Hi(adsbygoogle = window.adsbygoogle || []).push({});

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.

**Physics Forums - The Fusion of Science and Community**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Weak Law of Large Numbers versus Central Limit Theorem

Loading...

Similar Threads - Weak Large Numbers | Date |
---|---|

(weak) Law of Large Numbers and variance | Apr 27, 2012 |

Understanding the weak norm and it's notation | Apr 5, 2012 |

Weak convergence of the sum of dependent variables, question | Dec 13, 2011 |

What is the difference between weak and strong large number laws? | Nov 5, 2011 |

Weak limit of abs. continuous measures | Sep 23, 2011 |

**Physics Forums - The Fusion of Science and Community**