Abstract
This chapter discusses the special case of an infinite-dimensional Hilbert space. It also provides a proof for the individual and mean ergodic theorems for random variables with values in a Banach space. A more complicated procedure to establish them in a more general situation is adopted, even though simple proofs can be given in the case of a Hubert space. To carry out the proofs of limit theorems in a Hilbert space, it is necessary to get some generalization of the celebrated Kolmogorov's inequality. A Hilbert space has no nontrivial compact subgroups, and hence there are no nontrivial idempotent distributions. The chapter also explains accompanying laws and decomposition theorem in detail.
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