Some effects of trimming on the law of the iterated logarithm

We investigate some effects that the 'light' trimming of a sum S-n = X-1 + X-2 + (...) + X-n of independent and identically distributed random variables has on behaviour of iterated logarithm type. Light trimming is defined as removing a constant number of summands from S-n. We consider two versions: S-(r)(n), which is obtained by deleting the r largest X-1 from S-n, and S-(r)(n), which is obtained by deleting the r variables X-1 which are largest in absolute value from S-n. We summarise some relevant results from Rogozin (1968), Heyde (1969), and later writers concerning the untrimmed sum, and add some new results concerning trimmed Sums. Among other things we show that a general form of the law of the iterated logarithm holds for S-(r)(n), but not (completely) for S-(r)(n).