Dominating measures and boundedness conditions for certain stochastic processes. (English. Russian original) Zbl 0860.60027

Bagro, S. V.; Kozachenko, Yu. V.

Theory Probab. Math. Stat. 49, 31-37 (1994); translation from Teor. Jmovirn. Mat. Stat. 49, 45-54 (1993).

Let \((T,\rho)\) be a compact space with pseudometric \(\rho\). Let \(\zeta\) be the \(\sigma\)-algebra of Borel sets in \((T,\rho)\) and \(\mu\) be a probability measure on \((T,\zeta)\). Suppose \(X=\{X(t), t\in T\}\), \(EX=0\), is a stochastic process such that for every \(t\in T\), \(X(t)\) belongs to an Orlicz space \(L_U(\Omega)\) corresponding to an Orlicz \(N\)-function \(U(x)\). The authors obtain sufficient conditions for the boundedness of \(\sup_{t\in S}|X(t)-\int_S X(u)d\mu(u)|\), where \(S\subset T\) is a measurable set such that \(\mu\otimes\mu\{(u,v)\in S\times S:\rho(u,v)\neq 0\}>0\). Results obtained are akin to those of the second author and V. V. Ryazantseva [Theory Probab. Math. Stat. 41 (1990)].

Reviewer: B.L.S.Prakasa Rao (New Delhi)

MSC:

60G17

Sample path properties

Keywords:

compact space; probability measure; Orlicz space