The Study About Random Variable Sequence Convergence On The Basis Of Positive Part And Negative Part

In statistical analysis, Probability Limit Theory is often used to find the asymptotic distribution of the statistics. Generally speaking, the overall distribution of what we researched is unknown, so it is impossible to figure out the accurate distribution of the statistics through the collectivity distribution. In this case, we often figure out the asymptotic distribution of the sample whose size tends to infinity( n ??), and then conduct the statistic analysis. In the process of finding out the asymptotic distribution of the statistics, the Slutsky Theorem is often used, which is about the convergence in distribution of random variable sequence.In most of the current textbooks about probability and statistics, the proof process of Slutsky Theorem is too simplicity. It is lack of further elaborate analysis.Such as:Question 1: Convergence in distribution of random variable sequence refers to any consecutive point( x) on the distribution function are conformed to the equality: lim? ? ? ?nF x ?F x, and it doesn’t refer to each point converge on a limit distribution function. So, in the process of proving Slutsky Theorem, an arbitrary ? ?0, when ?0??, the function g??? about ? approaches to the consecutive points of the distribution function F(x). Under this condition, whether there always exists ? ?0 which can make g??? be the consecutive points between x and g??? on the function F? x?.Question 2: When there are both positive and negative values of a random variabl, the inequality symbols are lack of detailed analysis in the process of proving Slutsky Theorem 2 and 3.The paper embarks from indicative function which belongs to the measurable function, and makes one random variable derive into two random variables according to positive part and negative part, and then, this method is extended to one random variable sequence deriving two new random variable sequences. Through studying the convergence relation of the two derivative sequences with the original sequence, the two results are obtained: the necessary and sufficient condition of random variable sequences converges almost everywhere; the necessary and sufficient condition of the convergence in probability measure and a certain constant c. The results simplify the analysis process of the inequality symbols in the traditional proof of Slutsky Theorem 2 and 3.The paper also studies distribution of the consecutive points of the random variable distribution function in the set of real number R.Finally, Slutsky Theorem is proved in a new method.