Probability Estimations Of The Sum Of Independent Identically Distributed Random Variables

The probability estimations of the sum of independent random variables is an important research direction in probability statistics.This paper studies the several probability estimations of the sum of several independent identically distributed random variables.Firstly,we discussed the probability estimation of two independent random variables and proved that if the logarithm of their distribution function was greater than or equal to the power function,then the logarithm of the distribution function of the sum of these two random variables was also greater than or equal to some power function.Secondly,the probability estimations of three types of the sum(or weighted sum)of n independent random variables {Xi}i=1n were discussed,here P(Xi=1)=P(Xi=-1)=1/2,i=1,2,…,n.The first type is the probability estimation of the maximum value of the sum Sn:=∑i=1nXi.By means of the total probability formula,the recursive method and the symmetry of random variables,the expression(?)was given,where 1 ≤C<2 is a constant.For C≥1,by using the total probability formula and recursion method,the lower bounds of the logarithm were obtained for even number n and odd number n separately.The second type is the probability estimation of |∑i=1naiXil≤1,where a=(a1,…,an)is the random variable uniformly distributed on the hypersphere Sn-1:=｛(a1,…,an)∈Rn|∑i=1nai2=1} and is independent of {Xi}i=1n.The expression P(|∑i=1naiXi|≤1)is obtained by using polar coordinate transformation.When n≤7,we got that P(|∑i=1nai2Xi|≤1)≥1/2 by direct calculation.When n≥8,we also obtained P(|∑i=1naiXi|≤1)≥1/2 through R software.For n=3,4,with the help of Beta function,we directly proved P(|∑i=1naiXi|≤1)≥1/2.The third type is the error estimation of the distribution of the weighted sum X=∑i=1naiXi and the standard norm distribution.We discussed the upper bound |P(X ≤x)-P(Z≤x)| for any x∈R,where Z is the standard normal random variable.By R software and the symmetric property of the distributions of X and Z,it was proved that|P(X≤x)-P(Z≤x)≤P(Z∈(0,a1)),?x∈R.