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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Representation of optimal controls as diffusions

In reading this post I couldn't help but wonder the following question: Let $\sigma>0$ and suppose, as in the motivational post, we are given a stochastic optimal control problem: $$ \begin{...
ABIM's user avatar
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A randomized central limit theorem

Let $X_k$, $k = 1, 2, \dots$, be a sequence of i.i.d. random variables with finite second moments. Also, let $N_k \geq 1$, $k = 1, 2, \dots$, be a sequence of random variables taking integral values, ...
vassilis papanicolaou's user avatar
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Random matrix properties

Let $\mathbf{H}_{N,K}$ be a random matrix whose entries are i.i.d complex Gaussian random variables with variance $1$. Then, we know from the law of large number that if $N,K\rightarrow\infty$, we ...
Math_Y's user avatar
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Distance between two sample quantiles

Let $X_1,\dots X_n$ be i.i.d. samples from an unknown distribution. We know the distribution has uniformly bounded probability density function $f(x)$. Let $1>\tau_1>\tau_2>0$ be two quantile ...
aurora_borealis's user avatar
2 votes
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Small set in partition-large class

A collection $\mathcal{A}\subseteq \mathcal{P}(X)$ is $k$-large in $X$ if for every $k$-partition of $X$ namely $X_1,\cdots,X_k$, there exists an $i\leq k$ such that $X_i\in \mathcal{A}$; $\mathcal{...
Jiayi Liu's user avatar
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Confidence Intervals for strongly mixing stochastic processes

I have $s(t)$, a stationary stochastic process that we know is strongly mixing - and I also know that samples from $s(t)$ are definitely correlated over time. I want to estimate the mean of $s(t)$, $\...
Nithin Ramesan's user avatar
3 votes
1 answer
649 views

Characteristic function and moments

Let $X\in L^1(\Omega)$ and $\phi_X$ the corresponding characteristic function. We know that: $\phi_X$ is $n$ times differentiable (at $u=0$) iff $\mathbb{E}[X^n]<\infty$. (This depends a bit on ...
Alex's user avatar
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Sum of squares of middle binomial sums or 'Truncated mean' of binomial coefficients under binomial distribution

$\mu=1+\epsilon$ where $\epsilon>0$ holds. 1.Is there a good bound for $$T=\frac{\sum_{i=-\sqrt{\mu n\ln n}}^{\sqrt{\mu n\ln n}}\binom{n}{\frac n2 +i}^2}{2^n}?$$ This quantity can be ...
VS.'s user avatar
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What is the minimal $m$ for which the independence graph is $n$-universal?

Suppose, an $m$ sided die is rolled. Let's define the independence graph $I_m$ as a graph with the set of all possible events as vertices, and edges between two events iff they are independent. ...
Chain Markov's user avatar
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Less regular version of the Gaussian free field

One can define (continuous) Gaussian free field as follows: one can consider some orthonormal basis $(\psi_k)_{k=1}^{\infty}$ in the Sobolev space $H^1(\Omega)$ (here $\Omega \subset \mathbb{R}^d$) ...
truebaran's user avatar
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If $M$ is a manifold, $x∈M$ and $d(x,ω)=\inf\{t>0:x+tω∈M\}$, does the pushforward of the solid angle measure under $S^2∋ω↦x+d(x,ω)ω$ admit a density?

Let $S^2$ denote the unit 2-sphere, $M$ be a 2-dimensional oriented embedded $C^1$-submanifold of $\mathbb R^3$ with $$d_M(x,\omega):=\inf\left\{t>0:x+t\omega\in M\right\}<\infty\;\;\;\text{for ...
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2 answers
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Imprecise Definition of a $\sigma$-algebra

I'm reading some works on the hierarchical model in statistical mechanics and I came across an strange definition, which I need to clarify. Consider a finite set $\Lambda \subset \mathbb{Z}^{d}$. The ...
IamWill's user avatar
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4 votes
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Reconstructing probability distribution with high probability

Sample $m$ times from unknown probability distribution $p=(p_1,p_2,\cdots,p_n)$, we can construct a probability distribution $q=(q_1.q_2,\cdots,q_n)$. How large $m$ should be to achieve that the ...
gondolf's user avatar
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340 views

Relate the solid angle and surface measure of a surface

Let $M$ be a 2-dimensional embedded $C^1$-submanifold of $\mathbb R^3$ with a global chart$^1$ $(U,\phi)$. If $u\in U$ and $x=\phi^{-1}(u)$, let $\nu_M(x)$ denote the unique unit normal vector of $M$ ...
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Gaussian Property of the Renormalization Group

Let $\Lambda \subset \mathbb{Z}^{d}$ be a finite set and $\varphi = (\varphi_{x})_{x\in \Lambda} \in \mathbb{R}^{|\Lambda|}$. Let $F^{\Lambda}=F^{\Lambda}(\varphi)$ be a real-valued global function, ...
IamWill's user avatar
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7 votes
1 answer
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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
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1 answer
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Non-convergence to a Gaussian

Let $f_n: \mathbb R^2 \rightarrow \mathbb R$ be a family of probability distributions with the property that they vanish on the diagonal $f_n(x,x)=0.$ I would like to know: Can we show that a ...
Xin Wang's user avatar
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2 answers
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Two independent function when considered as random variable over $([0,1],\mathrm{Lebesgue},\mathcal{B}_{[0,1]})$

Does there exists two non-constant continuous functions $f,g:[0,1]\rightarrow \mathbb{R}$ so that they are in independent (in probability sense) when viewed as random variables over the measure space $...
De vinci's user avatar
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1 answer
142 views

Sphere inversion in Riesz potential

I am reading the paper: ``ON THE DISTRIBUTION OF FIRST HITS FOR THE SYMMETRIC STABLE PROCESSES" by Blumenthal, Getoor and Ray, (Trans. Amer. Math. Soc. 99 (1961), 540-554). On page 546, the authors ...
srg's user avatar
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Approximate Multivariate Gaussian Integration by Parts

When $Z$ is a $\mathcal{N}(0,1)$ random variable, $f$ smooth from $\mathbb{R} \to \mathbb{R}$ we have the Gaussian integration by parts formula $$ \mathbb{E}(Zf(Z)) = \mathbb{E} f'(Z). $$ One analog ...
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166 views

Weak convergence of $\mathcal{L}^2$ valued random variables

Consider two continuous functions $f,g: \mathbb{R}^{2} \rightarrow \mathbb{R}$ with $f(x,\cdot), g(x,\cdot) \in \mathcal{L}^2(\mathbb{R},\mathcal{B},\lambda)$ for all $x \in \mathbb{R}$ and a sequence ...
esner1994's user avatar
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1 answer
158 views

Visualization of the disintegration theorem [closed]

Where can I find a picture that gives a visualization of the disintegration theorem? If such reference does not exist, what would a nice visualization of this fundamental result look like?
Jay's user avatar
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612 views

is there a link with the probabilistic model for prime numbers?

Let $x \in \mathbb{R}_+$ and $k \in \mathbb{N}^{*}$. Let : $$\mathcal{A}(x)=\#\{(a_1, a_2, \ldots, a_k) \in \mathbb{P}^k \mid (a_1, a_2, \ldots, a_k \text{ verifying some properties}) \, , a_k \...
Lagrida Yassine's user avatar
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1 answer
115 views

Compute lower bound on $\min_{E} \mathcal N(0,\sigma^2 I_n)(E)$ subject to $vol(E \cap H_n(r)) / vol(H_n(r)) \ge p$ where $H_n(r)$ is $n$-hemisphere

Let $n \ge 2$ be an integer, which may be assumed to be very large. For $r > 0$, consider the hemi-sphere $H_n(r) := S_n(r) \cap (\mathbb R^+ \times \mathbb R^{n-1})$, where $$ S_n(r):= \{x \in \...
dohmatob's user avatar
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2 votes
1 answer
636 views

History of the name "subexponential distribution" in probability

In probability theory, the term subexponential distribution has historically been used for a distribution whose CDF $F(x)$ satisfies the relation $$ n(1-F(x)) \sim 1 - F^{*n}(x) $$ for any $n \ge 1$ ...
Greg Zitelli's user avatar
3 votes
1 answer
133 views

frequence of block of digits in Mobius sequence

Let $\mu$ be the Mobius function from $\mathbb{N}$ to $\{-1, 0, 1\}$. It is well known for the frequency of $-1, 1$, and $0$ for the sequence $(\mu(1), \mu(2), \mu(2), \dots, )$. For any $k\in \...
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1 vote
1 answer
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Proximity in terms of characteristic functions for $n$-dimensional distributions

Let $X\in \mathbb{R}^n$ and $Y\in \mathbb{R}^n$ be random variables with characteristic functions $\phi_X(t)$ and $\phi_Y(t)$, respectively. Suppose that \begin{align} \sup_{t \in \mathbb{R}^n} \...
Boby's user avatar
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2 votes
3 answers
731 views

Is unit ball in 2-Wassestein metric weakly compact?

This might be a trivial question, but I am trying to prove equi-coerciveness of some family of functions on the space of Probability measures on some space. I could reduce the problem to showing that $...
Raghav's user avatar
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6 votes
4 answers
534 views

Absolutely continuous coupling of probability measures

I have a Borel probability measure $\pi$ on $\mathbb{R}^{n+1}$ such that $\pi_1=\mu_1, \ldots, \pi_{n+1}=\mu_{n+1}$ for some fixed Borel probability measures $\mu_1, \ldots, \mu_{n+1}$ (where each $\...
Raghav's user avatar
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2 votes
1 answer
132 views

Reference request - parallel rectangles discrepancy theory

I've been reading about discrepancy theory and trying to understand some of the open problems in the field. Wikipedia has a list of some of the open problems, but the descriptions are terrible. In ...
DJA's user avatar
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2 votes
1 answer
168 views

Entropy rate problem of ergodic Markov process with non-ergodic joint

I have a problem with the entropy rate when two ergodic Markov processes who are independent of each other having a non-ergodic joint. More specifically let us consider two finite-state Markov ...
Yi Huang's user avatar
  • 333
13 votes
2 answers
629 views

Random matrix with given singular values

Let $\sigma_1\geq\sigma_2\geq...\geq\sigma_n\geq0$ be any deterministic sequence of positive real numbers such that $\sum_{i=1}^n\sigma_i^2=1$. Let $$D=diag\{\sigma_1,...,\sigma_n\}\in\mathbb{R}^{n\...
neverevernever's user avatar
1 vote
2 answers
270 views

Monotonicity of maximum of convex combination of two scaled concave functions

Let $f:R\rightarrow R$ be a concave function with a unique and finite maximum. Let $$g(x, \beta) = \beta f(\alpha \cdot x) + (1-\beta) f((1-\alpha) \cdot x), $$ where $\alpha \in [0,1/2]$ and $\beta \...
ACopt's user avatar
  • 13
0 votes
1 answer
375 views

Can I get away without using Arzela-Ascoli?

I am currently thinking of function-valued random variables. In order to prove a result, I need to approximate by (function-valued) step functions. This naturally leads to the idea of chopping up the ...
Daron's user avatar
  • 1,761
2 votes
1 answer
442 views

Lower-bound for $E[\min(X, k)]$ where $X$ is sum of Bernoulli random variables with $E[X]$ being a linear function of $k$

Given a real number $\alpha \in [0.5, 1.5]$, an integer number $k>1$, and a set of independent Bernoulli random variables $x_1, \dots, x_n$, I am interested to find a lower-bound for $F(\alpha, k)= ...
Melika's user avatar
  • 189
2 votes
2 answers
387 views

Concentration bound on maximum subset sum of standard Gaussians

Let $X_1, \dots, X_n$ be standard Gaussians. Let $\mathcal{S} \subseteq \{A \in 2^{\{1, \dots, n\}} : |A| = k\} $ be a family of subsets of $\{1,\dots, n\}$ with fixed size $k$. [Note that $\mathcal{S}...
Uthsav Chitra's user avatar
2 votes
1 answer
881 views

Marchenko-Pastur Law under general covariance structure

Let $x_1,...,x_n\in\mathbb{R}^p$ be i.i.d. random vectors with mean 0 and covariance $\Sigma_p$. Let $S_{n,p}=\sum_{i=1}^nx_ix_i^T/n$ be the sample covariance. We consider the asymptotics of the ...
neverevernever's user avatar
0 votes
1 answer
102 views

Functions that preserve the mixing of a stochastic process

Suppose we have a continuous-time stochastic process $s(t)$ that's mixing. What properties would a function $f$ have to have so that $f(s(t))$ would also be mixing? I'm sure this is a well-known ...
Nithin Ramesan's user avatar
0 votes
2 answers
217 views

Spectrum of a Markov kernel acting on $L^2$

Let $P$ be a Markov kernel on a measurable space $(E,\mathcal E)$ admitting an invariant probability measure $\pi$. $P$ acts on $L^2(\pi)$ via $$Pf:=\int\kappa(\;\cdot\;{\rm d}y)f(y).$$ The invariance ...
0xbadf00d's user avatar
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3 votes
3 answers
431 views

$H(p) \le H(q) + KL(p, q)$?

Let $H(p) = \sum_i p_i\log\frac{1}{p_i}$ be the entropy of $p$ and $KL(p, q) = \sum_i p_i\log\frac{p_i}{q_i}$ be the KL divergence between $p$ and $q$. Does it hold that $H(p) \le H(q) + KL(p, q)$? ...
Xi Wu's user avatar
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1 vote
0 answers
194 views

Karhunen-Loeve expansion of vector valued random processes

This thesis (https://www.semanticscholar.org/paper/Karhunen-Loeve-expansions-and-their-applications-Wang/f173dfb99ec4cbd08e779923770466cf1ef3f138) introduces multivariate KL expansion using a ...
Heisenberg's user avatar
1 vote
1 answer
106 views

Defining the conditional distribution of $Z$ as $E^{*}[Z| \mathcal{F}](f):=E[f(Z)| \mathcal{F}]$

I've been reading the first section Furstenberg's Noncommuting Random Products and I am confused with how he is defining conditional distribution. Here he is considering a group $G$ acting on a space ...
user135520's user avatar
6 votes
1 answer
225 views

Random walks: How many times does the largest component change?

My understanding is that for an unbiased random walk (starting at the origin) on $\mathbb R$ with $N$ steps that the expected number of sign changes is $O(\sqrt N)$. For a biased walk I believe the ...
Daron's user avatar
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2 votes
2 answers
183 views

Independence depth of linearly dependent random variables

Suppose, $\Xi$ is a collection of random variables. We call $\Xi$ $k$-independent, iff any $k$ distinct elements of $\Xi$ are mutually independent. For example, $2$-independence is pairwise ...
Chain Markov's user avatar
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2 votes
1 answer
264 views

Uniform upper bound on contraction coefficient w.r.t total-variation metric, of a certain set of block-diagonal Markov kernels

Disclaimer. This is related to another question I've asked on the TCS site https://cstheory.stackexchange.com/q/46097/44644. I'm new to information theory (and other relevant fields). It's even ...
dohmatob's user avatar
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2 votes
0 answers
87 views

Name for a probability density ''symmetrized'' by a permutation group?

Let $p$ be a probability density function over random variable $X$, and $G$ a compact permutation group over the outcomes of $X$. For each $g\in G$, let $p_g$ indicate the probability density ...
Artemy's user avatar
  • 650
0 votes
1 answer
110 views

PDF of $z = \exp(j\varphi)$, where $\varphi \sim \mathcal{U}[-a, +a]$ [closed]

How can I find the PDF of $z = \exp(j\varphi)$, where $\varphi \sim \mathcal{U}[-a, +a]$, $i.e.$, a uniformly distributed r.v.? My difficulty here is that it involves complex numbers and I don't know ...
Felipe Augusto de Figueiredo's user avatar
7 votes
1 answer
611 views

Do there exist three pairwise independent random variables, such that their sum is zero?

Do there exist such three non-constant pairwise independent random variables $X, Y, Z$ such that $X + Y + Z = 0$? I managed only to prove the following two facts: If such $X, Y, Z$ exist, they are ...
Chain Markov's user avatar
  • 2,618
0 votes
1 answer
191 views

Coupling between two distributions

Consider $s = \Theta(n^{\delta})$ for a $\delta\in (0,1)$ and let $p\in (0,1)$ with $m = \lfloor pn\rfloor$. Consider the random variable $Y$ which chooses $m$ elements from $\{1,\ldots,n\}$ such that ...
mssmath's user avatar
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1 vote
1 answer
113 views

$K(x,y)\in L^{\infty}(R^n\times R^n, m\times m)$, $K(x,y)=K(y,x)$, so $K(x,y)=\sum_{k=1}^{\infty}\lambda_k \phi_k(x)\phi_k(y)$, are $\phi_k$ bounded?

Consider a symmetric function $$ K(x,y):R^n \times R^n \to R $$ satisfying $K(x,y)=K(y,x)$ and $$ \int_{R^n\times R^n} K^2(x,y)dm(x) dm(y) <\infty. $$ Let $m$ be a probability measure on $R^n$. ...
mathmetricgeometry's user avatar

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