Law of large numbers for weakly dependent random variables with values in $D\left[0,1\right]$

Abstract

Limit theorems in Banach spaces are important, in particular, because of applications in functional data analysis. This paper is devoted to the law of large numbers for the random variables with values in the space $D\left[0,1\right]$. This space is not separable if we consider it with supremum norm and it is difficult to prove limit theorems in this space. The law of large numbers is well-studied for the sequences of independent $D\left[0,1\right]$-valued random variables. It is known that in the case of independent and identically distributed random variables with values in $D\left[0,1\right]$ the existence of the first moment of the norm of random functions is a necessary and sufficient condition for the strong law of large numbers. The law of large numbers for the sequences of independent and not necessarily identically distributed random variables with values in $D\left[0,1\right]$ were proved as well. Our main goal is to prove the law of large numbers for the weakly dependent random variables with values in the space $D\left[0,1\right]$. Namely, we consider the sequences of mixing random variables with values in $D\left[0,1\right]$. Mixing conditions for $D\left[0,1\right]$-valued random variables can be introduced in several ways. One can assume that random functions themselves satisfy mixing conditions. We consider a slightly different condition. In fact we assume that the finite dimensional projections of the $D[0,1]$-valued random variables satisfy mixing condition. This is a weaker condition than assuming that random functions themselves satisfy mixing condition. In the paper the law of large numbers for $\rho_{m}$-mixing sequences of $D\left[0,1\right]$-valued random variables are proved.