Limit theorems and some applications in statistical physics

Inhaltsverzeichnis1. Preliminary Probabilistic Results.1.1 Convergence of Probability Measures.1.2 Central Limit Theorem for Sums of Random Vectors.1.3 Invariance Principle.1.4 Stationary Random Processes and Fields on the Lattice.1.5 Slowly Varying Functions.1.6 Some Probabilistic Inequalities.1.7 Uniformly Integrable Sequences.2. Weak Dependence Conditions for Random Processes and Fields.2.1 Measures of Dependence.2.2 Weak Dependence Conditions for Random Processes. Classical Examples.2.3 Davydov’s Examples.2.4 Herrndorf’s Examples.2.5 Gaussian Random Sequences.2.6 Weak Dependence Conditions for Random Fields. Dobrushin’s Example.2.7 Additions.3. Asymptotic Behavior of the Variance, Estimates on the Moments and Some Probabilistic Inequalities for Sums of Weakly Dependent Random Variables.3.1 The Variance of the Sum of Random Variables.3.2 Estimates on Moments of Sums of Weakly Dependent Random Variables.3.3 Some Probabilistic Inequalities.4. Methods.4.1 Bernstein’s Method.4.2 Gordin’s Method.4.3 Stein’s Method.4.4 Moments’ (Semi-Invariants’) Method.5. Limit Theorems for Random Processes.5.1 Central Limit Theorem for ?-mixing Stationary Random Processes.5.2 Rate of Convergence in Central Limit Theorem for ?-mixing Stationary Random Processes.5.3 Invariance Principle for Stationary Random Processes with ?-mixing Condition.5.4 Law of Iterated Logarithm for Stationary Random Processes Satisfying the ?-mixing Condition.5.5 Limit Theorems for ?-mixing Stationary Random Processes.5.6 Limit Theorems for Stationary Random Processes Satisfying the ?-mixing Condition.5.7 Limit Theorems for ?-mixing Stationary Random Processes.6. Limit Theorems under Generalized Mixing Conditions.6.1 Distances in Space of Probability Measures, Kantorovioh-Vasershtein Metric.6.2 Generalized Mixing Conditions for Random Processes.6.3 Central Limit Theorem.6.4 Limit Theorems of Non-Commutative Theory of Probability.7. Limit Theorems for Random Fields.7.1 Sequences of Sets Tending to Infinity.7.2 Central Limit Theorem for Random Fields.7.3 Rate of Convergence in Central Limit Theorem for Random Fields.7.4 Law of Iterated Logarithm for Random Fields.8. Description of Random Fields by Means of Conditional Probability.8.1 Existence of Random Fields with Given Conditional Distribution.8.2 Uniqueness of Random Fields with Given Conditional Distribution.8.3 Decay of Correlation and Central Limit Theorem.9. Gibbs Random Fields.9.1 Existence and Uniqueness of Gibbs Random Fields, Weak Dependence of Components.9.2 Thermodynamical Limit, Existence of Free Energy.9.3 Strong Convexity of Free Energy, Linear Growth of Variance of Energy.9.4 Limit Theorems for Gibbs Random Fields.9.5 Cluster Properties and Mixing Conditions for Gibbs Random Fields with Vacuum Potential.9.6 Additions.Some Additional Remarks.References.