Let $\{X_n,n\ge 1\}$ be a sequence of independent random variables such that $E{X}_n=a_n$, $E{(X_n-a_n)}^2={\sigma }_n^2$, $n\ge 1$. Let $\{N_n,n\ge 1\}$ be a sequence od positive integer-valued random variables. Let us put $S_{N_n}={\sum }_{k=1}^{N_n}{X}_k$, $L_n={\sum }_{k=1}^n{a}_k$, $s_n^2={\sum }_{k=1}^n{\sigma }_k^2$, $n\ge 1$. In this paper we present necessary and sufficient conditions for weak convergence of the sequence $\{(S_{N_n}-L_n)/s_n,n\ge 1\}$, as $n\to \infty$. The obtained theorems extend the main result of M. Finkelstein and H.G. Tucker (1989).

A family of transformations on the set of all probability measures on the real line is introduced, which makes it possible to define new examples of convolutions. The associated central limit theorems are studied, and examples of the limit measures, related to the classical, free and boolean convolutions, are shown.

Sibley and Sempi have constructed metrics on the space of probability distribution functions with the property that weak convergence of a sequence is equivalent to metric convergence. Sibley's work is a modification of Levy's metric, but Sempi's construction is of a different sort. Here we construct a family of metrics having the same convergence properties as Sibley's and Sempi's but which does not appear to be related to theirs in any simple way. Some instances are brought out in which the metrics...

We consider “nonconventional” averaging setup in the form $\frac{\mathrm{d}{X}^{\epsilon }\left(t\right)}{\mathrm{d}t}=\epsilon B(X^{\epsilon }\left(t\right)$, $𝛯\left(q_1\left(t\right)\right),𝛯\left(q_2\left(t\right)\right),...,𝛯\left(q_{\ell }\left(t\right)\right))$ where $𝛯\left(t\right)$, $t\ge 0$ is either a stochastic process or a dynamical system with sufficiently fast mixing while $q_j\left(t\right)={\alpha }_{j}t$, ${\alpha }_{1}lt;{\alpha }_{2}lt;\cdots lt;{\alpha }_k$ and $q_j$, $j=k+1,...,\ell$ grow faster than linearly. We show that the properly normalized error term in the “nonconventional” averaging principle is asymptotically Gaussian.

We propose two methods to solve multistage stochastic programs when only a (large) finite set of scenarios is available. The usual scenario tree construction to represent non-anticipativity constraints is replaced by alternative discretization schemes coming from non-parametric estimation ideas. In the first method, a penalty term is added to the objective so as to enforce the closeness between decision variables and the Nadaraya–Watson estimation of their conditional expectation. A numerical application...

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,... .