(New page: 2.3 Weak law of large numbers Let <math>\left\{ \mathbf{X}_{n}\right\}</math> be a sequence of <math>i.i.d.</math> random variables with mean <math>\mu</math> and variance <math>\sigm...)
 
Line 24: Line 24:
  
 
There are also stronger forms of the law of large numbers. Strong one uses coveriance <math>\left(a.e.\right)</math> as well as weak one uses coveriance <math>\left(p\right)</math> .
 
There are also stronger forms of the law of large numbers. Strong one uses coveriance <math>\left(a.e.\right)</math> as well as weak one uses coveriance <math>\left(p\right)</math> .
 +
 +
----
 +
[[ECE600|Back to ECE600]]
 +
 +
[[ECE 600 Sequences of Random Variables|Back to Sequences of Random Variables]]

Revision as of 13:29, 22 November 2010

2.3 Weak law of large numbers

Let $ \left\{ \mathbf{X}_{n}\right\} $ be a sequence of $ i.i.d. $ random variables with mean $ \mu $ and variance $ \sigma^{2} $ . Define $ \mathbf{Y}_{n}=\frac{1}{n}\sum_{k=1}^{n}\mathbf{X}_{k},\quad n=1,2,\cdots $ . Then for any $ \epsilon>0 $ , $ p\left(\left\{ \left|\mathbf{Y}_{n}-\mu\right|>\epsilon\right\} \right)\rightarrow0 $ as $ n\rightarrow\infty $(convergence in probability).

$ \mathbf{Y}_{n}\longrightarrow\left(p\right)\longrightarrow\mu\text{ as }n\longrightarrow\infty. $

Proof

$ E\left[\mathbf{Y}_{n}\right]=E\left[\frac{1}{n}\sum_{k=1}^{n}\mathbf{X}_{k}\right]=\frac{1}{n}\sum_{k=1}^{n}E\left[\mathbf{X}_{k}\right]=\frac{1}{n}\cdot\left(n\mu\right)=\mu. $

$ Var\left[\mathbf{Y}_{n}\right] $

By the Chebyshev inequality,

$ p\left(\left\{ \left|\mathbf{Y}_{n}-\mu\right|\geq\epsilon\right\} \right)\leq\frac{\sigma^{2}}{n\epsilon^{2}}\longrightarrow\left(n\rightarrow\infty\right)\longrightarrow0. $

$ \Longrightarrow\mathbf{Y}_{n}\longrightarrow\left(p\right)\longrightarrow\mu\text{ as }n\longrightarrow\infty.\blacksquare $

Note

You can show this is true as long as the mean exists. The variance need not exist. Proof for this is harder and not responsible for this.

Note

There are also stronger forms of the law of large numbers. Strong one uses coveriance $ \left(a.e.\right) $ as well as weak one uses coveriance $ \left(p\right) $ .


Back to ECE600

Back to Sequences of Random Variables

Alumni Liaison

To all math majors: "Mathematics is a wonderfully rich subject."

Dr. Paul Garrett