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bio website math.wisc.edu/~janjigia
location Madison, WI
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visits member for 2 years, 9 months
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I am a graduate student in the probability group of the mathematics department at the University of Wisconsin - Madison.


23h
comment The ito integral is gaussian
The distributional limit normally distributed random variables is (possibly degenerate) normal. See math.stackexchange.com/questions/232540/…
Oct
23
comment In frequentism, does every event have a probability?
Not measurable with respect to what? Probability is not analysis. When you specify a sigma algebra on your underlying space in probability, you are saying something about what events are accessible to your experiment.
Oct
23
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.
Oct
23
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
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
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.