All Questions
Tagged with convolution pr.probability
27
questions
1
vote
1
answer
102
views
Small total variation distance between sums of random variables in finite Abelian group implies close to uniform?
Let $\mathbb{G} = \mathbb{Z}/p\mathbb{Z}$ (where $p$ is a prime). Let $X,Y,Z$ be independent random variables in $\mathbb G$.
For a small $\epsilon$ we have $\operatorname{dist}_{TV}(X+Y,Z+Y)<\...
- 23
4
votes
1
answer
361
views
Sufficient condition for a probability distribution on $\mathbb Z_p$ to admit a square-root w.r.t convolution
Let $p \ge 2$ be a positive integer, and let $Q \in \mathcal P(\mathbb Z_p)$ be a probability distribution on $\mathbb Z_p$.
Question. What are necessary and sufficient conditions on $Q$ to ensure ...
- 6,824
3
votes
1
answer
189
views
Is there a real/functional analytic proof of Cramér–Lévy theorem?
In the book Gaussian Measures in Finite and Infinite Dimensions by Stroock, there is a theorem with a comment
The following remarkable theorem was discovered by Cramér and Lévy. So far as I know, ...
- 647
4
votes
1
answer
224
views
Just how regular are the sample paths of 1D white noise smoothed with a Gaussian kernel?
Adapted from math stack exchange.
Background: the prototypical example of---and way to generate---smooth noise is by convolving a one-dimensional white noise process with a Gaussian kernel.
My ...
- 203
3
votes
1
answer
142
views
Convolution between normal distribution and the maximum over $m$ Gaussian draws
$\DeclareMathOperator\erf{erf}$
Let's consider the Gaussian distribution $P_X(x)= \frac{1}{\sqrt{2 \pi \sigma^2}} e^{- \frac{x^2}{2 \sigma^2}}$. Now consider the random variable $W \equiv \max \{ X_1, ...
- 185
0
votes
1
answer
288
views
When can a convolution be written as a change of variables?
Suppose $X$ is a random variable with a density $f(x)$ such that $f(x)$ is a convolution of some density $g$ with some other density $q$:
$$
f = g\ast q.
$$
Under what conditions does $X=h(Y)$, where $...
- 5
4
votes
1
answer
312
views
2-Wasserstein metric on convolution of probability distributions
I have two related questions. Let $\mu$ and $\nu$ be two distinct probability measures on $\mathbb{R}^n$ with finite second moments, and $W_2(\cdot,\cdot)$ be the $2$-Wasserstein metric. The question ...
- 161
2
votes
1
answer
104
views
Equivalent of a local limit theorem in the large deviation region and asymptotics of a convolution operator
Let $\{X_i \}_{i \in \mathbb{N}}$ be a sequence of i.i.d. random variables satisfying $\mathbb{E} X_1 = 0$ and $\mathbb{E} X_1 ^2 < \infty$. Assume that $\{S_n ��\}_{n \in \mathbb{N}}$ is a non-...
- 714
4
votes
1
answer
333
views
Recovering a function from its Gaussian convolution
Let $\varphi(x)=\frac{1}{\sqrt{2\pi}}\exp(-x^2/2)$ be the Gaussian density and
$f:\mathbb{R}\to\mathbb{R}$ another measurable function.
Under what conditions can $f$ be recovered from its convolution ...
- 43
2
votes
1
answer
319
views
Is $g(v)=\mathbb{E}[f(v+W)]$ a differentiable function of $v$ when $f$ is continuous and $W$ is multivariate normal?
Suppose $f$ is a continuous function on $\mathbb{R}^n$, and $W$ has a multivariate normal distribution on $\mathbb{R}^n$. If the expectation
$$g(v)=\mathbb{E}[f(v+W)]$$
is defined for all $v \in \...
user44143
3
votes
1
answer
593
views
Can we show that the characteristic function of an infinitely divisible probability measure has no zeros
Let $E$ be a normed $\mathbb R$-vector space, $\mu$ be a probability measure on $\mathcal B(E)$ and $\varphi_\mu$ denote the characteristic function$^1$ of $\mu$.
Assume $\mu$ is infinitely divisible, ...
- 157
2
votes
1
answer
303
views
Uniqueness of deconvolution after convolution?
I have the following question and I'd greatly appreciate any help!
Basically, I have an arbitrary probability distribution with pdf $f(x)$, we can assume it's continuous with support on $[0,\infty]$
...
- 31
0
votes
1
answer
133
views
How can we show this estimate for the convolution of two probability measures?
Let $(\delta_k)_{k\in\mathbb N}\subseteq(0,\infty)$ be nonincreasing with $\delta_k\xrightarrow{k\to\infty}0$ and $(\varepsilon_k)_{k\in\mathbb N}\subseteq(0,\infty)$ with $\sum_{k\in\mathbb N}\...
- 157
2
votes
2
answers
310
views
If $(\exp(\mu_n))_{n\in\mathbb N}$ is weakly convergent, is the normalized sequence convergent as well?
Let $E$ be a metric space and $\mathcal M(E)$ denoote the space of finite signed measures on $\mathcal B(E)$ equipped with the total variation norm $\left\|\;\cdot\;\right\|$.
I would like to know ...
- 157
6
votes
2
answers
779
views
Which random variables can be written as the difference of two independent positive random variables?
Can we characterize random variables $X$ that satisfy
$$
X\sim Y - Z
$$
for two independent positive random variables $Y$ and $Z$?
Are $Y$ and $Z$ unique in some sense?
Can (one possible choice of) $Y$...
- 292