zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Large deviations techniques and applications. (English) Zbl 0793.60030
Boston, MA: Jones and Bartlett Publishers. xiii, 346 p. $ 44.50/hbk (1993).

Various aspects of the theory and applications of large deviations principle (LDP) are considered. The book consists of seven chapters and Appendix.

Chapter 1 is devoted to the introduction and formulation of LDP. The finite-dimensional case is studied in Chapter 2. The main example is the empirical mean of a sequence of random variables taking values in d . Sanov’s and Cramer’s theorems are proved. The applications of the theory developed in Chapter 2 are presented in Chapter 3. The LDPs associated with the finite state irreducible Markov chains are derived and the asymptotic size of long rare segments in random walks is found. The asymptotics of the probability of error in hypothesis testing problems are analysed.

The LDPs for families of measures on general spaces are studied in Chapter 4. Relations between the topological structure of the space, the existence and uniqueness of LDP are explored. The transfer of LDP from one space to another is investigated. In Chapter 5 the probability that a path of a random process hits a particular set is found. The cases of random walk, Brownian motion and diffusion, the Frejdlin-Venttsel’ theory, are considered. The main results of Chapter 2 are generalized in Chapter 6 moving away from finite-dimensional case. The LDP for stationary processes satisfying a certain mixing condition is established as well. In Chapter 7 the applications considered in Chapters 2 and 3 are extended to the case of Polish space. Sanov’s theorem and projective limit approach is treated.

The appendix presents basic accompanying notions and facts making the book more self-contained. Each chapter contains historical notes of references.

60F10Large deviations
60-02Research monographs (probability theory)