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AuthorWentzell, A. D. author
TitleLimit Theorems on Large Deviations for Markov Stochastic Processes [electronic resource] / by A. D. Wentzell
ImprintDordrecht : Springer Netherlands, 1990
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Descript XVI, 176 p. online resource


0.1 Problems on large deviations for stochastic processes -- 0.2 Two opposite types of behaviour of probabilities of large deviations -- 0.3 Rough theorems on large deviations; the action functional -- 0.4 Survey of work on large deviations for stochastic processes -- 0.5 The scheme for obtaining rough theorems on large deviations -- 0.6 The expression: k (?) S (?) is the action functional uniformly over a specified class of initial points -- 0.7 Chapters 3 โ{128}{147} 6: a survey -- 0.8 Numbering -- 1. General Notions, Notation, Auxiliary Results -- 1.1. General notation. Legendre transformation -- 1.2. Compensators. Lรฉvy measures -- 1.3. Compensating operators of Markov processes -- 2. Estimates Associated with the Action Functional for Markov Processes -- 2.1. The action functional -- 2.2. Derivation of the lower estimate for the probability of passing through a tube -- 2.3. Derivation of the upper estimate for the probability of going far from the sets $$ {{\Phi }_{{{{x}_{0}};\left[ {0,T} \right]}}}\left( i \right),{{\bar{\Phi }}_{{{{x}_{0}};\left[ {0,T} \right]}}}\left( i \right) $$ -- 2.4. The truncated action functional and the estimates associated with it -- 3. The Action Functional for Families of Markov Processes -- 3.1. The properties of the functional $$ {{S}_{{{{T}_{1}},{{T}_{2}}}}}\left( \phi \right) $$ -- 3.2. Theorems on the action functional for families of Markov processes in Rr. The case of finite exponential moments -- 3.3. Transition to manifolds. Action functional theorems associated with truncated cumulants -- 4. Special Cases -- 4.1. Conditions A โ{128}{147} E of ยง 3.1. โ{128}{147} ยง 3.3 -- 4.2. Patterns of processes with frequent small jumps. The cases of very large deviations, not very large deviations, and super-large deviations -- 4.3. The case of very large deviations -- 4.4. The case of not very large deviations -- 4.5. Some other patterns of not very large deviations -- 4.6. The case of super-large deviations -- 5. Precise Asymptotics for Large Deviations -- 5.1. The case of the Wiener process -- 5.2. Processes with frequent small jumps -- 6. Asymptotics of the Probability of Large Deviations Due to Large Jumps of a Markov Process -- 6.1. Conditions imposed on the family of processes. Auxiliary results -- 6.2. Main theorems -- 6.3. Applications to sums of independent random variables -- References

Mathematics Probabilities Statistical physics Dynamical systems Mathematics Probability Theory and Stochastic Processes Statistical Physics Dynamical Systems and Complexity


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