Weak Law of Large Numbers

EC484_Econometrics_Analysis

Theorem Suppose $\{y_i{\}}_{i=1}^n$ is a sequence of i.i.d. random variables with finite expected value ($E|y|<\infty$), then

$$\bar{y}=\frac{1}{n}\sum _{i=1}^n{y}_i{\to }^p\mu =E[y_i]$$

as $n\to \infty$.

We could prove it by Chebyshev Inequality.

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Vector Case

WLLN can be extended to the case where $y\in {\mathbb{R}}^m$.

Definition A sequence of random vectors $z_n\in {\mathbb{R}}^k$ converges in probability to $z$ as $n\to \infty$, denoted as $z_n{\to }^{p}z$, if for every $\delta >0$,

$$\underset{n\to \infty }{\lim }\mathbb{P}(||z_n-z||>\delta )=0$$

The difference part is that we introduce $||\cdot ||$ here, which we called norm of a vector. It’s described as the “distance” between two vectors.

Here we only mention a very famous inequality, which is:

$$||a+b||\le ||a||+||b||$$