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bio website math.wisc.edu/~janjigia
location Madison, WI
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visits member for 2 years, 9 months
seen 4 hours ago

I am a graduate student in the probability group of the mathematics department at the University of Wisconsin - Madison.


4h
comment how to related a weakly convergent random variable with its k-th moment
Whenever you have convergence in distribution of real random variables, you can switch probability spaces and make it almost sure convergence. This is called the Skorokhod representation theorem. Your question is then whether or not convergence a.s. implies convergence of integrals, which is not true in general. A necessary and sufficient condition for $L^k$ convergence (which implies what you want) is uniform integrability. This is slightly stronger than just having convergence of the expectations though.
4h
comment how to related a weakly convergent random variable with its k-th moment
This result holds if the family $\{S_n^k\}_{n=1}^\infty$ is uniformly integrable.
Oct
17
comment Spotting mistake: unnecessary given condition
@PatrickDaSilva This is pretty fundamentally different from your example. Preference relations are the first topic covered in a first course in graduate microeconomics using the standard textbook (MWG). I've never heard of any physics program that starts with elliptic curves.
Oct
2
comment Can $\|f\|_p\to\infty$ arbitrarily slowly? (Looking for hints.)
This is a great classical analysis question. Hints: You may assume without loss of generality $\Phi(p)$ is as regular as you like (continuous for example). Consider a step function on a disjoint partition of $(0,1)$ and choose your coefficients and the measure of the sets in the partition carefully.
Sep
28
reviewed Close Can I use theorems that are beyond the scope of my class in my CBSE exams?
Sep
28
reviewed Close Is the product of i.i.d integrable random variables also integrable?
Sep
28
reviewed Leave Open Prove expression defines an inner product
Sep
24
awarded  Autobiographer
Sep
19
reviewed Close Has the abc conjuncture been proved by Shinichi Mochizuki?
Sep
5
comment CLT - infinite variance
I haven't thought much about this, but how do you know that the histogram is normal? Is it just the general shape or have you tried testing your samples with normality tests?
Aug
31
comment If $f(\mathbb{C})\subset \mathbb{C}-[0,1]$ then $f$ is constant
Hint: Think about the codomain of the entire function $\frac{1}{f(z)}$.
Aug
20
reviewed Leave Open Convergence of infinite series of complex numbers
Aug
19
reviewed Close why is (a proof of 1+1=2) necessary?
Aug
19
reviewed Close Integration of $e^{\cos x}\cos x$
Aug
19
reviewed Close When is Complex Normal Distribution equal to Normal distribution for real numbers
Aug
12
comment Show $\lim\limits_{n\to\infty}\mathbf E(f(X_n)g(Y))=\mathbf E(f(X)g(Y))$
I do not understand the second paragraph. Why is it clear that the limit as $R \to \infty$ of the supremum is zero? At some point, you need to use the fact that the hypotheses give you tightness, otherwise all you have from using compactly supported functions as your test functions is vague convergence.
Aug
10
comment How to talk about a random variable which only exists on an event
Have your new random variable be whatever you want on the nice set and be equal to the thing you are trying to bound off of it. That way it's defined everywhere and serves as a lower bound everywhere.
Aug
7
comment How to talk about a random variable which only exists on an event
The definition of a random variable is that it is a measurable function, so in particular it must be a function. That means you have to define it on the whole space.
Aug
2
comment The projective limit of probability spaces and the Kolmogorov-Daniell theorem
I think that the answer to your last question is that that is a peculiarity of the particular formulation of the extension theorem that you saw. Check the statement in Volume II of Bogachev's measure theory for example. The only thing you need for the proof to work is inner regularity (where the usual definition of compactness is replaced with the finite intersection property), which can be done in a purely measure theoretic framework.
Jul
31
comment Non-Probabilistic Argument for Divergence of the Simple Random Walk
You need an expectation and an absolute value in the statement, since $\frac{S_n}{\sqrt{n}}$ almost surely does not converge.