On strong law of large numbers for weakly stationary $\varphi$-mixing set-valued random variable sequences

In this paper we extend the notion of $\varphi$-mixing to set-valued random sequences that take values in the family of closed subsets of a Banach space. Several strong laws of large numbers for such $\varphi$-mixing sequences are stated and proved. Illustrative examples show that the hypotheses of the theorems are both natural and sharp.