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bio website math.wisc.edu/~janjigia
location Madison, WI
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visits member for 2 years, 7 months
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I am a graduate student in the probability group of the mathematics department at the University of Wisconsin - Madison.


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.
Jul
24
comment prove $\frac{e^{iuX_t}} {\mathrm{E}{[e^{iuX_t}]}}$ is martingale
Take a uniform random variable on the interval $[0,2\pi]$ or on the discrete set $\{0, \pi\}$ and set $u = 1$ for some simple examples. The exponential can have cancellations which sum to zero.
Jul
23
comment prove $\frac{e^{iuX_t}} {\mathrm{E}{[e^{iuX_t}]}}$ is martingale
Hint: for u sufficiently close to zero, the expectation in the denominator is bounded below and for all u, the expression in the numerator is bounded by $1$ in modulus. Notice also that it is possible that there exist u for which the expectation in the denominator is zero.
Jul
16
comment Do differentiable functions preserve measure zero sets? Measurable sets?
Ah sorry about that, you are right.
Jul
5
comment First hitting time Geometric Brownian motion
Hint: Taking logs, it suffices to find the distribution of the hitting time of a Brownian motion with drift (or the time that BM hits a line) math.stackexchange.com/questions/133628/…
Jun
30
comment If $X_{i}$ are I.I.D and $n^{-1}\sum_{i=1}^{n}X_{i}$ converges a.s/in-distribution to a constant $a$ is it true that $a=\mathbb{E}[X_{i}] $?
I think iid Cauchy random variables are a counterexample to your second "it is well-known" bullet. Are you sure you don't mean to have a limsup there?
Jun
20
comment Brownian Motion first hitting time distribution
You may want to re-read the question and make sure it isn't asking for this en.wikipedia.org/wiki/Inverse-gamma_distribution
Jun
14
comment Strange definition of Ergodicity
Take your sample space to be sequence space that you get for the joint law of all of the $X_i$ by using Kolmogorov's extension theorem and let your measure preserving transformation be the shift map taking you from coordinate $i$ to coordinate $i+1$. I think that this condition says that that shift map is ergodic in the usual definition. Proving that requires a bit of measure theory, since you need to show that the indicator of any invariant set is well approximated by a function of $k$ variables in the right sense. There may be some technical issues that make this a bit weaker though.
Jun
10
comment List of Advanced Math Text Books with answers
I have to disagree with this. There is more to an answer than a solution. You can learn a lot about how to approach problems and how to think about math by seeing how an expert in the field solves problem, even after writing up your own solutions. For example Don Passman used to write solutions to the algebra qualifying exams at Wisconsin. I learned more about algebra by solving old problems then looking at how he did it than I ever did by taking a class or reading a book. Most of my solutions were correct, but they were the wrong answer.
Jun
8
comment In Markov inequality proof, why is $\int_a^\infty xp(x) \, dx \ge \int_a^\infty ap(x) \, dx$
I added in bounds of integration to improve the formatting a bit.
Jun
3
comment conditional expectation of brownian motion
This really is a lovely paper, thank you for posting it.
May
31
comment Expression for $B_1$
There is a similar formula which you can obtain through an enlargement of filtration argument. That formula is $B_1 = W_1 + \int_0^1 \frac{B_1 - B_t}{1-t}dt$, where $W_t$ is Brownian motion with respect to the filtration of $B_t$ enlarged by $\sigma(B_1)$. I do not know whether or not that is what you are looking for.
May
27
comment Proof of a theorem about Baire categories
@Demons94 sorry for the bad hint.
May
27
comment Proof of a theorem about Baire categories
@DaveL.Renfro Yes thanks, I definitely did not think that through carefully.
May
19
comment Defining the scale function of a diffusion process
I suppose I should have said this in my previous comment. The scale function is not defined at zero for this process. If you want to define it consistently as a limit of $s(\epsilon)$ where $\epsilon \downarrow 0$ then the value is $\infty$, not $0$. Notice that if you formally assume $\frac{1}{\infty} = 0$ and $\frac{\infty}{\infty} = 1$, this still gives you the correct answer for this problem, which is that $P(\tau_0 < \tau_5) = 0$ and $P(\tau_5 < \tau_0) = 1$.
May
19
comment Frechet/Gateaux differentiability of an integral operator L^2 --> R
I don't think that this operator is well defined in general. Take a function which is just barely in $L^2$ and make it so that in any neighborhood of zero it is singular like $x^{-\frac{1}{2} + \epsilon}$ with terms that are both positive and negative. When you cube this function ($f(x) = x^3$), the positive part and the negative part will both be infinite, so the functional isn't well defined.