# Questions tagged [pr.probability]

Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.

7,283
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### A problem related to bivariate normal stochastic order

Let $\boldsymbol{X} = (X_1,X_2)^{\rm T}\sim \mathcal{N}_2(\boldsymbol{\mu}, \mathrm{\Sigma})$, where
\begin{eqnarray*}
\boldsymbol{\mu} = (\mu_1, \mu_2)^{\rm T}& = &(\sqrt{\xi_1\xi_2/(\xi_1+\...

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59 views

### Asymptotic moment of a multivariate normal distribution

Let the pdf of a multivariate normal distribution be
\begin{equation}
p_{Z}(\mathbf{z})=\frac{1}{\left(2\pi \sigma^2 \right)^{k/2}}\exp(-{\mathbf{z}}^{\text{T}}\mathbf{z}/2\sigma^2).
\end{equation}...

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236 views

### From biased coins (and nothing else) to biased coins

We're given a coin that shows heads with an unknown probability, $\lambda$. The goal is to use that coin (and possibly also a fair coin) to build a "new" coin that shows heads with a ...

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345 views

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### From coin flips to algebraic functions via pushdown automata

Background
We're given a coin that shows heads with an unknown probability, $\lambda$. The goal is to use that coin (and possibly also a fair coin) to build a "new" coin that shows heads ...

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74 views

### Jeffreys' priors as coefficients of a linear estimator

I asked the following question in a forum more suitable for statistics, but I didn't get any answer; I hope, someone could shed light on my question:
I have three random variables, $X_1$, $X_2$, and $...

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164 views

### A problem related to stochastic ordering

Let $\boldsymbol{X} = (X_1,X_2)^{\rm T}\sim \mathcal{N}_2(\boldsymbol{\mu}, \mathrm{\Sigma})$, where
\begin{eqnarray*}
\boldsymbol{\mu} = (\mu_1, \mu_2)^{\rm T}& = &(\sqrt{\xi_1\xi_2/(\xi_1+\...

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**1**answer

98 views

### Exact formula or non-trivial upper bound on p-norm of $f(x)=\|x\|_2$ in $[0,1)^d$

I wonder whether one can exactly calculate the following integral in terms of $d$ and $p\geq 1$ or not, or a better bound(than the trivial one I am going to give) in terms of $d,p$:
$$\left(\int_{[0,1)...

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**1**answer

59 views

### A periodically independent stochastic process

Does there exist a non constant almost surely continuous stochastic process $X$ on $[0, \infty)$ with $X_t$ independent of $X_{t+1}$ for all $t \geq 0$?

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60 views

### Uniform distribution on a manifold

To generate a uniform distribution on a sphere $S^n$ in $\mathbb R^{n+1}$, we can normalize a vector whose entries are $n+1$ i.i.d normal random variables. If $\rho$ is a correlation, $|\rho|<1$, ...

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87 views

### A question on the applicability Chebyshev inequality for sequence of random quantities

Let $(X_n)_n$ and $(Y_n)_n$ be two mutually independent sequences of random tensors (i.e scalars, vectors, matrices, etc.) defined on the same probability space, and let $f$ be a measurable function.
...

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177 views

### Regularity for the sum of iid random variables

Let $(X_i)_{i\in \mathbb{N}}$ iid random variables such that there exists $\alpha>0$ where $\mathbb{P}\left(X_1\in [x,x+1]\right)\leq \alpha$ for all $x\in \mathbb{R}$. Assume $\alpha$ small ...

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47 views

### Lower bound on likelihood of binary outcomes

I am wondering about the following: does there exist a stochastic process $(X_n)_{n \ge 1}$ with values in $\{0,1\}$ on a probability space $(\Omega, \mathcal F, \mathbb P)$ such that for all $n \ge 1$...

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584 views

### Explain seemingly non-random figures which arise from random Poisson points with normalization

Context Working with some biological datasets it was puzzling to see the patterns like Figure 2 (right) below. The first feeling was, that it corresponds to some biological effects like correlations ...

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159 views

### Expected value of $X^{\top}(XAX^{\top})^{-1}X$ for large random $X$

Let $X\in \mathbb{R}^{m\times n}$ be a random matrix where the entries are i.i.d. standard normal, and let $A\in \mathbb{R}^{n\times n}$ be a deterministic diagonal matrix with positive entries on the ...

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78 views

### If $\lambda_i$ is symmetric with $\lambda_i\{0\}=0$, why does $\int_B1-\cos\langle x,x'\rangle\:(λ_1-λ_2)({\rm d}x)=0$ imply $λ_1=λ_2$?

Let $E$ be a separable $\mathbb R$-Banach space and $\lambda_i$ be a finite symmetric measure on $\mathcal B(E)$ with $\lambda_i(\{0\})=0$ and $$\int_B1-\cos\langle x,x'\rangle\:\underbrace{(\lambda_1-...

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171 views

### How to prove the Fourier transform of $e^{-x^p}$ is positive [duplicate]

I wonder how to prove that
$$\int_0^\infty\exp(-x^p)\cos(tx)\,dt\geq 0, \quad \frac{1}{2}<p<1.$$
This conclusion is used in the answer to another question here
Looking for sufficient conditions ...

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**1**answer

264 views

### Fourier-positivity of a certain function

I am wondering how to prove the below Fourier transform is non-negative? I did much simulation and it seems to be non-negative.
$$\int_0^\inf (be^{-at^p}-ae^{-bt^p})\cos(tx)dt, 0<a<b, \frac{1}{2}...

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36 views

### Bounds on trace of ratio of rank-1 perturbed Wishart matrices

Let $a,b,d,e,f \ge 0$ and $c>0$. Let $X$ be a random $n \times m$ matrix with iid entries from $N(0,1/m)$ and set
$$
\begin{split}
W &:=XX^\top,\\
D &:= W \circ I_n = \mbox{diag}(w_{1,1},\...

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34 views

### Bounding the Jacobian determinant on the mapping given by Brenier’s Theorem

I have two distributions with small 2-Wasserstein distance between them that satisfy the conditions for Brenier's theorem. I am looking for conditions under which I can prove $p(x)\leq(1+a)\cdot q(x)+...

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62 views

### Relating sequence with or without replacement

I derived a relationship between sequences drawn with and without replacement for an application in genetics. The proof is easy enough, but I would rather find a source than provide a derivation of a ...

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56 views

### KL divergence between two sequences

Let us have a random sequence $(X_1, Y_1,\ldots,X_n,Y_n)$, where $X_t$ takes value in some set $\mathcal{X}$ and $Y_i$ are scalars. The sequence is generated by the following process: $X_i$ is chosen ...

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98 views

### Stochastic ordering of absolute multivariate normal random variables

Let $X\sim\mathcal{N}(\boldsymbol{\mu}_1,\mathrm{\Sigma}_1)$ and $Y\sim\mathcal{N}(\boldsymbol{\mu}_2,\mathrm{\Sigma}_2)$. Then it is know that $\mathbb{P}(X>\boldsymbol{t})\leq\mathbb{P}(Y>\...

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48 views

### Distrbution of points transformed by a family of polynomials

Consider a family of polynomials $\mathcal{F}$. Let $p$ be a single complex point or a finite set of complex points inside the unit disk.
I am interested in what can we say about the distribution of
$$...

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**1**answer

122 views

### Discrete random walk on polytope via involutions

Let $P$ be a convex polytope (or more generally convex body, I suppose) in $V=\mathbb{R}^n$. For each $v\in \mathbb{P}V$, we define an involution $\tau_v\colon P\to P$ by setting $\tau_v(p)$ to be the ...

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82 views

### Sampling uniformly in a ball of radius $\epsilon$ in the space of dicrete r.v. of m modalities for the total variation metric

I am looking for some reference or an algorithm that allows to sample uniformly in the ball centered at a discrete random variable of n modalities in the TV distance.
For the record for 2 discrete ...

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98 views

### An unnatural martingale

What is an example of a real valued stochastic process $X$, and a filtration $\mathcal F_t$ such that $X$ is a martingale with respect to $\mathcal F_t$ but not it’s natural filtration?
Either ...

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35 views

### Local well posedness for a stochastic wave equation

Suppose we have a stochastic wave equation, with Itô's derivative in the place of the usual Newtonian ones.
Does it make sense to talk about local well posedness?

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106 views

### Least positive value of a random polynomial

Fix a positive even integer $d$ and consider the polynomial $f(x)=c_d x^d+\ldots+c_1x+c_0$, where the $c_i$ are independent random variables that follow the uniform distribution in the interval $[-1,1]...

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84 views

### Levy Ito decomposition

I was having some difficulties understanding the Levy-Ito decomposition, so I summarised some results into one result. Could anyone please tell me if the following makes sense, i.e. is mathematically ...

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90 views

### Distribution of stopping time for a 2D random walk

Consider the following process on $\mathbb{C}$:
Start at the point 1.
At each step, move by adding $e^{i\theta}$, where $\theta$ is uniformly drawn from $\mathbb{S}^1$.
Stop at the first positive ...

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73 views

### SDE on an interval with non regular boundary points: weak uniqueness?

Let $\sigma,b\in\mathcal{C}^1\big(]0,1[,\mathbb{R}\big)$, with $\sigma$ non-vanishing, and consider the SDE
\begin{align}
& X_0=x\in\,]0,1[ \\
& \mathrm{d}X_t=\sigma(X_t)\,\mathrm{d}B_t+b(X_t)\...

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136 views

### Random walk always stays below a level $a$

Suppose we have a random walk $S_n$ with i.i.d. steps $X_i$. We assume that
$$\mathbb{E}[X_i] = -\mu, \text{Var}[X_i] = 1,$$
where $\mu$ is close (or going) to zero. We also assume that the moment ...

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36 views

### Generalization of a Gaussian measure continuity result from Hilbert to Banach space

Da Prato/Zabczyk "Second Order Partial Differential Equations in Hilbert Spaces" states the following lemma (this is a reformulation of proposition 1.3.11 in their book):
Let $\mu = \mathcal ...

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73 views

### Eigenvalues of Matérn covariance function

Recall that Matérn covariance function $C_\nu(d)$ is defined as
$$
C_\nu(d)=\sigma^2\frac{2^{1-\nu}}{\Gamma(\nu)}\left(\sqrt{2\nu}\frac{d}{\rho}\right)^\nu K_\nu\left(\sqrt{2\nu}\frac{d}{\rho}\right), ...

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57 views

### Does the compactness of parameter of distribution function imply the compactness of the distribution (or probability measure) in Wasserstein space?

For a family of probability measures sharing the same form of distribution function $F(x; p)$ with different parameters (i.e., $p$'s), if the parameter falls in a compact subset of real line, can we ...

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43 views

### Improving log-Sobolev inequalities via quadratic regularisation

Suppose that $\mu(dx) = \exp(-\psi(x)) \, \mathrm{dx}$ is a probability measure on $\mathbf{R}^d$.
For suitable functions $g \geqslant 0$, define
$$\text{Ent}(g) = \int \mu(dx) g(x) \log \left( \frac{...

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24 views

### Bound on the minimum cell frequency of a multinomial

Is there an upper bound to the cumulative distribution of the minimum cell frequency of a multinomial distribution that is tighter than the simple union bound?
Let $X \sim \mathrm{Multinomial}(n; p_1,\...

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64 views

### Expected diameter of a random point set

General problem: For a point set $S\subset X$ in a metric space $(X,d)$, let $\text{diam}(S)=\max_{x,y\in S}d(x,y)$. Given a distribution $P$ on $X$ and $m$ i.i.d. points $x_1,\ldots,x_m\sim P$, what ...

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26 views

### Commonly used ensembles of probability distributions

This is an open-ended question; hopefully specific enough for this forum.
If one is studying the properties of a 'typical' function, one can look for properties that hold for almost all functions ...

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113 views

### Why do we assume that a stopping time is a random variable?

Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t \geq 0},\mathbb{P})$ be a probability space and
$\tau: \Omega \rightarrow [0,\infty]$ be a stopping time. By definition this should be random variable so ...

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112 views

### Self study guide to Hamiltonian Monte Carlo

I was wondering if anybody has a suggested self-study path to understand the mathematical aspects on Hamiltonian Monte Carlo.
In this paper The geometric foundations of Hamiltonian Monte Carlo
it is ...

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79 views

### Expectation of the inverse of random principal submatrices

The goal of this question is finding the concentration point of the inverse of random principal submatrices, which is posed as follows. Consider $\mathbf{S}\in\mathbb{S}^{n}_{++}$ to be a strictly ...

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120 views

### An inequality in the optimality of Bayes' theorem

$\DeclareMathOperator\Ent{Ent}\newcommand{\prior}{\mathrm{prior}}\newcommand\Data{\mathrm{Data}}$I came across this paper on the optimality of Bayes' theorem
https://sinews.siam.org/Portals/Sinews2/...

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56 views

### Example(s) where replacing a multivariate, discrete RV with a single, univariate RV fail

Let $X_1,\ldots,X_n,Y,Z$ be $n+2$ binary random variables and define $X=(X_1,\ldots,X_n)$. In most problems, instead of treating $X$ as $n$ distinct binary random variables, there is no loss of ...

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71 views

### "Tails" of a multinomial distribution

Let $X_1,\dots,X_N$ denote a collection of independent samples of a uniform multinomial random variable in $\mathbb{Z}^k$, with the number of trials equal to $n\ll k$. (By "uniform", I mean ...

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63 views

### LLN of random nearest neighbor function

There are two samples of iid random variates: $X=\{X_1,X_2,...,X_n\}$ and $Y=\{Y_1,Y_2,...,Y_n\}$. Further, $\forall i,j: X_i$ is independent of $Y_j$. The probability distributions $P,Q$ are unknown ...

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82 views

### A random process with conserved momentum: 'particle decay'?

Consider a particle $p_1$ moving at unit speed along a straight line in $\mathbf{R}^2$, directed by some vector $v_1 \in \mathbf{S}^1$. Equid this particle with a Poisson clock $\tau_1$, with ...

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70 views

### Divergence-free Gaussian vector field with given mean magnitude and correlation function

My general question is how to construct an isotropic random vector field $\vec f: \mathbb{R}^3 \to \mathbb{R}^3$ with a given mean magnitude $\mathbb{E}[\|\vec f(\vec x)\|]=\mu$ and with vector ...

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163 views

### On the weak convergence of probability measures on $\mathbb R$

Let $\mathcal P(\mathbb R)$ be the set of probability measures. Set for $\mu,\nu\in\mathcal P(\mathbb R)$
$$d(\mu,\nu) := \inf\left\{\varepsilon>0:~ F_{\mu}(x-\varepsilon)-\varepsilon \le F_{\nu}(x)...

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121 views

### "Shape"/"norm" of a uniformly random set partition

Let $\mathcal{A}=\{A_1, A_2, \ldots, A_m\}$ be a uniformly random set partition of $[n]$.
What can we say about $||\mathcal{A}||_2 = \sqrt{\sum_{i=1}^m |A_i|^2}$? It is clearly upper bounded by $n$, ...