Linda Khachatryan (Erevan)
The problem of establishing classical limit theorems (the central and functional limit theorems, the law of iterated logarithm) is one of the main problems in probability theory. Investigation of stochastic processes with dependent components (mixing processes, Markov processes, martingales, etc.) has started in the last century and since that time led to construct the theory, which is of full value in many positions. The same is not true for random fields. The problem of extending the theory of limit theorems to multidimensional case (random fields) is dictated both by internal needs of the theory of random fields and by applications, in particular, in some problems of mathematical statistical physics.
The martingale method is one of the most useful in the theory of random processes, particularly in problems of convergence of sequences of random variables and in limit theorems for sums of random summands. At the same time in multidimensional case this method is not yet so productive. The standard explanation is the following. The notion of martingale is essentially based on the complete ordering property of the real line, while spatial structures do not possess this property. However recently published works show that by using the notion of martingale-difference random field one can significantly develop the martingale method in multidimensional case.
In our talk we discuss martingale-difference random fields and some of their applications in the theory of limit theorems. We present different approaches to construction of such fields which, in particular, can be applied to Markov and Gibbs random fields. Various limit theorems for martingale-difference random fields are presented based on the martingale method, including the central limit theorem (with rate of convergence), the law of iterated logarithm, the convergence of moments of sums of martingale-difference random field's components, the invariance principle.
