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

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.

### MSC:

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

### Keywords:

rate of convergence; Banach space