Approximation Methods in Probability Theory

Definitions and preliminary facts
The method of convolutions
Local lattice estimates
Uniform lattice estimates
Total variation of lattice measures
Non-uniform estimates for lattice measures
Discrete non-lattice approximations
Absolutely continuous approximations
The Esseen type estimates
Lower estimates
The Stein method
The triangle function method
Heinrich's method for m-dependent variables
Other methods
Solutions to selected problems
Bibliography
Index.
Machine generated contents note: 1.Definitions and Preliminary Facts
1.1.Distributions and Measures
1.2.Moment Inequalities
1.3.Norms and Their Properties
1.4.Fourier Transforms
1.5.Concentration Function
1.6.Algebraic Identities and Inequalities
1.7.The Schemes of Sequences and Triangular Arrays
1.8.Problems
2.The Method of Convolutions
2.1.Expansion in Factorial Moments
2.2.Expansion in the Exponent
2.3.Le Cam's Trick
2.4.Smoothing Estimates for the Total Variation Norm
2.5.Estimates in Total Variation via Smoothing
2.6.Smoothing Estimates for the Kolmogorov Norm
2.7.Estimates in the Kolmogorov Norm via Smoothing
2.8.Kerstan's Method
2.9.Problems
3.Local Lattice Estimates
3.1.The Inversion Formula
3.2.The Local Poisson Binomial Theorem
3.3.Applying Moment Expansions
3.4.A Local Franken-Type Estimate
3.5.Involving the Concentration Function
3.6.Switching to Other Metrics
Contents note continued: 3.7.Local Smoothing Estimates
3.8.The Method of Convolutions for a Local Metric
3.9.Problems
4.Uniform Lattice Estimates
4.1.The Tsaregradskii Inequality
4.2.The Second Order Poisson Approximation
4.3.Taking into Account Symmetry
4.4.Problems
5.Total Variation of Lattice Measures
5.1.Inversion Inequalities
5.2.Examples of Applications
5.3.Smoothing Estimates for Symmetric Distributions
5.4.The Barbour-Xia Inequality
5.5.Application to the Wasserstein Norm
5.6.Problems
6.Non-uniform Estimates for Lattice Measures
6.1.Non-uniform Local Estimates
6.2.Non-uniform Estimates for Distribution Functions
6.3.Applying Taylor Series
6.4.Problems
7.Discrete Non-lattice Approximations
7.1.Arak's Lemma
7.2.Application to Symmetric Distributions
7.3.Problems
8.Absolutely Continuous Approximations
8.1.Inversion Formula
8.2.Local Estimates for Bounded Densities
Contents note continued: 8.3.Approximating Probability by Density
8.4.Estimates in the Kolmogorov Norm
8.5.Estimates in Total Variation
8.6.Non-uniform Estimates
8.7.Problems
9.The Esseen Type Estimates
9.1.General Inversion Inequalities
9.2.The Berry-Esseen Theorem
9.3.Distributions with 1 + δ Moment
9.4.Estimating Centered Distributions
9.5.Discontinuous Distribution Functions
9.6.Problems
10.Lower Estimates
10.1.Estimating Total Variation via the Fourier Transform
10.2.Lower Estimates for the Total Variation
10.3.Lower Estimates for Densities
10.4.Lower Estimates for Probabilities
10.5.Lower Estimates for the Kolmogorov Norm
10.6.Problems
11.The Stein Method
11.1.The Basic Idea for Normal Approximation
11.2.The Lattice Case
11.3.Establishing Stein's Operator
11.4.The Big Three Discrete Approximations
11.5.The Poisson Binomial Theorem
11.6.The Perturbation Approach
Contents note continued: 11.7.Estimating the First Pseudomoment
11.8.Lower Bounds for Poisson Approximation
11.9.Problems
12.The Triangle Function Method
12.1.The Main Lemmas
12.2.Auxiliary Tools
12.3.First Example
12.4.Second Example
12.5.Problems
13.Heinrich's Method for m-Dependent Variables
13.1.Heinrich's Lemma
13.2.Poisson Approximation
13.3.Two-Way Runs
13.4.Problems
14.Other Methods
14.1.Method of Compositions
14.2.Coupling of Variables
14.3.The Bentkus Approach
14.4.The Lindeberg Method
14.5.The Tikhomirov Method
14.6.Integrals Over the Concentration Function
14.7.Asymptotically Sharp Constants.