Timeline for Does $X_{n}=o_{p}\left(Y_{n}\right)$ imply that $P\left(Y_{n}=0\right)=0$ for all $n$?
Current License: CC BY-SA 4.0
8 events
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| Nov 25, 2018 at 20:02 | comment | added | Emp Proc | @whuber That is very helpful. Thank you for all your help here and everywhere on stackexchange (I learn so much from your answers). As for what I mean, it is more that I have seen this notation elsewhere and assumed there was a well-defined notation. Now I understand that my confusion came from that there is no agreed-upon definition when $Y_n$ are random variables. Just understanding that is a big help to me. Now I know if I use that notation for random variables, I should first clearly define it. | |
| Nov 24, 2018 at 18:21 | comment | added | whuber♦ | Those remarks are not quite correct. $X_n/Y_n$ may have zero probability of being infinite even when $Y_n$ converges to zero. One simple example is the case $X_n=Y_n$ (where $\Pr(Y_n=0)=0$). Also, good accounts of measure theory permit random variables to have infinite values. (See Rudin's Real and Complex Analysis for instance.) But let's return to the heart of the matter: please explain what you mean by "$o(Y_n)$" when the $Y_n$ are random variables. | |
| Nov 24, 2018 at 15:38 | vote | accept | Emp Proc | ||
| Nov 24, 2018 at 1:04 | answer | added | guy | timeline score: 3 | |
| Nov 23, 2018 at 23:40 | history | edited | Emp Proc | CC BY-SA 4.0 |
Remove possible incorrect information (thank you to @whuber)
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| Nov 23, 2018 at 23:40 | comment | added | Emp Proc | Thank you very much for reply. First, I will delete the part about "obviously true", since I don't want to spread bad information. The reason why I think it is true is because if it does not go to 0 in the limit, then $\frac{X_n}{Y_n}$ has positive probability to be infinity, or undefined (if $X_n$ also is 0), which is not allowed for a random variable because random variables must as far as I know take real number values. | |
| Nov 23, 2018 at 20:11 | comment | added | whuber♦ | Since the little-o notation expresses something about a limiting value of a sequence, it makes no assertions at all concerning any finite set of values in the sequence. BTW, it isn't obvious that $\Pr(Y_n=0)\to 0:$ could you explain why that might be so? This might shed some light on what you mean by "$o_p(Y_n)$" when $Y_n$ is a sequence of random variables (as it seems to be here) instead of the usual sequence of positive real numbers, as is customary in the definition of this notation. | |
| Nov 23, 2018 at 19:35 | history | asked | Emp Proc | CC BY-SA 4.0 |