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

Research paper by Li Xin Zhang

Indexed on: 01 Apr '02Published on: 01 Apr '02Published in: Acta Mathematica Sinica, English Series


Let {Xn;n≥1} be a sequence of i.i.d. random variables and let \( X^{{{\left( r \right)}}}_{n} = X_{j} \) if |Xj| is the r-th maximum of |X1|, ..., |Xn|. Let Sn = X1+⋯+Xn 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)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)Sn are studied.