Timeline for Law of large numbers for stochastically chosen samples

8 events

when toggle format	what		by	license	comment
Nov 11, 2011 at 23:29	vote	accept	pharms
Nov 11, 2011 at 23:49
Nov 11, 2011 at 4:22	history	edited	John Jiang	CC BY-SA 3.0	added 409 characters in body
Nov 10, 2011 at 21:41	comment	added	pharms		No, that is not true. $X_{\sigma(1)},\ldots,X_\sigma(N)$ are not i.i.d given $\sigma(1),\ldots,\sigma(N)$. However, Ori Gurel-Gurevich's claim seems plausible to me. As an example for when your statement does not hold, consider conditioning on $\sigma(1)=1,\sigma(2)=2$. This implies $k_1=k_2=1$. Conditioning on $k_1$ does not have an influence because $k_1$ is deterministic. Conditioning on $k_2$ has no influence on the distribution of $X_2$, but it can have an influence on the distribution of $X_1$ because $k_2$ can be a function of $X_1$.
Nov 10, 2011 at 2:35	comment	added	John Jiang		See my updated solution above based on Ori's suggestion.
Nov 10, 2011 at 2:34	history	edited	John Jiang	CC BY-SA 3.0	added 1045 characters in body
Nov 9, 2011 at 23:31	comment	added	pharms		Thank you for your answer. I understand what you say. To use the Chebyshev inequality, I need to establish $var(Y_T/N_T) \to 0$. The problem with this is that $N_T$ now is a random variable, and that it is correlated with $Y_T$. Can you help me once more with this? PS: I sticked to my own notation in this comment, and I updated my question to make it more concise.
Nov 9, 2011 at 18:57	history	edited	John Jiang	CC BY-SA 3.0	added 98 characters in body
Nov 9, 2011 at 18:44	history	answered	John Jiang	CC BY-SA 3.0