ИСТИНА

All three gems in probability theory – the law of large numbers, the central limit theorem and the law of the iterated logarithm – concern the asymptotic behavior of the sums of random variables. It would be natural to extend the results to the functionals of the sums, in particular to quadratic forms. Moreover, in mathematical statistics there are numerous asymptotic problems which can be formulated in terms of quadratic or almost quadratic forms. We review the corresponding results with rates of convergence. Some of these results are optimal and could not be further improved without additional conditions. We focus also on approximations in high-dimensional case.