On the Rate of Convergence of Lp Norms in the Central Limit Theorem for Independent Random Variables

We present the upper and lower bounds of the L _p norms _np of order n ^-1 for all p, 1 p , in the central limit theorem for independent identically distributed random variables with the finite fourth and zero first and third moments of summands. The proof of the upper estimates of the norms _np is based on a linear differential equation formed from the characteristic function of the sum of independent random variables.