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

Abstract

Let {X n ;n≥1} be a sequence of i.i.d. random variables and let $$ X^{{{\left( r \right)}}}_{n} = X_{j} $$ if |X j | is the r-th maximum of |X 1|, ..., |X n |. Let S n = X 1+⋯+X n and $$ {}^{{{\left( r \right)}}}S_{n} = S_{n} - {\left( {X^{{{\left( 1 \right)}}}_{n} + \cdots + X^{{{\left( r \right)}}}_{n} } \right)}. $$ Sufficient and necessary conditions for (r) S n 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.