Convergence in Orlicz space norms and Lévy-Baxter theorems. (English. Russian original) Zbl 0634.60031

Besklinskaya, E. P.; Kozachenko, Yu. V.

Theory Probab. Math. Stat. 35, 1-4 (1987); translation from Teor. Veroyatn. Mat. Stat. 35, 3-6 (1986).

Let \(\xi\) (t) be a Gaussian random process on [a,b] with zero mean and let \(a=t_ 0^{(n)}\leq...\leq t_{k_ n}^{(n)}=b\) be partitions with \(\lambda_ n=\max | t^{(n)}_{k+1}-t_ k^{(n)}| \to 0\) as \(n\to \infty\). Let \(\Sigma_ n=\sum^{k_ n-1}_{k=0}[\xi (t^{(n)}_{k+1})-\xi (t_ k^{(n)})]\) 2.
The goal of the present paper is to show the equivalence of the mean square convergenc \(\| \Sigma_ n-{\mathbb{E}}\Sigma_ n\|_ 2\to 0\) and the convergence in terms of the Orlicz norm \(\| \Sigma_ n- {\mathbb{E}}\Sigma_ n\|_{L(U)}\to 0\), where \(U(x)=\cosh x-1\).

Reviewer: S.T.Rachev

MSC:

60F25	\(L^p\)-limit theorems
46E30	Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

Keywords:

Gaussian random process; convergence in terms of the Orlicz norm