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Let $\; X_0,X_1,X_2,X_3,...\;$ be independent and identically distributed (real-valued) random variables.

1. Suppose $\frac1n \cdot\sum\limits_{m=0}^n X_m$ converges in probability. Does it follow that $\operatorname{E}(X_0)$ exists?

2. Suppose $\operatorname{E}(X_0) = 0$ and that $\frac1{\sqrt n} \cdot\sum\limits_{m=0}^n X_m$ converges in distribution to a normal random variable. Does it follow that $\operatorname{E}((X_0)^2)$ is finite?

(I already found that the converse of the strong law of large numbers holds.)

edited Aug 25 2011 at 12:49

Didier Piau
4,193●1●10●15

asked Aug 25 2011 at 9:57

Ricky Demer
6,411●8●31

a remark. The weak law fails for the Cauchy distribution. – Gerald Edgar Aug 25 2011 at 12:01

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A classical example for 1'. is a symmetric integer-valued X with P(X=n)=P(X=-n)=c/(n^2log(n)). Then phi is C^1 but X is not integrable. On the other hand, if phi is C^2 then X^2 is integrable. – Didier Piau Aug 25 2011 at 12:44

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Didier, this gives a counterexample to 1, right? I think the last line in the question means that if one replaces, in 1, convergence in probability by a.s. convergence, then the answer is yes (by say the converse to Borel-Cantelli). – Mikael de la Salle Aug 25 2011 at 12:59

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Necessary and sufficient conditions (in terms close to those you want) for the WLLN and the CLT can be found, e.g., in "Foundations of modern probability" by Kallenberg (Theorems 4.16 and 4.17). – Yvan Velenik Aug 25 2011 at 14:09

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@Ricky in the 2nd Edition it's Theorems 5.16 and 5.17 – pgassiat Aug 25 2011 at 23:48

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Yvan Velenik · Accepted Answer · 2011-08-27 08:06:36Z UTC

(As suggested, I promote my comment to an answer, with pgassiat's complement.)

Necessary and sufficient conditions (in terms close to those you want) for the WLLN and the CLT can be found, e.g., in "Foundations of modern probability" by Kallenberg (Theorems 4.16 and 4.17 in the first edition, Theorems 5.16 and 5.17 in the second edition).

Do the converses of [weak law of large numbers / central limit theorem] hold?

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