| One of the most common techniques used in proving limit theorems for dependent random sequences is the "Bernstein blocking argument", where "big blocks" of random variables are separated from each other by "small blocks" in between. If these "small blocks" are large enough, then the "big blocks" have almost no influence on each other. Using this blocking argument, I derived a central limit theorem for strictly stationary random fields of real-valued random variables, satisfying a strong mixing condition. Using Cramer-Wold Device Theorem, I extended this case to a strictly stationary random field of finite dimensional real-valued random vectors in the presence of the same mixing condition, which enabled us to prove both a central limit theorem for strictly stationary (infinite-dimensional) Hilbert-space valued random field, and also a functional central limit theorem for empirical processes. In the presence of the interlaced mixing and the strong mixing condition, I also studied the asymptotic normality of the normalized partial sum of a multivariate strictly stationary random field. |
