The law of large numbers for weakly correlated random elements in Hilbert spaces

The first rigorous proof of the law of large numbers (LLN) belongs to Bernoulli. More than hundred years later, Bernoulli’s theorem was generalized by Poisson, who was the first to use the term “law of large numbers”. Significant contribution in this direction belongs to Chebyshev, who presented the problem in terms of random variables and gave a very simple and precise proof of the generalized LLN. The LLN was a popular area of research and has been intensively studied by many authors for the sequences of independent (or uncorrelated) random variables. The study of the LLN for dependent random variables started later. One of the first results in this direction was obtained in 1928 by Khinchin. we proved an analogue of the Khinchin’s theorem for weakly correlated random elements with values in Hilbert spaces.