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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.

60G17 Sample path properties
60G07 General theory of stochastic processes