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Aug
19
comment $L^2(\mathbb{R})$ sequence such that $\sum_{n=1}^{\infty}\int_{\mathbb{R}}f_n(x)g(x)d\mu(x)=0$
I think this holds if it is sum for the L^2 norm converges, rather than their squares? Just apply c-s followed by minkowski?
Aug
13
comment What can you tell me about backward Brownian motion?
My supervisor is an expert on this but i do not understand any of this :(. Bm reversed in time is still bm, conditioning on hitting a point is like a brownian bridge... Do not get the difference from your description...
Aug
13
comment What space is the set of all CDFs?
@Math1000 you mean point wise maximum? clearly that does not work if you take [0,1] interval to be the sample space and a uniform distribution on [0,1/2] and [1/2,1] respectively...
Aug
13
comment What space is the set of all CDFs?
Surely this is not a vector space because a (non identical) multiple of a CDF is no longer a CDF?
Aug
11
comment Need help with an Elementary Math question
Does the second equation hold for every $x$? This is a poorly written question with no attempt.
Aug
11
comment Optimal algorithm for guessing random variable
@Tad see my comment above. I disagree.
Aug
11
comment Optimal algorithm for guessing random variable
@Aaron I do not think your guess is correct. Whilst the transformation can transform it into a uniform distribution, the expectation is now different? Note one is maximising the biggest expected pay off, not minimising the distance to the actual value.
Aug
11
comment Show that $\mathbb{E}\left(\int_0^1X_n(t)dW(t)-\int_0^1X(t)dW(t)\right)^2\to 0$
the mistake is that you should not integrate between 0 and 1, but between $(k/n, (k+1)/n)$ in sum of integral.
Aug
11
comment Show that $\mathbb{E}\left(\int_0^1X_n(t)dW(t)-\int_0^1X(t)dW(t)\right)^2\to 0$
@s1047857 I am pretty sure it is correct... Your derivation is wrong.
Aug
10
comment Power method for calculating dominant eigenvalue and eigenvector
Okay, thank you. I will read it through again.
Aug
10
comment Power method for calculating dominant eigenvalue and eigenvector
through a more careful reading of the theorem of the wikipedia page, it seems like the problem can only occurs if the matrix is complex, since for a real matrix, the dominant eigenvalue cannot be complex. The non convergence comes from the fact different subsequence converge to different eigenvectors.
Aug
10
comment Power method for calculating dominant eigenvalue and eigenvector
If the dominant eigenvector $v$, You mean as long as we pick an initial vector $u_0$. As long as $v.u_0 \neq 0$, this is convergent?
Aug
10
comment Power method for calculating dominant eigenvalue and eigenvector
en.wikipedia.org/wiki/Power_iteration but this suggests the sequence of vectors may not converge?
Aug
8
comment Optimal algorithm for guessing random variable
This problem falls in the class of Markovian optimal control problems. it is useful to look at Bellman's principle.
Aug
8
comment Optimal algorithm for guessing random variable
Hmm, this is not Markovian, but still...
Aug
8
comment Optimal algorithm for guessing random variable
The simple answer to this question is work backward. I imagine no analytical solution in general, like most optimal control problems.
Aug
3
comment derivative of a linear operator
thank you guys. This is from Zeidler's applied functional analysis. I imagine there is an errata somewhere...
Jul
8
comment Can I show that a process which a supermartingale above a certain value and a submartingale below it converges?
Limit must exists by martingale convergence theorem. The limit is almost sure and $L^1$
Jul
8
comment Are $n$-dimension cubes $C^k$ manifolds with boundaries?
@AlexS can you just confirm for me that my understanding for boundary is correct.
Jul
8
comment Are $n$-dimension cubes $C^k$ manifolds with boundaries?
@AlexS can you please provide me with a reference. My source is Wikipedia: en.wikipedia.org/wiki/…