The employment of ‘Strong Laws of Large Numbers ’ is instrumental to the analysis of system estimation and identification strategies. However, the vast bulk of such laws, as presented in the wider literature, assume independence or at least uncorrelatedness of random components and these assumptions are quite restrictive from an engineering point of view. By way of contrast, this paper shows how to establish strong laws for possibly non-stationary random processes with very general dependence structure. Brief examples are provided that illustrate the utility of the Strong Law of Large Numbers presented.

The 2--party communication complexity of Boolean function f is known to be at least log rank(M f ), i.e. the logarithm of the rank of the communication matrix of f [19]. Lov'asz and Saks [17] asked whether the communication complexity of f can be bounded from above by ( log rank(M f )) c , for some constant c. The question was answered affirmatively for a special class of functions f in [17], and Nisan and Wigderson proved nice results related to this problem [20], but for arbitrary f , it remained a difficult open problem. We prove here an analogous poly-logarithmic upper bound in the stronger multi--party communication model of Chandra, Furst and Lipton [6], which, instead of the rank of the communication matrix, depends on the L 1 norm of function f , for arbitrary Boolean function f .