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2013/10/15

weak law of large numbers

Let random variablesX_1,X_2,\cdots,X_n be i.i.d. and any E(X_i)=\mu<\infty. If we put
Y_n=\frac{1}{n}\sum_{i=1}^nX_i
then,
P(\lim_{n\rightarrow\infty}Y_n=\mu)=1
It is called strong law of large numbers.

On the other hand, weak law of large numbers has a weaker assertion in which Y_n approches \mu than strong law of large numbers. It says, for any \epsilon>0,
P(\lim_{n\rightarrow\infty}|Y_n-\mu|>\epsilon )=0\quad 
Weak law of large numbers does not say that Y_n approaches \mu with probability 1.

On mathematical terms, strong law of large numbers is a almost everywhere convergence. Weak law of large numbers is a convergence in probability. Therefore, in the random variables, if strong law of large numbers is true, then weak law of large numbers is true. However the converse is not true.


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