On The Convergence in a Local Limit Theorem for th Densities of Sums of Independent Random Variables

 For a class of sequences with independent and identi-cally distributed random variables a theorem is proved for the convergence concerning a local limit theorem for the densities of sums of independent random variables.
Suppose we have a sequence of independent and identi-cally distributed random variables Xi with E(X1) = o and E(X1)2 = 1, E denotes the mathematical expectation. Let
f(t) = E exp(it X1), - 00 < t <
be the characteristic function of X1. The probability den-
_u n sity function of S = n E X4is denoted by Pn(x), pro-n i=l vided it exists. If 0(x) = (1/,7T) exp(-x2/2) is the stan-
dard normal probability density function, we set and
n(t) = fn(t/ii),
An(x) = 1Pn(x) = ("x".
A constant will be denoted by C. Thought constants appearing in the article have different values, we see that this convention makes the paper more transparent and less – ambiguous. The following two lemmas will be used in the sequal