Let $(X_i,i=1,2,...)$ be the normalized gaussian system such that $X_i\in N(0,1)$, i = 1,2,... and let the correlation matrix ${\rho }_{ij}=E\left(X_i{X}_j\right)$ satisfy the following hypothesis: $C=su{p}_{i\ge 1}{\sum }_{j=1}^{\infty }\left|{\rho }_{i,j}\right|<\infty$. We present Gebelein’s inequality and some of its consequences: Borel-Cantelli type lemma, iterated log law, Levy’s norm for the gaussian sequence etc. The main result is that (f(X₁) + ⋯ + f(Xₙ))/n → 0 a.s. for f ∈ L¹(ν) with (f,1)ν = 0.

This paper studies a portfolio optimization problem in a discrete-time Markovian model of a financial market, in which asset price dynamics depends on an external process of economic factors. There are transaction costs with a structure that covers, in particular, the case of fixed plus proportional costs. We prove that there exists a self-financing trading strategy maximizing the average growth rate of the portfolio wealth. We show that this strategy has a Markovian form. Our result is obtained...