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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.

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Discrepancy between probability measures, tested against bounded functions of bounded variance

When studying some concentration inequalities, it became relevant to consider the following discrepancy between two probability measures $\pi$ and $\nu$ (treating $\sigma \in \left( 0, \frac{1}{2} \...
πr8's user avatar
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1 vote
0 answers
156 views

Nontrivial nonrandom properties of prime numbers

What are some nontrivial nonrandom properties of prime numbers. Consider the simple model where each number is prime with probability 1/log(n) by Montgomery and extensions of it. Once you add some ...
ericf's user avatar
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4 votes
0 answers
87 views

Statistics of random Voronoi S-tessellations

Given a locally finite set of points $\{x_1,x_2,\dots\}\subset\mathbb{R}^d$, the Voronoi cell of a point $x_{i}$, denoted by $C(x_{i})$, consists of all the points in $\mathbb{R}^d$ that are closer to ...
Qidong He's user avatar
3 votes
0 answers
58 views

Infinitesimal generators of random evolutions

Consider two state spaces $X$ and $Y$ and infinitesimal generators of Markov processes $(A_y)_{y\in Y}$ and $B$, on $X$ and $Y$ respectively. We assume that $A_y$ share the same domain $D(A)$, and ...
Gabriel's user avatar
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2 votes
0 answers
158 views

Conformally mapping between the upper half complex plane, and the plane with a tree on spatial points removed

A stochastic process such as SLE$_{\kappa}$ can be defined by taking the scaling limit of a curve in the upper half complex plane: put simply, one removes a line segment, then another, $n$ times, each ...
apg's user avatar
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0 answers
134 views

Asymptotics of a ratio on the unit sphere

Let $(a_n)_{n \geq 1}$ be a nonnegative, strictly decreasing sequence with $a_n \to 0$ as $n \to \infty$. Consider the ratio (for $k \geq n$) $$ R_{n, k} = \mathbb{E}_{u \sim \text{Unif}(\mathbb{S}^{k-...
Drew Brady's user avatar
2 votes
0 answers
106 views

When is there a Lipschitz Kantorovich Potential?

Let $c:\mathbb{R}^d\times \mathbb{R}^d\to [0,\infty)$ be a Lipschitz cost function and consider the optimal transport problem $$ C(\mu,\nu):=\inf_{\pi}\, \int c(x,y)\,\pi(dxdy) $$ where, as usual, the ...
LittleQuestionBoy's user avatar
5 votes
2 answers
792 views

How to calculate an integral over the complex unit sphere

We want to calculate the following integral over the complex unit sphere $S^{2n-1}$: $$\int_{S^{2n-1}} \frac{1 }{|1 - \langle z, \zeta \rangle|^2} \, d\sigma(\zeta),$$ where $ z $ is a fixed point in ...
Ryo Ken's user avatar
  • 109
1 vote
0 answers
114 views

An urn model with weighted objects and replacement

Consider the following game: In an urn, there are $K$ balls, $x_0$ of them are blue and light (mass $m_0$), $x_1$ are blue and heavy ($m_1$), $x_2$ are red and light ($m_2$), the rest $x_3$ are red ...
PontyMython's user avatar
2 votes
0 answers
61 views

Characterisation of Bessel process

Let $\delta \in (0, 2)$; $(X_t)_{t \ge 0}$ a nonnegative continuous Markov process. Suppose that For each $T \ge 0$, if we write $\tau \overset{\mathrm{def}}= \inf\{t \ge T : X_t = 0\}$, then $(X_{T +...
Focus's user avatar
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1 vote
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45 views

Inequality Involving Concave Monotonic Function

Assume that $ f: \mathbb{R} \to \mathbb{R}_+ $ is a concave, non-decreasing and positive function. Let $\mathbb{X}$ be a finite set consisting of $ 0\leq x_1 \leq x_2 \leq x_3 \leq \ldots \leq x_n$. ...
Alireza Bakhtiari's user avatar
0 votes
0 answers
93 views

Distance between binomial and normal distributions

I want to compare binomial distribution $Bin(n,p)$ with a constant $p$ when $n\rightarrow \infty$, to a normal distribution with $\mu=np,\sigma^2=np(1-p)$. How close are they with the discrete ...
Tomer Ezra's user avatar
6 votes
1 answer
370 views

Convergence of iterated conditional expectations

Notation: We write $\mathbb E_{\mathcal F} X$ for the conditional expectation $\mathbb E[X|\mathcal F]$ of a random variable $X$ with respect to a $\sigma$-algebra $\mathcal F$. Let $X$ be an ...
Nate River's user avatar
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0 votes
1 answer
158 views

Techniques for bounding the operator norm of the expectation of random matrix?

Let $\mu$ be a distribution on the unit sphere in $\mathbb{R}^n$. Let $u \sim \mu$ and consider the random matrix $$ A = I_n - uu^T. $$ Question: What techniques are available to provide (reasonably ...
Drew Brady's user avatar
5 votes
1 answer
139 views

Dispersion of random walk with scaled step sizes

Let $Y_j$ be a sequence of independent Gaussian random variables with mean zero and unit variance ($\mathbb{E} Y_j = 0$ and $\mathbb{E} Y_j^2 = 1$) and let $\sigma:\mathbb{R}\to [1,2]$. We define the ...
felipeh's user avatar
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1 vote
0 answers
69 views

Simulating binomial distribution

$\DeclareMathOperator\Bin{Bin}\DeclareMathOperator\Pr{Pr}$I have a series of distributions $D_k=\Bin(3k,\frac{1+k^{-1/3}}{3})$, and a distribution $D_{k,\ell} = k +\Bin(k,\ell)$ parametrized by $\ell\...
Tomer Ezra's user avatar
0 votes
0 answers
73 views

Tight tail bounds for sums of random variables

Let $X_1, X_2, \dots$ be iid uniformly on $[0,1]$. Define $Z_i^{(a)} = (X_i - a)^2$. Let $Y_n = \sum_{k=1}^n Z_k^{(1/k)}$. I am interested in matching tail bounds for $Y_n$ as $n \to \infty$. In ...
user14097523067's user avatar
1 vote
1 answer
148 views

An inequality about binomial distribution

Statement Assume that $\sigma,R\in (1,+\infty)$, $N\in\mathbb{N}^*$, $p\in (0,1)$, $n_1\in\{0,1,2,\cdots,N-1\}$. Prove or disprove that $$B^\frac{1}{\sigma}(n_1)-B^\frac{1}{\sigma}(n_1+1)<1 .$$ ...
John_zyj's user avatar
3 votes
1 answer
175 views

Convergence rate of the sum of squares of inverse distances of random points which become dense in a region

$n$ points $\{X_i\}$ are drawn at random from a uniform distribution over a domain $\Omega\subset \mathbb{R}^m$ with a Lipschitz boundary. $D_n$ is defined as $$D_n = \sqrt{\frac{1}{\sum\limits_{1\le ...
Rajesh D's user avatar
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2 votes
1 answer
105 views

Inequality for Gaussian measures

Let $\mu$ denote a centered Gaussian measure on $\mathbb{R}^k$, $K=(-\infty, a] \times \mathbb{R}^{k-1}$ ($a\ge 0$) and $L=\mathbb{R}\times C$ where $C$ is a convex set in $\mathbb{R}^{k-1}$, ...
bdx77's user avatar
  • 197
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0 answers
31 views

What is the Fisher information matrix of the von Mises-Fisher distribution?

Assuming the von Mises-Fisher distribution as $$f_{p}(\mathbf{x}; \boldsymbol{\mu}, \kappa) = C_{p}(\kappa) \exp \left( {\kappa \boldsymbol{\mu}^\mathsf{T} \mathbf{x} } \right),$$ where $\kappa \ge 0$,...
Math_Y's user avatar
  • 287
19 votes
2 answers
2k views

Higher or lower?

Consider the following game - I draw a number from $[0, 1]$ uniformly, and show it to you. I tell you I am going to draw another $1000$ numbers in sequence, independently and uniformly. Your task is ...
Nate River's user avatar
  • 6,215
5 votes
1 answer
192 views

Non-equivalent definitions of Markov process

As far as I know, there are three definitions of Markov processes (or of Markov chains). DEFINITION 1 (WEAKER). A process $(X_t)_{t\in[0,\infty)}$ on $(\Omega,\mathcal{F},\mathbb{P})$ with values in ...
No-one's user avatar
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8 votes
1 answer
522 views

One step forward, one step back

$N \geq 2$ players play a cooperative game on the integers $\mathbb Z$. All of them start from $0$. At each turn, they are simultanously given the same yes or no question to answer. The questions ...
Nate River's user avatar
  • 6,215
1 vote
0 answers
80 views

Moments from characteristic function for matrices

When $x$ is a random variable with the smooth characteristic function $\phi_x(t) = \mathbb{E}e^{itx}$, we can easily compute the moments as $\mathbb{E}[x^k] = i^{-n}\phi_x^{(n)}(0)$. There is no magic ...
user3826143's user avatar
2 votes
0 answers
56 views

Sum of independent Wisharts

Suppose random vectors $y_1,y_2,\ldots,y_m$ are independent and the distribution of each $y_i$ is a $d$-dimensional complex Gaussian with mean $0$ and covariance $\Gamma_i$, that is $y_i \sim \mathcal{...
user3826143's user avatar
2 votes
0 answers
29 views

Steiner symmetrization of smooth function on non-simply connected regions

Given a smooth function $u$ defined on $\mathbb{R}^2$, restrict $u$ to a subset $\Omega \subset \mathbb{R}^2$ (possibly not simply connected) foliated by level sets of a smooth function $\psi: \Omega \...
MathLearner's user avatar
1 vote
0 answers
66 views

Random lattice always has trivial automorphism group?

In example 2.5 of a paper [LS17] written by Lenstra and Silverberg, it is written that “Random” lattices have $Aut(L) = \{ \pm 1 \}$, I guess the 'Random' here refers to the distribution in Siegel's ...
constantine's user avatar
1 vote
0 answers
55 views

Limit process of a sequence of Gaussian variables on mesh grid going to zero

Consider the interval $[0,1]$ and a partition $\mathscr{P}_n = \{ [t_i,t_{i+1}), \, i=1,\ldots,N_n \, : \, 0=t_0 < \ldots < t_{N_n} = 1\}$. Suppose that for all $i$ and $t \in [t_i,t_{i+1})$, we ...
Grandes Jorasses's user avatar
3 votes
1 answer
194 views

Dynamics of a random stretch map

Notation: Here $S^1$ denotes the circle, which we view as the unit sphere in $\mathbb C$. We equip the circle with its natural length metric. Let $\{\epsilon_n\}_{n \geq 1}$ be iid uniformly ...
Nate River's user avatar
  • 6,215
34 votes
7 answers
3k views

A hat puzzle question—how to prove the standard solution is optimal?

I am currently writing an essay on hat puzzles, and for the warm-up section I introduce some of the standard finite hat puzzles. One of these proceeds as follows: You and two friends are each given a ...
Joel David Hamkins's user avatar
9 votes
1 answer
155 views

How to sample exactly k indices given the inclusion probabilities of all indices?

Let $k<d$ two positive integers, and $\{p_i\}_{i=1}^d$ a series of probabilities, with $p_i \in (0,1)$ and $\sum_{i=1}^d p_i = k$. We wish to sample exactly $k$ distinct indices $\mathcal{I}\...
Daniel Soudry's user avatar
2 votes
1 answer
276 views

Probability of visible permutations

A visible permutation $\sigma$ of $[1,2, ...,n]$ has a permutation matrix such that all "1" locations are visible from the origin $(0,0)$. Two "1" locations are visible if the two ...
Mohammad Al-Turkistany's user avatar
0 votes
0 answers
30 views

Why is the $\alpha$-divergence unique in positive measure space $\mathcal{M}$?

In this article https://bsi-ni.brain.riken.jp/database/file/298/303.pdf (S. Amari 2009), it is said that a $f$-divergence (eq. 17) which can be written by a decomposable Bregman divergence (eq. 53) ...
aaaa's user avatar
  • 1
4 votes
0 answers
127 views

A "resampling identity" for the Bessel(3) process

I've come across the following resampling identity and was wondering if this is known since it seems rather natural. Take $X$ a two-sided Brownian motion conditioned to always stay below $1$. (So if ...
Martin Hairer's user avatar
3 votes
1 answer
232 views

Bounds on relative entropy for MLE in Bernoulli coin tosses

In the context of estimating the parameter $p$ from a dataset of $n$ i.i.d Bernoulli coin tosses, we often use the relative entropy $D(p \parallel \hat{p})$ to measure the performance of an estimator $...
entropy07's user avatar
4 votes
2 answers
312 views

What is the expected size of the smallest hitting set?

Suppose we pick $n$ subsets of size $j$ of an $N$-element set $S$ uniformly at random. A hitting set is a subset of $S$ that intersects all our subsets. I am interested in the smallest size of an ...
HenrikRüping's user avatar
1 vote
0 answers
43 views

Moments on the Stiefel manifold

Let $S_{n, k} = \{V \in \mathbb{R}^{n \times k} : V^T V = I_k\}$ denote the Stiefel manifold, $1 \leq k \leq n$. Let $P \in \mathbb{R}^{n \times n}$ denote a symmetric real, positive definite matrix, ...
Drew Brady's user avatar
2 votes
1 answer
202 views

Strong Liouville property of virtually abelian groups

Let $G$ be a finitely generated group and let $\mu$ be a symmetric non-degenerate measure on $G$. By strong Liouville property for $(G, \mu)$, we mean that every positive $\mu$-harmonic function on $G$...
SMS's user avatar
  • 1,407
7 votes
5 answers
514 views

Probability of $\operatorname{Bin}(n,p)=\operatorname{Bin}(n,q)$ is decreasing when $n$ increases

$\newcommand{\Bin}{\operatorname{Bin}}$I would like to show that $\mathbb P(\operatorname{Binomial}(n,p) = \operatorname{Binomial}(n,q))$ decreases when $n$ increases for a fixed pair $(p,q)$. This ...
YuiTo Cheng's user avatar
3 votes
1 answer
99 views

Intersection of IID fractal sets

Let $A, B \subset \mathbb R$ be IID random closed subset. Suppose that there exists $d \in (1/2, 1]$ such that the Hausdorff dimension of $A$ is equal to $d$ almost surely. Is it true that $\mathbf P\...
Focus's user avatar
  • 177
3 votes
0 answers
131 views

Matrix-Gaussian distributions

The point of this question is to ask for references on matrix-variate Gaussian distributions. But I will explain what I mean by a matrix-variate Gaussian with an example (the notion I have in mind is ...
user3826143's user avatar
3 votes
0 answers
353 views

Moments of normalized multivariate Gaussians (and Wick's/Isserlis theorems)

Suppose $x = \begin{bmatrix}x_1 \\ x_2\end{bmatrix}$ is distributed according to the real two-dimensional Gaussian with mean-$0$ and covariance matrix $\Sigma$. I am interested in a closed form for ...
user3826143's user avatar
-1 votes
1 answer
103 views

Convergence in $\mathbb{L}_1$ implies convergence "perturbed" conditional expectations

Consider a sequence of conditional pdf's $p_n(y | x)$ on a Polish space $X \times Y$, endowed with its Borel sigma algebra. Suppose, as $n\rightarrow \infty$, in $\mathbb{L}_1$ (the following ...
Grandes Jorasses's user avatar
3 votes
1 answer
70 views

Multiplicative approximation for a negative moment of the binomial distribution

Let $X$ be a binomial random variable with parameters $n,p$. Define the function $f(n, p, t) = E\frac{1}{1 + t X}, $ where $t > 0$. Question: Can we find an elementary function $F(n, p, t)$ such ...
Drew Brady's user avatar
-2 votes
1 answer
43 views

$E(\mathbf{y}|\mathbf{x}+\mathbf{z})=g(\mathbf{x})$ almost surely, if $\mathbf{z}\perp \!\!\! \perp \{\mathbf{y},\mathbf{x}\}$ jointly?

Let $\mathbf{y},\mathbf{x}$ and $\mathbf{z}$ be real-valued random vectors with possibly different dimensions. If $\mathbf{z} \perp\!\!\!\perp \{\mathbf{y},\mathbf{x}\}$ (i.e., $\mathbf{z}$ is ...
John's user avatar
  • 193
2 votes
1 answer
526 views

What are some (popular) references on variants of the classical gambler's ruin problem that exists in literature?

It is fascinating that the gambler's ruin problem which is so ubiquitous in modern probability theory (cf. the Levin-Peres text on Markov chain and Mixing Times) actually dates back to a letter from ...
Aditya Guha Roy's user avatar
3 votes
1 answer
279 views

Bounds on hitting time of sum of i.i.d. random variables

I have a sequence $(X_i)_{i\geq 1}$ of i.i.d. random variables taking values in $\mathbb Z$. I know that each $X_i$ has mean $0$ and finite variance $\sigma^2$. Let $S_n=X_1+\cdots+X_n$. Then I can ...
Colin Defant's user avatar
0 votes
0 answers
149 views

Reference book for a probability course

In the next months I am planning to deliver a (more-or-less) advanced course in probability theory. My students will have had already a first encounter with discrete probability theory (discrete ...
Johnny Cage's user avatar
  • 1,561
2 votes
0 answers
70 views

Poisson process subordinated by a gamma process

I am working on a problem and I encountered the following situation: $(N(t): t \ge 0)$ is a Poisson process with parameter $\lambda t $. If $T_{n} = \sum_{i=1}^n W_i$ represents the $n^\text{th}$ ...
Rosy's user avatar
  • 21

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