Timeline for Do the converses of [weak law of large numbers / central limit theorem] hold?
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16 events
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| Aug 28, 2011 at 8:37 | vote | accept | CommunityBot | ||
| Aug 27, 2011 at 8:06 | answer | added | Yvan Velenik | timeline score: 8 | |
| Aug 26, 2011 at 6:48 | comment | added | Yvan Velenik | @pgassiat: Yes, thank you. I hadn't realized that there was a newer edition ;) . | |
| Aug 25, 2011 at 23:48 | comment | added | pgassiat | @Ricky in the 2nd Edition it's Theorems 5.16 and 5.17 | |
| Aug 25, 2011 at 20:09 | comment | added | user5810 | @Yvan: Am I missing something, or do those not actually address either of my questions? | |
| Aug 25, 2011 at 17:26 | comment | added | user5810 | Oh, yeah. I was confusing that with the moment-generating function. | |
| Aug 25, 2011 at 17:19 | comment | added | Mikael de la Salle | @Ricky: the characteristic function of a real random variable $X$ is $\phi(t)=E(e^{itX})$, and is defined for any real $t$. | |
| Aug 25, 2011 at 15:24 | comment | added | Did | @Mikael, I did not write counterexample because 1. asks a question. But you are right, the example I recalled proves that the answer to 1. is "no". | |
| Aug 25, 2011 at 14:09 | comment | added | Yvan Velenik | 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). | |
| Aug 25, 2011 at 12:59 | comment | added | Mikael de la Salle | 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). | |
| Aug 25, 2011 at 12:49 | history | edited | Did | CC BY-SA 3.0 |
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| Aug 25, 2011 at 12:46 | comment | added | Did | Ricky, does the last line of your post mean that you have a proof of 1.? | |
| Aug 25, 2011 at 12:44 | comment | added | Did | 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. | |
| Aug 25, 2011 at 12:29 | comment | added | Mikael de la Salle | A remark: your question can be also rephrased in the following way. Let $\varphi$ be the characteristic fonction of $X_0$, and assume for simplicity that $X_0$ is symmetric. 1': If $\varphi(t) = 1+ o(t)$ as $t\to 0$, does it follow that $E(X_0)$ exists (and is zero)? 2' If $\varphi(t)=1- t^2/2 + o(t^2)$, does it follow that $E(X_0^2)$ exists (and is $1$)? | |
| Aug 25, 2011 at 12:01 | comment | added | Gerald Edgar | a remark. The weak law fails for the Cauchy distribution. | |
| Aug 25, 2011 at 9:57 | history | asked | user5810 | CC BY-SA 3.0 |