Properties of orthogonal random variables sums and their applications

Inequality is derived for mean of maximum of squared accumulated orthogonal random variables sums with perturbations. Estimates for means of squared accumulated sums supremums by sums of second moments with multiplier are proved for finite and infinite orthogonal random sequences. Estimates are derived for probabilities of deviation. Analogues of such statements for functions in arbitrary space with finite measure are proved. Convergence with probability 1 is established for series of orthogonal random variables. Limit equalities for moments of sums and for their variances are proved. These results have visible applications in stochastic modeling. Example of adjacent-orthogonal random variables sequence is presented, that has finite sum of second moments, while its series diverges for sure. It shows, that inequalities of the above sort are not true for general adjacent-orthogonal random variables. Comparison is performed for experiments, where samples of orthogonal random variables have finite and infinite sums of second moments correspondingly.