In distribution convergence (convergence in distribution)

Convergence in distribution

Distribution converge in a convergence of random variables, provided {ξn, n≥1} is the probability space (Ω, F, P) random variable in the column, as the corresponding distribution function {Fn (x), n≥1}, if Fn (x) converges to a weak distribution function F (x) of the random variable [xi], called random variables ξn converge in distribution to the random variable ξ. 

definition

Definition 1

Said sequence of random variables in distribution convergence (convergence in distribution) to a random variable X, if for any continuous point x, there 

Definition 2

Weak Convergence 

Provided is a distribution function column, if there is a distribution function , so that there is point on each successive set up, called weak convergence to , and referred to as

Convergence in Distribution 

Set random variable sequence, it is the distribution function corresponding to the column, if there is a distribution function having a random variable , so called converges in distribution , and denoted .

We must point out that only when the distribution function sequence converges to a distribution function, we say that it is the convergence in distribution, this explanation is necessary because the distribution function sequences may converge to a function, and this function is not necessarily a distribution function.

Convergence in probability, convergence will be useless, convergence in distribution

Note that although we define a sequence of random variables converges in distribution, and its essence is the cumulative distribution function of convergence rather than a random variable, and therefore the convergence in probability convergence in distribution, the Scots will be convergence (convergence almost everywhere: almost surely convergent) different in nature, but the other two are respectively convergence implies convergence in distribution.

 

Theorems

Theorem 1

If the sequence of random variables converges in probability random variable X, the sequence also converges in distribution X.

Theorem 2

Sequence of random variables converges in probability constant , if and only if the sequence converges in distribution , i.e., equivalent to