STRONG LIMIT-THEOREMS OF EMPIRICAL FUNCTIONALS FOR LARGE EXCEEDANCES OF PARTIAL-SUMS OF IID VARIABLES

Let (X(i), U(i)) be pairs of i.i.d. bounded real-valued random variables (X(i) and U(i) are generally mutually dependent). Assume E[X(i)] < 0 and Pr{X(i) > 0} > 0. For the (rare) partial sum segments where SIGMA-i = k(l)X(i) --> infinity, strong limit laws are derived for the sums SIGMA-i = k(l)U(i). In particular a strong law for the length (l - k + 1) and the empirical distribution of U(i) in the event of large segmental sums of SIGMA-X(i) are obtained. Applications are given in characterizing the composition of high scoring segments in letter sequences and for evaluating statistical hypotheses of sudden change points in engineering systems.