## New answers tagged pr.probability

0
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### About Palm distribution

C. Palm's theory of spatial point processes relies heavily on measure theory in an abstract setting. A more gentle introduction is given in the lecture notes Conditioning in spatial point processes. ...

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2
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### Why does non-decreasing entropy imply actual convergence to that max entropy distribution?

$\newcommand{\de}{\delta}\newcommand{\al}{\alpha}\newcommand{\be}{\beta}\newcommand{\R}{\mathbb R}\newcommand{\vpi}{\varphi} $Without loss of generality, the variance of $X_1$ is $1$.
Let $H:=\mathrm ...

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2
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### Proof of Krylov-Bogoliubov theorem

If you still need a reference:
You can find a clear exposition of the result and the proof of the theorem in "Ergodic Theory" by Einsiedler 2013 (free PDF here).
Or see the original work by ...

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2
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Accepted

### Reference request: “A random integral and Orlicz spaces”

I could not find it on the internet so I uploaded it here: https://www.transfernow.net/dl/20221126nnUfCto7

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### Sub-exponential tail bound for Poisson multiplied by cosine of an independent uniform random variable

$\newcommand\la\lambda\newcommand\de\delta\newcommand\si\sigma\newcommand\Th\Theta$There is hardly a reason for having the parameter $n$. So, let $\la:=\la_n\to0$, $X:=X_n$, $\Th:=\theta_n$, $Y:=X/\la$...

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### Conditional expectation of linear combination of Rademacher RVs

I give a crude lower bound, which does not use the distribution of $Z$, but only that $|Z|=\sqrt{d}$. It relies on the triangle inequality for the angular distance on the unit sphere.
I assume $c_1$ ...

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### Is there an inverse Lamperti transformation for diffusions?

Girsanov's applies to state dependent states. So it actually gives simpler answer if initially the volatility didn't depend on the state.
But for changing to a different volatility, the result is ...

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### Differentiability of characteristic functions and moments of the corresponding measure

Zygmund's example is the discrete random variable $X$ where
$\mathbb{P}(X=\pm n) = \frac{C}{n^2\log(n)}$ for integer $n \geq 2$. $C$ is a unique constant that makes this a probability distribution. ...

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### On conditions for the existence of $h\in L^1$ such that $h>0$ a.e

This is a characterization of $\sigma$-finiteness: If $h$ is an $L^1$-function with values in $(0,\infty)$ then $A_n=\{h\ge 1/n\}$ are measurable sets with $\bigcup_{n\in\mathbb N} A_n=\Omega$ and $\...

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### Can an a.s. non constant continuous martingale be differentiable with nonzero probability?

Almost surely, we can write for every $t \ge 0$, $M_t=M_0+\beta_{\langle M,M \rangle_t}$, where $\beta$ is some Brownian motion.
By Kahane's theorem, almost surely, for every $s \ge 0$, $\limsup_{\...

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2
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### How to get the lower bound of the following $\tau$?

By way of illustration, to get some insight, I plot $H_1(t)$ versus $t$ for a particular realization of the random matrix, with $n=1000$ [link to Mathematica notebook]
For any realisation with $H_1(0)...

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6
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### Quadratic variation of supremum of brownian motion

The quadratic variation is identically $0$, i.e.
$$\langle S, S \rangle_t = 0$$
for all $t$, almost surely.
To see this, note that $S$ is almost surely increasing, hence has bounded variation almost ...

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1
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### Concentration inequality for the spectral norm of the product of normalized Gaussian (and subgaussian) matrices in high dimensions

Vershynin, R. Spectral norm of products of random and deterministic matrices. Probab. Theory Relat. Fields 150, 471–509 (2011).
This result is a sharp bound on the spectral norm of $W=BA$, where $A$ ...

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### Concentration inequality for the spectral norm of the product of normalized Gaussian (and subgaussian) matrices in high dimensions

The Marcenko-Pastur resut (see, e.g., https://www.sciencedirect.com/science/article/pii/S0047259X85710512) gives you the Stieljes equation of the limiting spectral distributions of matrices of the ...

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2
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### Comparing diffusion processes in different metrics

The generator of the diffusion corresponding to a Riemannian metric (i.e., the diffusion process which the limit of the random walks such that their increments go along geodesics and the ...

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Accepted

### A question related to the CDFs of multivariate normal distribution

The joint cdf of a multivariate distribution uniquely determines the distribution.
So, if the cdf's of multivariate normal distributions are the same, then their mean vectors must be the same (and ...

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2
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### Joint irreducibility and aperiodicity of two independent Markov chains

$\newcommand{\X}{\mathcal X}\newcommand{\Y}{\mathcal Y}\newcommand{\si}{\sigma}\newcommand{\B}{\mathscr B}$Since $\X$ and $\Y$ are Polish spaces, the corresponding Borel $\si$-algebras $\B(\X)$ and $\...

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0
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Accepted

### On conditions for the existence of $h\in L^1$ such that $h>0$ a.e

It is false without assuming that the measure is $\sigma$-finite. Let $\mu$ be the counting measure on $\mathbb{R}$. Then $h>0$ a.e. means $h>0$ everywhere and clearly
$$
\int_{\mathbb{R}}hd\mu=\...

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2
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### Mutual information in large deviation theory

There's a few results. First of all there is the classical Sanov's Theorem.
One other result is about Gaussian measures.
For a centered Gaussian measure $\mu_0$ on Banach space $\mathcal B$ we can ...

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### Is there an inequality relation between KL-divergence and $L_2$ norm?

Now, I am trying to answer this question.
Proposition. If $p$ and $q$ are two probability densities, and (upper) bounded by $\tau_1$ and $\tau_2$, respectively, then
$$
KL(p,q) \ge \frac{1-\log(2)}{\...

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1
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Accepted

### Probability of accurate sparse recovery

A good starting point is "Mathematics of sparsity (and a few other things)" by E. Candes, or a book on compressed sensing such as "A Mathematical Introduction to Compressive Sensing&...

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4
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Accepted

### Origin of the term "connective constant"

Q: Is there some application where $\mu$ plays a role in some kind of "connectedness" which would excuse the name?
A: The application is to crystalline structure. The name originates from ...

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0
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### Forgery theorem: the Brownian motion stays close to any curve with positive probability

There was part of the hint that you ignored, namely using Levy's construction. To add some detail, the approximation of $S^{n}$ by BM is done recursively in each dyadic interval, rather than globally. ...

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### Vertex degree on random graphs

Just an amateur answer, I would assume there's a paper or known approach out there from experts.
I would partition the vertices into equal sets $U,V$ and delete edges to make it a bipartite graph. It ...

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Accepted

### A question about the proof of the Levy-Khintchine representation Theorem

$\newcommand\R{\mathbb R}\newcommand\ip[1]{\langle #1 \rangle}$In these notes, two related definitions of truncation functions are given. In Definition 5.6, a truncation function is defined as a ...

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Accepted

### The ratio of spectral edge of the GOE matrix

$|\lambda_1|=2+\delta_1$, $|\lambda_2|=2+\delta_2$, with $\delta_1$ and $\delta_2$ both of order $n^{-2/3}$, so
$$|\lambda_1|/|\lambda_2|=1+(\delta_1-\delta_2)/2+{\cal O}(n^{-4/3})=1+{\cal O}(n^{-2/3})...

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2
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### What areas of algebra could be interesting to probability theorists?

This is just an extension of the (5) point in the answer by @Henry.L. The basic notion is the algebraic probability space, an abstraction from the classical ones, Tao has a post on them, https://...

3
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### Quasi-random vs pseudo-random graphs

Someone else will probably have a better answer, but I can't leave a comment. In my experience "quasi-random graph" (almost?) always refers to the Chung Graham Wilson type graphs you ...

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2
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Accepted

### Decreasing tail integrals for nonnegative random variable $X$

This iff statement is false. Indeed, this iff statement can be restated as follows:
If
$$\ell(x):=\frac1{xS(x)}\int_x^\infty S(u)\,du\quad\text{and}\quad
m(x):=\frac1{S(x)}\int_x^\infty\frac{S(u)}{u}...

- 85k

0
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### Malliavin differentiability of solutions to SDEs

As Bass and Nualart mention (Theorem 2.2.1 in "MC and related topics"), they are talking about the matrix derivative DX, not the Malliavin derivative $D_{h}X$. That's what Bass means by &...

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0
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Accepted

### Weak solutions of linear parabolic PDEs and corresponding SDEs

One of the latest conditions on $\mu,\sigma$ are from "A Numerical Method for SDEs with Discontinuous Drift":
This result states that the SDE (1) ($dX_t = \mu(X_t)dt + \sigma(X_t)dW$) ...

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2
votes

### Hitting time of an Ornstein-Ulhenbeck process

It turns out that finding its density is still open as mentioned here: On the First Hitting Time Density of an Ornstein-Uhlenbeck Process.
The Laplace transform has been computed though as mentioned ...

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0
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### Using compactness method to prove the existence of a pathwise solution to an SPDE

This type of existence of first fixing an omega and then obtaining a solution is pretty standard eg. see "Regularization by noise and flows of solutions for a stochastic heat equation".
Is ...

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0
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### Expected value of a stochastic integral expression

Based on the comment of being interested in the hitting time for OU, it turns out that finding its density is still open as mentioned here: On the First Hitting Time Density of an Ornstein-Uhlenbeck ...

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0
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### Joint law of a standard Brownian motion and its local time at a nonzero level

In "Trivariate Density of Brownian Motion, Its Local and Occupation Times, with
Application to Stochastic Control"
at section 4 "The trivariate density with nonzero initial condition&...

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2
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### Vague convergence: confusion about the regularity of a signed Radon measure and that of its variation

First we state here some general results about Radon measures following Bichteler, K. Integration: A Functional Approach, Birkhauser, 1991 and Bichteler, K. Integration theory: With Special Attention ...

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### Vague convergence: confusion about the regularity of a signed Radon measure and that of its variation

$\newcommand{\Om}{\Omega}\newcommand{\Th}{\Theta}\newcommand{\B}{\mathscr B}\newcommand{\M}{\mathcal M}\newcommand\ep\varepsilon\newcommand{\de}{\delta}\newcommand{\R}{\mathbb R}$Take any $\mu\in\M(\...

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10
votes

Accepted

### Sum of a Poisson point process

The density function of $Z$ is the Dickman $\rho$ function, normalized (that is, divided by its mass, $e^{\gamma}$).
This function, $\rho\colon [0,\infty)\to (0,\infty)$, is defined via $\rho(t)=1$ ...

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5
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### Sum of a Poisson point process

There is a formula for the Laplace transform of any additive functional of a Poisson process with intensity measure $\lambda$. Specifically, for any non-negative measurable function $f$,
$$E\left[e^{-\...

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3
votes

Accepted

### Lower bound $L_{1}$-metric with $L_{2}$-metric for bounded pdfs, on common support

This is false. If (let's say) $f-g=1$ on a set of measure $\epsilon$ and $|f-g|\simeq \epsilon$ otherwise (and note that we can still normalize them both), then $\|f-g\|_1^2\simeq \epsilon^2$, $\|f-g\|...

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0
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### Minimal conditions on random vector $X \in R^n$ to ensure that $\lim_{t\to 0^+}\sup_{\|w\|_p = 1}\sup_{u \in \mathbb R}\mathbb P(|X'w-u| \le t)=0$

Claim. If $X$ has density, then $L(S_p^n,t) \longrightarrow 0$ in the limit $t \to 0^+$.
Indeed, if $X$ has density, then so does $F(X)$, for any continuous function $F:\mathbb R^n \to \mathbb R^m$. ...

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2
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### Minimal conditions on random vector $X \in R^n$ to ensure that $\lim_{t\to 0^+}\sup_{\|w\|_p = 1}\sup_{u \in \mathbb R}\mathbb P(|X'w-u| \le t)=0$

In Hengartner, W. and Theodorescu, R. (1973). Concentration Functions. Academic Press, New York. MR033144
they study the continuity properties and when the concentration function is zero even for the ...

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2
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Accepted

### When are the total variation distance and Hellinger distance comparable?

$\newcommand{\R}{\mathbb R}\newcommand{\al}{\alpha}\newcommand{\be}{\beta}
\newcommand{\De}{\Delta}$What you want is impossible for any reasonable class of probability distributions, including the ...

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1
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### Abstract Wiener spaces for pinned processes (e.g., Brownian Bridge)

A Gaussian process can be thought of as a "standard Gaussian" on its Cameron-Martin space $\mathcal{M}$. That is, given an orthonormal basis $\psi_1,\psi_2, \dots$ of the Cameron-Martin ...

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4
votes

Accepted

### A linearly distributed version of the balls into bins problem

From the referenced paper, I am writing in terms of their variables, $k$ is the number of bins or type of coupons:
Let $n_1$ be the time where the last of the missing events is observed. Let $n_2$ ...

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2
votes

Accepted

### Infinite-variance associated processes are (BL, $\theta$)-dependent

The answer is no. E.g., Let $X(t)=Z$ for all $t$, where $Z$ is any random variable with infinite variance. Then the process $(X(t)\colon t\in\mathbb Z)$ is positively associated. On the other hand, ...

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1
vote

Accepted

### Deduce that a function is zero on interval $[0,M]$

As you say, it is not difficult to prove that $g=0$ when there are finitely many fluctuations in sign.
More precisely, suppose we have points $0=a_0,a_1,\dots,a_n=M$ such that $g\geq0$ or $g\leq0$ in ...

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0
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### Ito-Levy decomposition for $\alpha$-stable processes?

$\newcommand\al\alpha\newcommand\R{\mathbb R}$Let $(X_t)_{t\ge0}$ be a nondegenerate $\al$-stable Lévy process (so that $P(X_t=a)\ne1$ for all $t\in(0,\infty)$ and all $a\in\R$).
According to (say) ...

- 85k

2
votes

Accepted

### Distribution of scaled Johnson-Lindenstrauss transforms

$\newcommand\ep\epsilon\newcommand{\de}{\delta}\newcommand{\R}{\mathbb R}$We have
\begin{equation*}
P((1-\ep)\|x\|\le\|Ax\|\le(1+\ep)\|x\|)\ge\de \tag{1}\label{1}
\end{equation*}
for some $\ep,\de$...

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