A multidimensional version of the results of Komlos, Major and Tusnady for the Gaussian approximation of the sequence of successive sums of independent random vectors with finite exponential moments is obtained.

Let P1, ..., Pd be commuting Markov operators on L∞(X,F,μ), where (X,F,μ) is a probability measure space. Assuming that each Pi is either conservative or invertible, we prove that for every f in Lp(X,F,μ) with 1 ≤ p < ∞ the averagesAnf = (n + 1)-d Σ0≤ni≤n P1n1 P2n2 ... Pdnd f (n ≥ 0)converge almost everywhere if and only if there exists an invariant and equivalent finite measure λ for which the Radon-Nikodym derivative v = dλ/dμ is in the dual space Lp'(X,F,μ). Next we study the case in...

Let $(X_i,i=1,2,...)$ be a Gaussian sequence with $X_i\in N(0,1)$ for each i and suppose its correlation matrix $R={\left({\rho }_{ij}\right)}_{i,j\ge 1}$ is the matrix of some linear operator R:l₂→ l₂. Then for $f_i\in L²\left(\mu \right)$, i=1,2,..., where μ is the standard normal distribution, we estimate the variation of the sum of the Gaussian functionals $f_i\left(X_i\right)$, i=1,2,... .

This note deals with the orthogonality between sequences of random variables. The main idea of the note is to apply the results on equidistant systems of points in a Hilbert space to the case of the space $L^2(\Omega ,ℱ,\mathbb{P})$ of real square integrable random variables. The main result gives a necessary and sufficient condition for a particular sequence of random variables (elements of which are taken from sets of equidistant elements of $L^2(\Omega ,ℱ,\mathbb{P})$) to be orthogonal to some other sequence in $L^2(\Omega ,ℱ,\mathbb{P})$. The result obtained is interesting...

Mark Kac gave an example of a function f on the unit interval such that f cannot be written as f(t)=g(2t)-g(t) with an integrable function g, but the limiting variance of $n^{-1/2}{\sum }_{k=0}^{n-1}f\left(2^{k}t\right)$ vanishes. It is proved that there is no measurable g such that f(t)=g(2t)-g(t). It is also proved that there is a non-measurable g which satisfies this equality.

An exponential inequality for Choquet expectation is discussed. We also obtain a strong law of large numbers based on Choquet expectation. The main results of this paper improve some previous results obtained by many researchers.

Consider a $N\times n$ non-centered matrix ${𝛴}_n$ with a separable variance profile: $${𝛴}_n=\frac{D_n^{1/2}{X}_n{\tilde{D}}_n^{1/2}}{\sqrt{n}}+A_n.$$ Matrices $D_n$ and ${\tilde{D}}_n$ are non-negative deterministic diagonal, while matrix $A_n$ is deterministic, and $X_n$ is a random matrix with complex independent and identically distributed random variables, each with mean zero and variance one. Denote by $Q_n\left(z\right)$ the resolvent associated to ${𝛴}_n{𝛴}_n^{*}$, i.e. $$Q_n\left(z\right)={\left({𝛴}_n{𝛴}_n^{*}-z{I}_N\right)}^{-1}.$$ Given two sequences of deterministic vectors $\left(u_n\right)$ and $\left(v_n\right)$ with bounded Euclidean norms, we study the limiting behavior of the random bilinear form: $$u_n^{*}{Q}_n\left(z\right)v_n\forall z\in ℂ-{ℝ}^{+},$$ as the dimensions...