The law of Large number

Created
Created
2023 Mar 7 2:41
Editor
Creator
Creator
Seonglae Cho
Edited
Edited
2024 Dec 5 14:22

LLN

When
iid
, which E[X1]==μ\mathbb{E}[X_1] = \dots = \mu
Xˉn:=1ni=1nXinμ\bar{X}_n := \frac{1}{n} \sum_{i=1}^n X_i \xrightarrow{n \to \infty} \muVar(Xˉn)=σ2nVar(\bar{X}_n) = \frac{\sigma^2}{n}

Condition

Basically requires
iid
however
Ergodicity
or

Variance

V(f^)=V(1mi=1mf(Xi))=1m2V(i=1mf(Xi))\mathbb{V}(\hat{f}) = \mathbb{V}(\frac{1}{m}\sum_{i=1}^mf(X_i)) \\ = \frac{1}{m^2} \mathbb{V}(\sum_{i=1}^mf(X_i))
Since it is
iid
=1m2mV(f(X))=1mV(f(X)) = \frac{1}{m^2} m \mathbb{V}(f(X)) \\ = \frac{1}{m} \mathbb{V}(f(X))
In other words, as mm\rightarrow \infty, the variance goes to zero.
 
 
 
 

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