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Stochastic processes in Orlicz function spaces. (English. Ukrainian original) Zbl 0955.60037
Theory Probab. Math. Stat. 60, 73-85 (2000); translation from Teor. Jmovirn. Mat. Stat. 60, 64-76 (1999).
The author considers a random process $$X(t)$$, $$t\in T$$, where $$(T,\rho)$$ is a pseudometric space and $$\mu$$ is a Borel measure on $$T.$$ Two Orlicz spaces are considered: the Orlicz space $$L_V(\Omega)$$ of random variables and the Orlicz space $$L_U(T)$$ of functions on $$(T,\mu)$$. Conditions are obtained under which $$\|\|X(\cdot)\|_{L_U(T)}\|_{L_V(\Omega)}<B<\infty,$$ where $$\|\cdot\|$$ denotes the Luxemburg norm in the corresponding space. Constants $$B$$ are estimated. E.g. if $$T=[0,1]$$, $$\mu(dt)=dt$$, $$V(x)=\exp\{|x|^\alpha\}-1,$$ $$U(x)=\exp\{|x|^\beta\}-1,$$ $$\beta>\alpha,$$ and $$\sup_{|t-s|<h}\|X(t)-X(s)\|_{L_V(\Omega)}\leq \sigma(h)$$ for an increasing function $$\sigma$$ with $$\sigma(h)\to 0$$ as $$h\to 0$$, $$\Gamma=\sup\|X(t)\|_{L_V(\Omega)}$$, then for any integer $$m$$ and $$p\in(0,1),$$ $\begin{split} \|\|X(t)\|_{L_U(T)}\|_{L_V(\Omega)}\leq\\ \alpha_V \Biggl[\Gamma(\ln(1+\sigma^{(-1)}(p^m\sigma(1)))^{-1})+ {1+p\over p(1-p)}\int_0^{\gamma p^{m+1}} (\ln(1+\sigma^{(-1)}(u)^{-1})^{1/\alpha-1/\beta})du \Biggr]=B, \end{split}$ where $$\alpha_V$$ is a constant defined in the paper. Spaces $$L_p$$ are considered too.

##### MSC:
 60G17 Sample path properties 60G07 General theory of stochastic processes