# Publishing house / Banach Center Publications / All volumes

## Gebelein's inequality and its consequences

### Volume 72 / 2006

Banach Center Publications 72 (2006), 11-23 MSC: 47B34, 60F15, 60F20. DOI: 10.4064/bc72-0-1

#### Abstract

Let $(X_i, i=1,2,\ldots)$ be the normalized gaussian system such that $X_i\in N(0,1)$, $i=1,2,\ldots$ and let the correlation matrix $\rho_{ij}=E(X_iX_j)$ satisfy the following hypothesis: $$C=\sup_{i\geq 1}\sum_{j=1}^\infty |\rho_{i,j} | < \infty .%\leqno (R)$$ 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 $$\frac{f(X_1)+\cdots+f(X_n)}{n}\to 0\ \rm a.s.$$ for $f\in L^1(\nu)$ with $(f,1)_\nu =0$.

#### Authors

• M. BeśkaFaculty of Applied Mathematics,
Gdańsk University of Technology
Narutowicza 11/12,
80-952 Gdańsk, Poland
e-mail
• Z. CiesielskiInstitute of Mathematics,