Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

Stable non-Gaussian random processes , by G. Samorodnitsky and M. S. Taqqu. Pp. 632. £49.50. 1994. ISBN 0-412-05171-0 (Chapman and Hall).

Concentration functions and inequalities Isoperimetric and functional examples Concentration and geometry Concentration in product spaces Entropy and concentration Transportation cost inequalities Sharp bounds of Gaussian and empirical processes Selected applications References Index

/pdf/the-concentration-of-measure-phenomenon-27qyoor2ef.pdf

The concentration of measure phenomenon

Convex bodies: the brunn–minkowski theory

1. General Questions of the Distribution Theory in Banach Spaces.- 1.1. Random Elements in Banach Spaces.- 1.2. Main Facts on Gaussian Measures in Banach Spaces.- 1.3. Weak Convergence of Probability Measures and Some Inequalities.- 2. Some Questions of Non-Linear Analysis.- 2.1. Differentiation in Normed Spaces.- 2.2. Smooth Approximation in a Banach Space.- 2.3. Supplements.- 3. The Central Limit Theorem in Banach Spaces.- 3.1. Operators of Type p.- 3.2. The Central Limit Theorem.- 3.3. Supplements.- 4. Gaussian Measure of ?-Strip of Some Sets.- 4.1. General Remarks.- 4.2. Distribution Density of the Norm of a Gaussian Element in Lp.- 4.3. Distribution Density of the Norm of a Gaussian Element in c0.- 4.4. Supplements.- 5. Estimates of Rate of Convergence in the Central Limit Theorem.- 5.1. Estimates of the Rate of Convergence on Sets.- 5.2. Estimates of the Rate of Convergence in the Central Limit Theorem in Some Banach Spaces.- 5.3. Estimates of the Rate of Convergence in the Prokhorov Metric and the Bounded Lipschitz Metric.- 5.4. Supplements.- Bibliographical Notes.- References.

Approximation Theory in the Central Limit Theorem: Exact Results in Banach Spaces

Approximation Theory in the Central Limit Theorem

Let B be a real separable Banach space with norm || · || = || · || B . Suppose that X, X 1, X 2, … ∈ B are independent and identically distributed (i.i.d.) random elements (r.e.’s) taking values in B. Furthermore, assume that EX = 0 and that there exists a zero-mean Gaussian r.e. Y ∈ B such that the covariances of X and Y coincide.

The Accuracy of Gaussian Approximation in Banach Spaces

We study the asymptotic behavior and limit distributions for sums Sn =bn-1 ∑i=1n ξi,where ξi, i ≥ 1, are i.i.d. random convex compact (cc) sets in a given separable Banach space B and summation is defined in a sense of Minkowski. The following results are obtained: (i) Series (LePage type) and Poisson integral representations of random stable cc sets in B are established; (ii) The invariance principle for processes Sn(t) =bn-1 ∑i=1[nt] ξi, t∈[0, 1], and the existence of p-stable cc Levy motion are proved; (iii) In the case, where ξi are segments, the limit of Sn is proved to be countable zonotope. Furthermore, if B = Rd, the singularity of distributions of two countable zonotopes Yp1, σ1,Yp2, σ2, corresponding to values of exponents p1, p2 and spectral measures σ1, σ2, is proved if either p1 ≠ p2 or σ1 ≠ σ2; (iv) Some new simple estimates of parameters of stable laws in Rd, based on these results are suggested.

https://link.springer.com/content/pdf/10.1023/A:1007726708227.pdf

Vygantas Paulauskas

Papers

Approximation Theory in the Central Limit Theorem: Exact Results in Banach Spaces

Approximation Theory in the Central Limit Theorem

The Accuracy of Gaussian Approximation in Banach Spaces

More on p -stable convex sets in Banach spaces

On the Rate of Convergence in the Central Limit Theorem in Certain Banach Spaces