Stanisław Kwapień

Random series and stochastic integrals : single and multiple

0 Preliminaries.- 0.1 Topology and measures.- 0.2 Tail inequalities.- 0.3 Filtrations and stopping times.- 0.4 Extensions of probability spaces.- 0.5 Bernoulli and canonical Gaussian and ?-stable sequences.- 0.6 Gaussian measures on linear spaces.- 0.7 Modulars on linear spaces.- 0.8 Musielak-Orlicz spaces.- 0.9 Random Musielak-Orlicz spaces.- 0.10 Complements and comments.- Bibliographical notes.- I Random Series.- 1 Basic Inequalities for Random Linear Forms in Independent Random Variables.- 1.1 Levy-Octaviani inequalities.- 1.2 Contraction inequalities.- 1.3 Moment inequalities.- 1.4 Complements and comments.- Best constants in the Levy-Octaviani inequality.- A contraction inequality for mixtures of Gaussian random variables.- Tail inequalities for Bernoulli and Gaussian random linear forms.- A refinement of the moment inequality.- Comparison of moments.- Bibliographical notes.- 2 Convergence of Series of Independent Random Variables.- 2.1 The Ito-Nisio Theorem.- 2.2 Convergence in the p-th mean.- 2.3 Exponential and other moments of random series.- 2.4 Random series in function spaces.- 2.5 An example: construction of the Brownian motion.- 2.6 Karhunen-Loeve representation of Gaussian measures.- 2.7 Complements and comments.- Rosenthals inequalities.- Strong exponential moments of Gaussian series.- Lattice function spaces.- Convergence of Gaussian series.- Bibliographical notes.- 3 Domination Principles and Comparison of Sums of Independent Random Variables.- 3.1 Weak domination.- 3.2 Strong domination.- 3.3 Hypercontractive domination.- 3.4 Hypercontractivity of Bernoulli and Gaussian series.- 3.5 Sharp estimates of growth of p-th moments.- 3.6 Complements and comments.- More on C-domination.- Superstrong domination.- Domination of character systems on compact Abelian groups.- Random matrices.- Hypercontractivity of real random variables.- More precise estimates on strong exponential moments of Gaussian series.- Growth of p-th moments revisited.- More on strong exponential moments of series of bounded variables.- Bibliographical notes.- 4 Martingales.- 4.1 Doobs inequalities.- 4.2 Convergence of martingales.- 4.3 Tangent and decoupled sequences.- 4.4 Complements and comments.- Bibliographical notes.- 5 Domination Principles for Martingales.- 5.1 Weak domination.- 5.2 Strong domination.- 5.3 Burkholders method: comparison of subordinated martingales.- 5.4 Comparison of strongly dominated martingales.- 5.5 Gaussian martingales.- 5.6 Classic martingale inequalities.- 5.7 Comparison of the a.s convergence of series of tangent sequences.- 5.8 Complements and comments.- Tangency and ergodic theorems.- Burkholders method for conditionally Gaussian and conditionally independent martingales.- Necessity of moderate growth of ?.- Comparison of Gaussian martingales revisited.- Comparing H-valued martingales with 2-D martingales.- The principle of conditioning in limit theorems.- Bibliographical notes.- 6 Random Multilinear Forms in Independent Random Variables and Polynomial Chaos.- 6.1 Basic definitions and properties.- 6.2 Maximal inequalities.- 6.3 Contraction inequalities and domination of polynomial chaos.- 6.4 Decoupling inequalities.- 6.5 Comparison of moments of polynomial chaos.- 6.6 Convergence of polynomial chaos.- 6.7 Quadratic chaos.- 6.8 Wiener chaos and Herrnite polynomials.- 6.9 Complements and comments.- Tail and moment comparisons for chaos and its decoupled chaos.- Necessity of the symmetry condition in decoupling inequalities.- Karhunen-Loeve expansion for the Wiener chaos.- ?-stable chaos of degree d ? 2.- Bibliographical notes.- II Stochastic Integrals.- 7 Integration with Respect to General Stochastic Measures.- 7.1 Construction of the integral.- 7.2 Examples of stochastic measures.- 7.3 Complements and comments.- An alternative definition of m-integrability.- Bibliographical notes.- 8 Deterministic Integrands.- 8.1 Discrete stochastic measure.- 8.2 Processes with independent increments and their characteristics.- 8.3 Integration with respect to a general independently scattered measure.- 8.4 Complements and comments.- Stochastic measures with finite p-th moments.- Bibliographical notes.- 9 Predictable Integrands.- 9.1 Integration with respect to processes with independent increments: Decoupling inequalities approach.- 9.2 Brownian integrals.- 9.3 Characteristics of semimartingales.- 9.4 Semimartingale integrals.- 9.5 Complements and comments.- The Bichteler-Dellacherie Theorem.- Semimartingale integrals in Lp.- ?-stable integrals.- Bibliographical notes.- 10 Multiple Stochastic Integrals.- 10.1 Products of stochastic measures.- 10.2 Structure of double integrals.- 10.3 Wiener polynomial chaos revisited.- 10.4 Complements and comments.- Multiple ?-stable integrals.- Bibliographical notes.- A Unconditional and Bounded Multiplier Convergence of Random Series.- A.2 Almost sure convergence.- A.3 Complements and comments.- A hypercontractive view.- Bibliographical notes.- B Vector Measures.- B.1 Extensions of vector measures.- B.2 Boundedness and control measure of stochastic measures.- B.3 Complements and comments.- Bibliographical notes.

A Remark on the Median and the Expectation of Convex Functions of Gaussian Vectors

Ten years ago A. Ehrhard published an important paper, [1], in which he proved that if γn is a gaussian measure on R n, Φ is the normal distribution function, i.e \(\Phi (t)=\frac{1}{\sqrt{2\pi}}\int _{-\infty}^{t}e^{s^2/2}ds\) for t ∈ R 1 then

On the Rademacher Series

\Phi ^{-1}(\gamma _n(\lambda ^1K^1+\lambda ^2K^2))\geq \lambda ^1\Phi ^{-1}(\gamma _n(K^1))+\lambda ^2\Phi ^{-1}(\gamma _n(K^2))

On the Limit Set in the Law of the Iterated Logarithm for U -statistics of Order Two

for each K 1, K 2 convex subsets of R n and λ1,λ2 ≥ 0, λ1 + λ2 = 1.

Sample Hölder Continuity of Stochastic Process and Majorizing Measures

Let X; Xi ;i 2 N, be independent identically distributed random variables and let h(x;y)= h(y;x) be a measurable function of two variables. It is shown that the bounded law of the iterated logarithm, lim supn(n log log n) 1 P 1i<jn h(Xi;Xj) <1 a.s., holds if and only if the following three conditions are satised: h is canonical for the law of X (that is, Eh(X;y)=0 for almost all y) and there exists C<1 such that, both, E(h 2 (X1;X2)^u)C log log u for all large u and supfEh(X1;X2)f(X1)g(X2):kf(X)k21;kg(X)k21;kfk1<1;kgk1<1gC.

MACKEY CONTINUITY OF CHARACTERISTIC FUNCTIONALS

In this note we give simple proofs of some of the inequalities on Rademacher series given by M. Ledoux and M. Talagrand, [6], ch.4.1, S.J. Montgomery -Smith, [8], and by P. Hitczenko, [3]. We obtain better constants with proofs which can be useful in some other cases. As a corollary we prove a theorem of Kolmogorov on the lower estimates of the tail of sums of symmetric, independent random variables.

On a.s. Unconditional Convergence of Random Series in Banach Spaces

It is proved that in Hilbert spaces a single Hilbert–Schmidt operator radonifies cylindrical semimartingales to strong semimartingales. This improves a result due to Badrikian and Ustunel (also L. Schwartz), who needed composition of three Hilbert–Schmidt operators.

Some combinatorial and probabilistic inequalities and their application to Banach space theory

We find the cluster set in the Law of the Iterated Logarithm for U-statistics of order 2 in some interesting special cases. The lim sup is an unusual function of the quantities that determine the Bounded LIL.