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7 votes
1 answer
1k views

Properties of convolutions

Consider the function $$f_{n}(x)=e^{-x^2}x^n.$$ and the function $$h_p(x):=e^{-\vert x \vert^p}.$$ My goal is to analyze $$ F_p(y):=\frac{(f_2*h_p)(y)}{(f_0*h_p)(y)}- \left(\frac{(f_1*h_p)(y) }{(f_0*...
Landauer's user avatar
  • 173
1 vote
2 answers
194 views

Continuity of the densities of a stochastic process

Let $X=(X_t)_{t\in I}$ ($I\subset\mathbb{R}$ an interval) be a stochastic process with continuous sample paths and such that $X_t$ admits a continuous Lebesgue density $\chi_t\in C(\mathbb{R}^d)$ for ...
fsp-b's user avatar
  • 463
0 votes
1 answer
557 views

Is the limsup or liminf of n-wise independent events independent?

Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space. Consider events indexed by $m, n \in \mathbb N$: $ \ \ \ \ \ \ \ \ \ \ \ A_{1,n}, A_{2,n}, A_{3,n} ...$ are n-wise independent. $A_{m,1}...
BCLC's user avatar
  • 247
32 votes
4 answers
4k views

Is a random subset of the real numbers non-measurable? Is the set of measurable sets measurable?

One might say, "a random subset of $\mathbb{R}$ is not Lebesgue measurable" without really thinking about it. But if we unpack the standard definitions of all those terms (and work in ZFC), it's not ...
Gene S. Kopp's user avatar
  • 2,200
9 votes
1 answer
950 views

Sort-of converse of Kolmogorov zero-one theorem

Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space. The Kolmogorov zero-one theorem states that Suppose we have independent random variables $X_1, X_2, ...$. Then $\forall \ A \in \bigcap_n ...
BCLC's user avatar
  • 247
6 votes
1 answer
1k views

About the generating structure of Borel field

This is a graduate-level measure theory problem. I have thought throught it and asked on math.SE but received no satisfying answer. On P.32 of [P.Billingsley] Probability and Measure, 3ed, 1993, the ...
Henry.L's user avatar
  • 8,071
3 votes
1 answer
304 views

Question abouth Skorokhod representation of random variables

It is known that for any two probability measures $\mu$ and $\nu$ on $\mathbb R$ that are close in the Prokhorov metric $\rho$, i.e. $$\rho(\mu,\nu)<\varepsilon,$$ then there exist two random ...
CodeGolf's user avatar
  • 1,835
3 votes
0 answers
237 views

Reference request: Darboux properties of real-valued set functions (measures, densities, etc.)

Fix a set $S$ and let $f: \mathcal P(S) \rightharpoonup \mathbf R$ be a real-valued partial function on the power set of $S$; denote by $\mathcal D$ the domain of $f$. We say that $f$ has: (i) the ...
Salvo Tringali's user avatar
2 votes
1 answer
144 views

Do we have independence if we let the indices of the events increase?

Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space. Consider events indexed by $m, n \in \mathbb N$: $ \ \ \ \ \ \ \ \ \ \ \ A_{1,n}, A_{2,n}, A_{3,n} ...$ are n-wise independent. $A_{m,1}...
BCLC's user avatar
  • 247
2 votes
1 answer
450 views

Show that the absolute value of this function is twice differentiable except on a set of Lebesgue measure $0$

Let $f\in C^3(\mathbb R)$ with $f>0$ and $$\int f(x)\:{\rm d}x=1\tag1$$ $g:=\ln f$ and assume that $g'=\frac{f'}f$ is Lipschitz continuous (note that this implies that $f'(x)\xrightarrow{|x|\to\...
0xbadf00d's user avatar
  • 167
2 votes
1 answer
437 views

If $g$ is differentiable, how can we show that $z\mapsto1\wedge e^{g(z)}$ is differentiable except on a countable set

If $g:\mathbb R\to\mathbb R$ is differentiable, how can we show that $$h(z):=\min\left(1,e^{g(z)}\right)\;\;\;\text{for }z\in\mathbb R$$ is also differentiable, except at a countable number of points, ...
0xbadf00d's user avatar
  • 167
1 vote
0 answers
416 views

When does a proper Zariski closed set have measure zero with respect to a conditional measure?

Assume we have a probability measure $\mu$ over $\mathbb{R}^d$ that is absolutely continuous with respect to Lebesgue measure. Given $m$ polynomials $p_1,\ldots,p_{m}\in \mathbb{R}[x_1,\ldots,x_d]$ ...
Ron's user avatar
  • 61
1 vote
1 answer
166 views

Question abouth Skorokhod representation of random variables (II)

This is a continuation of Question abouth Skorokhod representation of random variables Let $\mu$ and $\nu$ be two probability measures on $\mathbb R$ such that $$\int_{\mathbb R}|x|^pd\mu(x),~ \...
CodeGolf's user avatar
  • 1,835
1 vote
1 answer
632 views

Does sequence almost sure convergence imply almost sure convergence?

This is a cross-post of this and this questions from math.stackexchange.com since I have not received any response there. I would like to seek help here. Suppose $x(t,\omega): [0,T]\times\Omega\...
Hans's user avatar
  • 2,239