Some Korovkin-type theorems for stochastic processes (Q2740457)

The classical Banach-Steinhaus theorem states that a sequence of linear operators in linear normed spaces tends to some linear operator pointwise if and only if their norms are bounded and they converge strongly to this operator. \textit{P. P. Korovkin} [``Lineare Operatoren und Approximationstheorie'' (1959; Zbl 0094.10201)] proved that the condition on convergence can be weaken when the sequence of operators is positive. This paper deals with the classical Korovkin theorem which is extended to the class of finite second-order stochastic processes. Applying this result to the construction of a sequence of linear positive operators by \textit{I. I. Ibragimov} and \textit{A. D. Gadzhiev} [Sov. Math., Dokl. 11, 1092-1095 (1970); translation from Dokl. Akad. Nauk SSSR 193, 1222-1225 (1970; Zbl 0217.17302)] the author generates the Bernstein-Kholodovski, Baskakov and Hille-Szász-Mirakjan operators on the class of second-order processes. The quadratic mean truncation error upper bound is established for some of these operators.