**4**

votes

**1**answer

77 views

### What class of probability distributions do probabilistic turing machines induce? [closed]

What class of probability distributions is induced by the class of probabilistic turing machines? Is there a precise characterization?

**0**

votes

**0**answers

49 views

### methods to analyze martingale conditioned on return in the future

Consider a martingale $S_t$ on $\mathbb{Z}$ starting from 0. Assume that for any $t$, $Var[s_t\, | \, \mathcal{F}_{t-1}] < V$, where $V$ is some positive constant. Fix an $n$ and for $t \leq n$, ...

**10**

votes

**1**answer

296 views

### Extension of Dynkin's formula, conclude that process is a martingale

This question was asked here, but it did not get enough attention, so I'm crossposting it to MO.
Let $u: \mathbb{R}_+ \times \mathbb{R}^d$ be a bounded $C^2$ function whose first and second partial ...

**4**

votes

**0**answers

78 views

### Concluding that the Poisson kernel is indeed the Cauchy distribution?

See here.
Let $d = 2$, and consider the domain $D = \mathbb{H}$, the upper half-plane. Let $W_t = (X_t, Y_t)$. We see that for any $\theta \in \mathbb{R}$ and any $t \ge 0$, we have$$E^{(x, ...

**3**

votes

**1**answer

138 views

### Poisson kernel, expectation, an absolute value comes in

See here.
Let $d = 2$, and consider the domain $D = \mathbb{H}$, the upper half-plane. Let $W_t = (X_t, Y_t)$. We see that for any $\theta \in \mathbb{R}$ and any $t \ge 0$, we have$$E^{(x, ...

**2**

votes

**1**answer

119 views

### Poisson kernel, $E^{(x, y)}\text{exp}\{i\theta X_t - \theta Y_t\} = e^{i\theta x - \theta y}$

Let $d = 2$, and consider the domain $D = \mathbb{H}$, the upper half-plane. Let $W_t = (X_t, Y_t)$. How do I see that for any $\theta \in \mathbb{R}$ and any $t \ge 0$, we have$$E^{(x, ...

**8**

votes

**2**answers

270 views

### Distribution of $\max_{n \ge 0} S_n$, random walk

Say we have a random walk that is a nearest neighbor random walk on the integers where at each step the probability of moving one step to the right is $p$ and the probability of moving one step to the ...

**3**

votes

**1**answer

116 views

### Is the set of multiple points of the Brownian path $W[0, \infty)$ dense in the plane almost surely?

Let $d = 2$. With probability $1$, is the set of multiple points of the Brownian path $W[0, \infty)$ dense in the plane?

**1**

vote

**1**answer

144 views

### Formula for maximum of two Gumbel distributions?

I have parameters of two Gumbel distributions ($\mu_1, \beta_1)$ and $(\mu_2, \beta_2)$. Since max of 2 Gumbels is a Gumbel, I'd like to compute $\mu_m, \beta_m$, so that:
$Gumbel(\mu_m,\beta_m)$ = ...

**2**

votes

**1**answer

128 views

### Poisson kernel is the Cauchy distribution, reference?

Let $d = 2$, and consider the domain $D = \mathbb{H}$, the upper half-plane. Can someone give me a reference to a proof that the Poisson kernel is the Cauchy distribution?

**1**

vote

**0**answers

29 views

### Importance sampling for bernoulli-sequence, favouring long sequences of ones

Assume we have a sequence of i.i.d. bernoulli-distributed random variables of length $n$.
I'm interested in doing rare event simulation and my event depends, among other random factors, on the ...

**7**

votes

**1**answer

200 views

### Brownian motion, crossing intervals, possible usage of second moment method?

This is a followup to my question here.
Let $B_t$ be a standard Brownian motion. Let $E_{j, n}$ denote the event$$\left\{B_t = 0 \text{ for some }{{j-1}\over{2^n}} \le t \le ...

**11**

votes

**4**answers

424 views

### Number of intervals needed to cross, Brownian motion

Let $B_t$ be a standard Brownian motion. Let $E_{j, n}$ denote the event$$\left\{B_t = 0 \text{ for some }{{j-1}\over{2^n}} \le t \le {j\over{2^n}}\right\},$$and let$$K_n = \sum_{j = 2^n + 1}^{2^{2n}} ...

**0**

votes

**0**answers

39 views

### Order statistics of the diagonal terms of an inverse Wishart matrix

I have a question about the inverse Wishart matrix. In my understanding, consider $\mathbf H$ is a $n\times n$ matrix with each elements are complex Gaussian with zero mean unit variance. Then ...

**8**

votes

**1**answer

218 views

### If $X∼F_1$, $Y∼F_2$, under what conditions on $F_1$, $F_2$ can we construct $Y=E(X\mid\mathscr{G})$ for some $\mathscr{G}$?

Suppose that we have distributions $F_1 $ and $F_2$. Under what conditions on $F_1,F_2$ is it possible to construct random variables $X\sim F_1,Y\sim F_2$ such that $Y=E(X|\mathscr{G})$, that is, $Y$ ...

**6**

votes

**2**answers

122 views

### For which $r > 0$ is it the case with probability one, for all $n$ sufficiently large $M_n \le r\sqrt{\log n}$?

Let $B_t$ be a standard Brownian motion. Let$$M_n = \max\{|B_t - B_{n-1}| : n - 1 \le t \le n\}.$$For which $r > 0$ is it the case with probability one, for all $n$ sufficiently large$$M_n \le ...

**0**

votes

**1**answer

140 views

### Discrete random walk with uniformly distributed transition p, set initially

I've been working on a discrete version of the "unreliable friend" distribution. It would seem that what I've come up with is equivalent to the following random walk:
Choose $p$ from $U(0,1)$
Start ...

**1**

vote

**2**answers

128 views

### Non-normality of limit of random variables

I have encounter the following difficulty in the study of limits of random variables. Assume that $\{X_n\}_{n\geq 1}$ is a sequence of real-valued random variables such that
...

**2**

votes

**0**answers

70 views

### Construct a sequence of i.i.d random variables with a given distribution function, diagonalization? [closed]

Assume we have a sequence of i.i.d. random variables $X_1, X_2, \dots,$ on a probability space $(\Omega, \mathcal{F}, P)$ with$$P(X_n = 1) = P(X_n = -1) = {1\over2}.$$Given a distribution function ...

**3**

votes

**1**answer

128 views

### Weak convergence of random variables in $L^2$ and vague convergence

Dumb question: Let $X_n:\Omega \to \mathbf{R}$ be a sequence of $L^2(\Omega,\Sigma,\mathbf{P})$ random variables that has a weak limit $X$ in $L^2$.
Suppose also that $\mu_n$, the distributions of ...

**5**

votes

**0**answers

72 views

### Joint cumulants of $Z_2^n$ characters

Let $f_{c}:Z_2^n \rightarrow \{-1,1\}$ be the character defined as $f_c(x) = (-1)^{<x,c>}$, where $c,x \in Z_2^n$. It is easy to see that since $f_{c_1}\cdot\ldots\cdot f_{c_k} = f_{c_1 \oplus ...

**10**

votes

**1**answer

260 views

### Normal approximation of tail probability in binomial distribution

My problem: From the Berry--Esseen theorem I know, that $$\sup_{x\in\mathbb R}|P(B_n \le x)-\Phi(x)|=O\left(\frac 1{\sqrt n}\right),$$ where $B_n$ has the standardized binomial distribution and $\Phi$ ...

**3**

votes

**2**answers

147 views

### splitting exponential random variable into independent components

$X$ follows Exponential $(\lambda)$. Can we split $X$ into two independent r.v.'s, i.e.,
do there exist functions $g$ and $h$ such that $g(X)$ and $h(X)$ are independent for any fixed $\lambda$? ...

**7**

votes

**2**answers

215 views

### Estimating entropy conditional to an event

Take for example the measure $\mu(n)=n^2$ on $\{1, \ldots, N\}$ and a random variable $X$ distributed according to the probability obtained by normalizing $\mu$.
Does there exists a constant ...

**3**

votes

**1**answer

95 views

### Reference request for a result regarding density of induced probability measure under a submersion

Let $\pi: M \to N$ be a smooth submersion from a bounded open subset of $\mathbb{R}^m$ onto $ N \subset \mathbb{R}^n$, $m \geq n$. Further, let $M$ be given a probability measure $\mu$. Then the map ...

**6**

votes

**1**answer

136 views

### Zeta zeros standard normal distribution about $\vartheta (\gamma_n)$

Asked at MSE here without response.
I realise that this resembles Odlyzko's famous nearest neighbours plot, and was wondering whether this is simply a manifestation of the same phenomenon.
That ...

**0**

votes

**0**answers

66 views

### Is this probability distribution studied in literature?

Let $\theta_1,\theta_2,\theta_3$ be 3 non-negative random variables such that $\theta_1+\theta_2+\theta_3=1$ with the joint probability distribution
\begin{align}
...

**2**

votes

**0**answers

47 views

### What is the Blumenthal-Getoor index of Student's distributions?

For infinitely divisible random variables, Blumenthal and Getoor introduced in [1] an index that allow to study for instance the local Hölder regularity of Lévy processes. For an infinitely divisible ...

**1**

vote

**2**answers

284 views

### The Levy measure of the compound Poisson distribution

The compound Poisson distribution is defined as(see Levy processes and infinitely divisible distributions page: 18):
Let $c>0$ and $\sigma$ be a measure on $\mathbb{R}$ with $\sigma(\{0\})=0$, a ...

**1**

vote

**0**answers

55 views

### Conditions for Mellin inversion

Under which conditions is the function
$$
g(s)=a^{c(s-1)}\Gamma(s),\qquad a>0,c\in \mathbb{R}
$$
the Mellin transform of a probability density function $f$? If $c=-1$, then $f$ is the exponential ...

**0**

votes

**0**answers

50 views

### Log-concave distributions: Weighted sum of pdfs

Assuming $f_n(\cdot)$ is a log concave function (e.g., pdf of Gaussian distribution) and $0\le q_n\le 1$ for all $n\le N$, I am trying to find conditions under which the following holds
...

**0**

votes

**0**answers

32 views

### Interpolating a polynomial when we permute part of $y_i$'s

Let $\vec{x}=[x_1,...,x_n]$ be elements of field $\mathbb{Z}_p$, where $p$ is a large prime. $x_i \neq x_j$, $x_i \in \mathbb{Z}_p$. Note $x_i$ values are NOT picked uniformly random and they are ...

**2**

votes

**1**answer

123 views

### Orthogonal polynomials with respect to the lognormal distribution

I am currently doing some inspection on the orthogonal polynomials with respect to the lognormal distribution. Does anyone already work on that or know some cool references?
All the best,
Pierre-O.

**0**

votes

**0**answers

54 views

### Quotient of cumulative binomial distribution functions

Given to integers $n < m \in \mathbb{N}_0$ and a probability $p$, I'm struggling to calculate (or at least get an upper bound for) the quotient
$$Q = \frac{F(n+1;m,p)}{F(n;m,p)}$$
where $F$ denotes ...

**5**

votes

**1**answer

83 views

### when does elementwise-log preserve positive-semidefiniteness?

Let $Z$ be a positive semidefinite matrix with nonnegative entries, and define $X=\log(1+Z)$, where the $\log$ is taken entrywise, i.e., $X_{ij}=\log(1+Z_{ij})$. Are there some simple sufficient ...

**0**

votes

**0**answers

38 views

### Validating a probability density distribution forecast model for a Markov process

Let's say we have a Markov process $X_t$, and we come up with a forecast model that takes some information from outside world and says: "value $X_{t+1}$ has probability density distribution $P_t(x)$". ...

**0**

votes

**0**answers

58 views

### $\exists \mathcal{A},\mathcal{B}:X\sim \mathcal{A}\Rightarrow \frac{p}{\sqrt{q+rX}}\sim \mathcal{B}$?

Does there exist a parametric distribution $\mathcal{A}$, such that
$$
X\sim \mathcal{A}\Rightarrow\frac{p}{\sqrt{q+rX}}\sim \mathcal{B}
$$
for some parametric distribution $\mathcal{B}$, where ...

**1**

vote

**0**answers

123 views

### What is a two-sided geometric distribution?

I found in some articles (such as this) references to two-sided geometric distribution. But I went through texts of probability and did not find anything called "two-sided geometric distribution". ...

**2**

votes

**0**answers

42 views

### A canonical example of the non-existence of predictive probability distribution

Section 3 of Fortini et al. (2000) states that
Given $(X^\infty, \mathcal X^\infty,P)$, a predictive probability distribution of $x_n$ given $(x_1, \dots, x_{n-1})$ with respect to $P$ need not ...

**0**

votes

**0**answers

102 views

### Probability of substring given string production probabilities

I originally posted this question on the Math StackExchange, but have not received answers there and thought it might be more appropriate to post it here.
Let $\Sigma$ be an alphabet and let $y = x_1 ...

**4**

votes

**0**answers

103 views

### Dimension reduction for low-order moments of Rademacher-weighted sums of vectors

Let $x_1,\dots,x_n$ be vectors in a Euclidean space $H$. Let $\varepsilon_1,\dots,\varepsilon_n$ be independent Rademacher random variables (r.v.'s), so that $P(\varepsilon_i=\pm1)=1/2$ for all $i$.
...

**-2**

votes

**1**answer

146 views

### expected value of cosine wirh Gaussian phase

Is there a solution to the expected value/variance for a Gaussian with random phase:
$$\cos(\omega_0 t + \phi), \qquad \phi \sim \cal{N}(0,\sigma^2) $$
?
For $t=0$, the solution is for example ...

**1**

vote

**0**answers

28 views

### PDF of points at the intersection of a sphere and hyperboloid in n dimensions

I'm studying a statistical mechanics problem and I have two conserved quantities:
$$ E = \sum_{k=0}^{M} \left[ a_1^2(k) + a_2^2(k) + b_1^2(k) + b_2^2(k)\right] $$
$$ H = \sum_{k=0}^{M} 2 k \left[ ...

**6**

votes

**1**answer

152 views

### Is there an $\infty$ version of the Wasserstein distance between two distributions?

If I have two probability distributions $\mu$ and $\nu$ defined on $X$ and $Y$ respectively, then the $p$-th Wasserstein distance between the two of them is defined as $$W_p(\mu,\nu) = ...

**7**

votes

**2**answers

303 views

### Maximal entropy distribution with given conditionals

It is well known that of all the joint distributions $p(x,y)$ with fixed marginals $p(x),p(y)$, the one with the highest entropy is:
$$
p(x,y)=p(x)p(y).
$$
Suppose instead that we have conditionals. ...

**7**

votes

**1**answer

169 views

### Distribution of infinity-norm over the unit sphere

I need to compute probabilities of the form
$P( \Vert X \Vert_\infty < r ),$
where $X$ is a random variable of dimension $n$, drawn with a uniform distribution on the unit sphere ...

**0**

votes

**0**answers

189 views

### Conditional probability of dependent random variables

Let $ X \sim f_X(x), Y \sim f_Y(y) $ are two dependent random variables and their corresponding PDFs. I want to find a probability $$ P(Y\ge 0 | X+Y\ge 0) .$$ If these variables were independent I'd ...

**5**

votes

**0**answers

144 views

### Extrapolation between longest increasing and longest alternating subsequences

The question
When should we expect Tracy-Widom?
motivated me to post the following question, in which I have been
interested for a while. Let $f(n)$ be a function from the positive
integers to ...

**13**

votes

**2**answers

487 views

### A probability distribution in n dimensional space which its projection on any line is a uniform distribution?

Does there exist, for any natural $n$, a probability distribution in $\mathbb{R}^n$ whose projection on any line is a uniform distribution?

**0**

votes

**0**answers

82 views

### Why is this distribution exponential?

Take the interval $[0, 1]$.
Now sample 10000 points in this interval randomly according to the uniform distribution.
The fact is that the distribution of the distances between adjacent points on ...