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Apr
12
comment Limit w/o L'hopital
@black multiply the fraction by $$ \frac{\sqrt x + \sqrt c}{\sqrt x + \sqrt c} $$
Apr
7
comment $P^n$ transition matrix of a Markov chain
Not sure what you're asking right now. Maybe this will help: $P^n$ actually is the $n$-th power of the transition matrix $P$, only I am trying to get to it through a combinatorial argument, rather than attempting to multiply the matrix $n$-times.
Apr
7
comment $P^n$ transition matrix of a Markov chain
If I stay in state $2$, it means I put a ball inside one of the two boxes which were already nonempty, thus I changed nothing about the number of nonempty boxes and the transition to state $3$ still has the same probability. (Also: It is a homogenous Markov chain)
Apr
7
comment $P^n$ transition matrix of a Markov chain
To add to how $P$ was constructed: the first row, for example, says that with probability $1$ we'll go from state zero to state $1$ - that is, the ball has to be put somewhere, thus making one box nonempty. The second row says that we might hit the one nonempty box with probability $\frac 1k$ or an empty one with probability $\frac {k-1}{k}$
Apr
7
comment $P^n$ transition matrix of a Markov chain
Sorry, will try to add details to clarify.
Apr
6
comment Proving completeness of $L^p$
@Potato I was not aware of the definition, nor do I see the need for such definition, Am I missing something, or is this perhaps not needed if you start with the measure theoretic definition of $L^p$?
Apr
6
comment Proving completeness of $L^p$
The measure theoretic definition: $$ L^p(X,\mu):=\{f:X\to\mathbb R:\ \|f\|_{L^p}<\infty\},$$
Apr
6
comment Ito's Integral's definition: Importance of isometry
Ah, I see. This just shows how I lack confidence and intuition in a topic I know (so far) very little about. I think I understand both arguments. This one seems clearer, we just show that $\phi_n$ form a Cauchy sequence in a complete space, fine. So what was wrong/unrigorous with the previous approach?
Apr
5
comment Ito's Integral's definition: Importance of isometry
Thanks, now it's completely clear!
Apr
2
comment how to solve this limit
Hint: Try factoring out the $\sqrt{x}$ and use limit arithmetic
Apr
1
comment Markov chains: Condtitional independence implies independence?
Edited, hopefully making the question clearer
Apr
1
comment History and early development of Mathematics
Some help may also be found in these series of lectures (highly recommended, by the way!) youtube.com/playlist?list=PL34B589BE3014EAEB
Mar
31
comment Markov chains: Condtitional independence implies independence?
Not sure what you mean, but $T_i=\tau_j(i)-\tau_j(i-1)$ where $\tau_j(i)$ is the time of i-th return to the state $j$
Mar
31
comment Eigenvalues of nilpotent matrices
I will have to read a bit more on $\mathbb R^0$! Thanks.
Mar
30
comment Eigenvalues of nilpotent matrices
Point 2 is false for $k=0$? I didn't even know there could be $k=0$! What is a $0\times 0$ matrix? Also, what do you mean by "anything implies 1"?
Mar
28
comment Eigenvalues of nilpotent matrices
@Batman and the implication $\Leftarrow$ is in these two claims, too? I don't see that.
Mar
28
comment Eigenvalues of nilpotent matrices
and sorry for perhaps a bit nitpicky stream of claims to validate, but I just wanted to make sure my understanding is 100% correct
Mar
28
comment Eigenvalues of nilpotent matrices
added the obligatory question, to make things clear
Mar
27
comment How can a set be a subset of a $\sigma$-algebra
Just to clarify, if I wrote $S\subset 2^X$, it would be fine, right?
Mar
27
comment How can a set be a subset of a $\sigma$-algebra
Good. Everything alright with the world and the impostor syndrome is quiet again. Thanks!