Strong Approximation Theorems for Sums of Random Variables when Extreme Terms are Excluded

摘要

Let [X_n; n≥1] be a sequence of i. i. d. random variables and let X~(r)_n=|X_j| is the r-th maximum of |x_1|,...,|X_n|. Let S_n=X_1+...+X_n and (r) S_n=S_n-(X~(1)_N...+x~(r)_N). Sufficient and necessary conditions for (r)~Sn approximating to sums of independent normal random variables are obtained. Via approximation results, the convergence rates of the strong law of large numbers for (r)S_n are studied.