On The Convergence in a Local Limit Theorem for th Densities of Sums of Independent Random Variables | ||||
| The Egyptian Statistical Journal | ||||
| Article 5, Volume 28, Issue 1, June 1984, Page 85-89 PDF (2.5 MB) | ||||
| Document Type: Original Article | ||||
| DOI: 10.21608/esju.1984.428412 | ||||
 View on SCiNiTO | ||||
| Author | ||||
| M.M. El-Fahham | ||||
| Abstract | ||||
| 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 | ||||
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