Stability theorems for sequences \(\eta _{n+1}=f(\eta _ n,\xi _{n+1})\) in Banach and metric spaces. (English. Russian original) Zbl 0639.60043

Kukush, A. G.

Theory Probab. Math. Stat. 33, 47-57 (1986); translation from Teor. Veroyatn. Mat. Stat. 33, 42-52 (1985).

Summary: For a sequence of random elements in a Banach space with unconditional basis \(\eta_{n+1}=f(\eta_ n,\xi_{n+1})\), \(n\geq 0\), where \(\{\xi_ n\}\) is a stationary sequence of random elements, theorems containing explicit restrictions on the function f are proved about the existence, uniqueness, and stability of stationary distributions. The case when \(\{\xi_ n\}\) is a sequence of independent identically distributed elements is also considered.

MSC:

60G10	Stationary stochastic processes
34D20	Stability of solutions to ordinary differential equations
46B15	Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
60F05	Central limit and other weak theorems
60K25	Queueing theory (aspects of probability theory)

Keywords:

strict-sense stationary sequence; ergodic properties; method of Lyapunov functions; stability of stationary distributions