Gebelein's inequality and its consequences

Volume 72 / 2006

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


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 $(f,1)_\nu =0$.


  • M. BeśkaFaculty of Applied Mathematics,
    Gdańsk University of Technology
    Narutowicza 11/12,
    80-952 Gdańsk, Poland
  • Z. CiesielskiInstitute of Mathematics,
    Polish Academy of Sciences
    Abrahama 18,
    81-125 Sopot, Poland

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