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maverick280857

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Hi

I was studying the WLLN and the CLT. A form of WLLN states that if [itex]X_{n}[/itex] is a sequence of random variables, it satisfies WLLN if there exist sequences [itex]a_{n}[/itex] and [itex]b_{n}[/itex] such that [itex]b_{n}[/itex] is positive and increasing to infinity such that

[tex]\frac{S_{n}-a_{n}}{b_{n}} \rightarrow 0[/tex]

[convergence in probability and hence convergence in law] where [itex]S_{n} = \sum_{i=1}^{n}X_{i}[/itex]. For now, suppose the random variables are independent and identically distributed and also have finite variance [itex]\sigma^2[/itex].

The Lindeberg Levy Central Limit Theorem states that

[tex]\frac{S_{n}-E(S_{n})}{\sqrt{Var(S_{n})}} \rightarrow N(0,1)[/tex]

[convergence in law]

Now, if we take [itex]a_{n} = E(S_{n})[/itex] and [itex]b_{n} = \sqrt{Var(S_{n})} = \sigma\sqrt{n}[/itex], conditions of both the theorems are satisfied. But, the limiting random variables are different. In the first case, the normalized random variable tends to a random variable degenerate at 0 (in law/distribution) whereas using CLT, it tends to a Normally distributed random variable with mean 0 and variance 1.

Does this mean that convergence in law is not unique? What are the implications of these results?

Thanks.

I was studying the WLLN and the CLT. A form of WLLN states that if [itex]X_{n}[/itex] is a sequence of random variables, it satisfies WLLN if there exist sequences [itex]a_{n}[/itex] and [itex]b_{n}[/itex] such that [itex]b_{n}[/itex] is positive and increasing to infinity such that

[tex]\frac{S_{n}-a_{n}}{b_{n}} \rightarrow 0[/tex]

[convergence in probability and hence convergence in law] where [itex]S_{n} = \sum_{i=1}^{n}X_{i}[/itex]. For now, suppose the random variables are independent and identically distributed and also have finite variance [itex]\sigma^2[/itex].

The Lindeberg Levy Central Limit Theorem states that

[tex]\frac{S_{n}-E(S_{n})}{\sqrt{Var(S_{n})}} \rightarrow N(0,1)[/tex]

[convergence in law]

Now, if we take [itex]a_{n} = E(S_{n})[/itex] and [itex]b_{n} = \sqrt{Var(S_{n})} = \sigma\sqrt{n}[/itex], conditions of both the theorems are satisfied. But, the limiting random variables are different. In the first case, the normalized random variable tends to a random variable degenerate at 0 (in law/distribution) whereas using CLT, it tends to a Normally distributed random variable with mean 0 and variance 1.

Does this mean that convergence in law is not unique? What are the implications of these results?

Thanks.

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