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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
23
reviewed Close Find $dy/dx$ where $(7x+2y)^2=6x^4y^3$
Jul
16
comment Do differentiable functions preserve measure zero sets? Measurable sets?
Ah sorry about that, you are right.
Jul
6
reviewed Leave Open A year having more than one Friday the 13th?
Jul
3
reviewed Close For a real valued function $f(x,y)$ on $\mathbb{R}^2$ which is in $L_2$, show that $f(x+ε,y+ε) → f(x,y)$ in $L_2$ when $ε → 0.$
Jul
3
reviewed Leave Open Extending holomorphic function to neighborhood of square
Jul
3
revised how to prove Exponential is strictly positive?
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Jul
3
revised How to prove Exponential is monotone?
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Jul
2
awarded  Curious
Jul
1
reviewed Close Solve logarithmic equation: $2\log_7 (x+2) - \log_7 (3x+10) = 0$
Jul
1
reviewed Close Prove that the norm of $E$ is generate by the inner product $\langle x,y \rangle =\frac{1}{4}\left(||x+y|^2-||x-y||^2\right)$
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.