Timeline for A "random variable" with infinite value
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
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| Feb 15, 2011 at 19:04 | comment | added | Did | @Bogdan: You are welcome. Some keywords here are end-compactification and Freudenthal compactification. (And one could use $\delta(x)=x/(1+|x|)$, which identifies $E$ with $[-1,1]$.) | |
| Feb 15, 2011 at 18:12 | comment | added | Bogdan | Yes, clearly, with such a metric everything becomes simple. I did not think about extended real line as about metric space. Now I need to think if this (or similar) metric is natural for my application. Its a little bit unnatural that some negative numbers are "closer" to $-\infty$ than to, say, $0$... Anyway, thank you! | |
| Feb 15, 2011 at 16:59 | comment | added | Did | .../... positive $u$. (Hence yes, $X_n=-n$ does converge, almost surely and in probability and in distribution, to $X=-\infty$.) The continuous mapping theorem is still valid, naturally, if the continuity hypothesis refers to the metric $d$, as it should. And so on. As far as convergences in probability, in distribution or almost sure are concerned, everything works fine. But expectations and convergences in $L^p$ are another matter, of course. Books about probability measures on metric spaces would explain this, for example Billingsley's classic Convergence of Probability Measures. | |
| Feb 15, 2011 at 16:58 | comment | added | Did | @Bogdan: OK, let me expand on my previous answer. Consider the distance $d$ on $E=\mathbb{R}\cup\\{-\infty,+\infty\\}$ defined by $d(x,y)=|\arctan(x)-\arctan(y)|$ if $x$ and $y$ are both in $\mathbb{R}$ and extended in the obvious way to $E\times E$ (or, more generally, $d(x,y)=|\delta(x)-\delta(y)|$ for any bounded increasing function $\delta:\mathbb{R}\to\mathbb{R}$). Then $(E,d)$ is a separable metric space hence (even according to the wikipedia page you refer to), the convergence of $(X_n)$ to $X$ in probability is defined by the condition that $P(d(X_n,X)\ge u) \to 0$ for every .../... | |
| Feb 15, 2011 at 14:53 | comment | added | Bogdan | For example, sequence of constant random variables $X_n=-n$ does not converge in Probability to constant $X=-\infty$ (at least with definition in wikipedia),but $Y_n=arctan(X_n)$ converges in probability to $Y=arctan(X)=-\pi/2$. So, this trick already does not help with all theorems involving convergence in probability. Continuous mapping theorem states that $X_n -> X$ implies $g(X_n) -> g(X)$ for cont. function g(.) on metric space, but it is not applicable because $R\cup{-\infty}$ is not a metric space. So, many theorems stop working! Rather than check them one by one, I would want a book.. | |
| Feb 15, 2011 at 13:37 | comment | added | Did | @Bogdan: Take your favorite probability textbook and replace every random variable $X$ with values in $\mathbb{R}\cup\{-\infty,+\infty\}$ you are interested in, by $Y=\arctan(X)$. Then $Y$ takes values in $[-\pi/2,+\pi/2]$, hence you are back to a setting which is even less general than the one needed for real valued random variables, and the convergence theorems of your textbook hold for your $Y$ random variables. | |
| Feb 15, 2011 at 13:02 | comment | added | Bogdan | Thank you for answer. I would prefer to work with $R\cup\{-\infty}$ only - this is also a measureable space. But I would not say that I am in standard probability theory - most books that I know works with real-valued r.v. Can you recommend a good book where all the standard probability results are proved in this more general settings? For example, here en.wikipedia.org/wiki/Convergence_of_random_variables there are many theorems (like Portmanteau lemma) about convergence of r.v.s - are they valid for r.v. from $\Omega$ to E? | |
| Feb 15, 2011 at 10:51 | history | answered | Did | CC BY-SA 2.5 |