Covergence in Distribution and Limiting Distribution

Limiting Distribution

Introduction

* Exact distributions can be difficult to obtain for large sample.
*Limit Theorems can be used to obtain properties of estimators as the sample sizes tend to infinity, and to get a limiting distribution instead the original distribution for same properties.
*Convergence in Probability ( Limit of an estimator),
and using this convergence for verification
*Convergence in Distribution ( Limit of a CDF)
* Central Limit Theorem
(Large Sample Distribution of the Sample Mean of a Random Sample)
Convergence in Distribution
Definition (1)
A sequence of random variables, X1, X2, . . ., converges in distribution to a random variable X if limn??FXn(x) = FX(x)at all points x where FX(x) is continuous.

Example (Maximum of Uniform)
If X1, X2, . . . are iid Uniform(0,1) and
X(n) =max1?i?n Xi ,find the limiting distribution of X(n) as n ? ?.
Solution
let us examine if X(n) converges in distribution.
As n ? ?, we have for any ? > 0,
P (|Xn ? 1| ? ?) = P (X(n) ? 1 ? ?)
= P (Xi ? 1 ? ?, i = 1, . . . , n) = (1 ? ?)n
,which goes to 0. However, if we take ? = t/n, we then have
P (X(n) ? 1 ? t/n) = (1 ? t/n)n? e?t,
which, upon rearranging, yields
P (n(1 ? X(n)) ? t) ? 1 ? e?t;
that is, the random variable n(1?X(n)) converges in distribution to an exponential(1) random
variable.
We note that it is not a sequence of random variables converge in distribution, but it is really the cdfs that converge. This very fundamental way convergence in distribution is quite different from convergence in probability.