Eric T. Bell O. H ea viside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non Iinearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service. topology has rendered mathematical physics . . .':: 'One service logic has rendered com puter science . . .'; 'One service category theory has rendered mathematics . . .'. All arguably true. And all statements obtainable this way form part of the raison d 'e1:re of this series.
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.
