Conditions for convergence and rate of convergence of random series in \(K_\sigma\) spaces of random variables. I. (English. Ukrainian original) Zbl 0923.60041

Kozachenko, Yu. V.

Theory Probab. Math. Stat. 56, 115-128 (1998); translation from Teor. Jmovirn. Mat. Stat. 56, 112-125 (1997).

The Banach space \((K,\| \cdot\|)\) of random variables is called \(K_\sigma\)-space if: i) \(\xi,\eta\in K\) imply \(\max(\xi,\eta)\in K\), \(\min(\xi,\eta)\in K\); ii) for any \(\xi,\eta\in K\), \(| \xi| \leq| \eta| \) a.s. implies \(\| \xi\| \leq\| \eta\| \); iii) if \(\xi_n,\eta\in K\), \(\sup_{n\geq 1}| \xi_n| \leq\eta\), then \(\sup| \xi_n| \in K\). Rates of convergence of the series \(R(x)=\sum_{l=1}^\infty\xi_l g_l(x)\), where \(\xi_l\) are in \(K\), \(g_l(x)\) are continuous bounded functions of \(x\in X\) (\(X\) being a separable metric space) are investigated. Inequalities of the type \[ \Biggl\| \sup_x| c(x)R(x)| \;\Biggr\| \leq\sum_{l=1}^\infty s_l\sup_x \Biggl\| \sum_{l=1}^\infty\xi_i\psi_l g_l(x)\Biggr\| \] are obtained for some nonrandom \(c(x)\), \(\psi_l\) and \(s_l\). The obtained results are used for the investigation of solutions of partial derivative equations with random boundary conditions.

Reviewer: R.E.Maiboroda (Kyïv)

MSC:

60G07	General theory of stochastic processes
60G17	Sample path properties
60G10	Stationary stochastic processes

Keywords:

rate of convergence; Banach space