Some effects of trimming on the law of the iterated logarithm

We investigate some effects that the ‘light’ trimming of a sum S_n = X₁ + X₂ + . . . + 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: ^(r)S_n, which is obtained by deleting the r largest X_i from S_n, and ^rS_n, which is obtained by deleting the r variables X_i 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 newresults concerning trimmed sums. Among other things we show that a general form of the law of the iterated logarithm holds for ^(r)S_n but not (completely) for ^(r)S_n.