THE LIL FOR THE PRODUCT LIMIT ESTIMATOR

Let （X_i, Y_i）,i=1, 2, …, be iid. random sequence with unknown distribution function. For i=1, 2,…, n, set δ_i=1 if X_i≤Y_i and 0 if X_i＞Y_i,and Z_i= min（X_i, Y_i）. One way to estimate the distribution function of X from the sample （Z_i,δ_i）, i= 1, 2, …, n is by product limit estimator. We give the if and only if condition for product limit estimator to be uniformly strong consistent and the LIL under this condition.