THE EFFECT OF TRIMMING ON THE STRONG LAW OF LARGE NUMBERS

'Trimmed' sample sums may be defined for r = 1, 2, ..., by ((r))S-n = S-n - M(n)((1)) - M(n)((2)) - ... - M(n)((r)) and ((r))(S) over tilde( n) = S-n - X(n)((1)) - X(n)((2)) - ... - X(n)((r)), where S-n = X(1) X(2) + ... + X(n) is the sum of independent and identically distribut ed random variables X(i), M(n)((1)) greater than or equal to ... great er than or equal to M(n)((n)) denote X(1), ..., X(n) arranged in decre asing order, and X(n)((f)) is the observation with the jth largest mod ulus. We investigate the effects of these kinds of trimming on various forms of convergence and divergence of the sample sum. In particular, we provide integral tests for ((r))S-n/n --> +/- infinity, and analyt ical criteria for almost sure relative stability when the number of po ints trimmed, r, is fixed, but n --> infinity. Some surprising results occur. For example, when r = 0, 1, 2,..., (r)S-n may be almost surely negatively relatively stable (((r))S-n/B-n --> -1 a.s. as n --> infin ity for some non-stochastic sequence B-n up arrow infinity) only if -i nfinity < EX(1) less than or equal to 0, and a striking corollary of t his is an example of a random walk S-n which is recurrent (even has me an 0), but for which ((r))S-n and ((r))(S) over tilde(n) are transient when r greater than or equal to 1.