Convergence in Distribution

EC484_Econometrics_Analysis

While Weak Law of Large Numbers is useful to understand the Convergence in Probability, it does not tell us how to give approximation to distribution of estimator. Asymptotic Approximation can be obtained by this, Convergence in Distribution.

Definition

Let $z_n$ be a random vector with cdf $F_n(u)=\mathbb{P}(z_n\le u)$. We say that $z_n$ converges in distribution to $z$, if

$$F_n(u)\to F(u)\text{as}n\to \infty$$

for all $u$ where $F(u)=\mathbb{P}(z\le u)$ is continuous.

We could denote as $z_n{\to }^{d}z$.

Intuitively, as $n$ becomes larger, the shape of the sample distribution of $z_n$ becomes closer to that of $z$.

The very classical example is the Central Limit Theorem.