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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12 views

expectation of the function of Wishart matrix eigenvalues

For Given a $N×M$ random complex gaussian matrix $X$ where $M=XX^H$, let $\lambda_1>\lambda_2>....>\lambda_N$ be the ordered eigenvalues of $M$ my objective is to get an estimation of \begin{...
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1answer
42 views

Intuition behind Gubinelli derivative

I apologise for the confusion of the following sentences. I'm lazy to give more information about Rough path theory as Is a fairly broad subject. On page 14 of "A Course on Rough Paths With an ...
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0answers
25 views

How to understand the distance between two edges on a graph in the bond percolation? [closed]

onsider the Euclidean lattice $\mathbb{Z}^d$ and the graph $G=(V(\mathbb{Z}^d), E(\mathbb{Z}^d))$, I try to define $\{f:d(e,f)>k\}$ for a constant $k$ where $e, f$ are two edges in $E(\mathbb{Z}^d)$...
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1answer
64 views

Test for OU-Process

Suppose that I'm given a sample from time-series $(x_n)_{n=1}^N$ and want to decide if it comes from an OU process or not. Is there a (rigorous) test I can use? So far, everything I've seen is hand-...
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49 views

How to solve this stochastic optimization problem?

How one can solve the following stochastic optimization problem? \begin{align} \max\quad& \mathbb{E}[\mathbf{1}^{\mathrm{T}}X]\\ \text{s.t.} \quad& \mathbb{E}[\mathbf{A}X]\leq\mathbf{1}_{m\...
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1answer
80 views

Diffeomorphism for mapping one SDE into another

Let $Y_t,X_t$ be $(\Omega,\mathcal{F},\mathcal{F}_t,\mathbb{P})$-adapted Markov diffusion processes with valued in $\mathbb{R}^n$. (When) does there exist a diffeomorphism $\phi:\mathbb{R}^n\to \...
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1answer
66 views

On the concentration of Lipschitz functions near its expectation, where the vector has identical but not independent, components

Consider the random vector $X:=(X_1\dots X_1) \in \mathbb{R}^n, X_1 \sim \mathcal{N}(0,1).$ Notice the identical components, they're identically distributed but not independent. Now, I was wondering ...
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2answers
163 views

Connections between two constructions of infinite dimensional Gaussian measures

Let me discuss two possible constructions of Gaussian measures on infinite dimensional spaces. Consider the Hilbert space $l^{2}(\mathbb{Z}^{d}) := \{\psi: \mathbb{Z}^{d}\to \mathbb{R}: \hspace{0.1cm} ...
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0answers
87 views

On a concentration bound without i.i.d. assumptions

Pick uniform independent random integers $x_1,\dots,x_k\in\{0,1,\dots,2^t-2,2^t-1\}$ and let $x_1$ be minimum and $x_k$ be maximum. Pick $k-1$ non-negative integers $g_1=x_1-x_2$ to $g_{k-1}=x_{k-1}-...
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1answer
49 views

How to get the E[XY] when X and Y are both binary variables? [closed]

Suppose $W_i \in \{0,1\}$, then the textbook said $$E[W_iW_{i'}]=Pr(W_i=1)Pr(W_{i'}=1|W_i=1)$$ Why this equation holds?
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1answer
44 views

Distribution of a stopped random sum, with subexponential stopping time

I am trying to find a reference (or, if it's false, a counterexample) for the following sort-of-intuitive fact: if $\tau$ is a stopping time with a subexponential probability distribution, and $(X_n)_{...
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55 views

Distribution of gaps between uniform random variables

Pick $k$ uniform independent random integers $x_1,\dots,x_k\in\{0,1,\dots,2^t-2,2^t-1\}$ and denote $y_{\sigma(i)}=x_i$ where $\sigma$ is a permutation in $S_n$ such that $y_1\leq y_2\leq\dots\leq y_{...
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45 views

Probability distribution of sum of squares of sum/difference of uniform random variables

If we pick $k$ uniformly random integers $x_1,\dots,x_k\in\{0,1,\dots,2^t-2,2^t-1\}$ then what is the probability distribution of the quantities $$\sum_{\substack{i,j=1\\i\leq k}}^n(x_i-x_j)^2$$ $$\...
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20 views

(Conditional) Independence of additive Gaussian noise disturbed sensor

Let's assume a random variable $Z\in\mathbb{R}^n$ being $Z_{i} = X + Y_i$, where $X\in\mathbb{R}^n$, $Y_i\in\mathbb{R}^n$ and $i\in\mathbb{N}$ being an index. It is assumed that $Y_i\perp Y_j$ (where $...
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72 views
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Correlation of stopping times for integral of Brownian motion increment

Let $\mu(x):=\int_{\epsilon}^{x}\exp\{B_{s+\epsilon}-B_{s-\epsilon}\}ds$, where $(B_{s})_{s\geq 0}$ is a Brownian motion (starting at $B_{0}=0$) and epsilon is small $0<\epsilon\ll 1 $. Consider ...
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1answer
83 views

Scalar product of random unit vectors

Let $X,X'$ be two random vectors on the sphere $S^{d-1}$. What is the distribution of their dot product $X\cdot X'$ in the following cases: $X,X'$ independent with uniform distribution on the sphere $...
3
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1answer
145 views

Do Lyapunov functions imply exponential integrability of hitting times?

I have a question of some integrability of hitting times. Let $X=(\{X_t\}_{t \ge0},\{P_x\}_{x \in E})$ be a diffusion process on a locally compact separable metric space $E$. We assume that there ...
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1answer
90 views

Reference request: The transform of a bounded random variable has a zero in the complex plane

Together with coauthors I'm working on a paper where we use the following Proposition: If a real-valued random variable $X$ has bounded support, then except in the trivial case that $X$ has all ...
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If X and Y are dependent, can we prove that P(XY) can be approximated by P(X)*P(Y)? [closed]

Given two variables X and Y, which are not independent. Can we provide some derivations that $P(XY)$ can be approximated by $P(X)\cdot P(Y)$, i.e., $P(XY)\approx P(X)\cdot P(Y)$. This confused me ...
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1answer
53 views

$ \int_{E}^{*}{\psi (t) d\mu(t)}=\int_{E}{\phi (t) d\mu(t)} $

Let $(T, \mathcal{A}, \mu)$ be an arbitrary measure space. The outer integral over $(T, \mathcal{A}, \mu)$ of a (possibly nonmeasurable) function $\psi: T\to (-\infty, +\infty]$ is defined by: $$ \...
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22 views

On full rank submatrices of a construction

Take two matrices $T_1$ and $T_2$ in $\mathbb Z^{n\times n}$ with entries uniformly in $[-b,b]\cap\mathbb Z$ at some $b>0$. The matrices will be of rank $n$ each with probability at least $1-\frac1{...
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61 views

2-d geometric Brownian motion hitting time distribution

I am trying to solve following problem: Given two independent geometric Brownian motions $\frac{d x_t}{x_t}=\mu_x dt + \sigma_x dw_t^x$ and $\frac{d y_t}{y_t}=\mu_y dt + \sigma_y dW_t^y$ and ...
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Can someone explain why we have this for a GP regression conditioned on the observations

Consider a Gaussian Process (GP) regression $y_t=f(x_t)+\epsilon_t$ with iid noise $\epsilon_t \sim N(0,\sigma^2)$. Can someone please explain why conditioned on $(y_1, \ldots, y_{t-1})$, $\{x_1, \...
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25 views

Dominance convergence theorem to compute expectation of a sequence of random variables defined by their time derivatives

Let $ (X_t) $ be a stochastic process, and define a new stochastic process by $ Y_t = \int_0^t f(X_s) ds $. Then consider a sequence $X_t^0,X_t^1,\ldots, X_t^n$ for which we get $Y_t^0,Y_t^1,\ldots, ...
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58 views

Can we say that $ \frac{1}{n}\sum_{i=1}^{n}{f_i(t)\to 0 }~\text{a.e} $ [closed]

Let $(E,\mathcal {A},\mu)$ be a finite measure space and $\{f_n\}$ be a sequence bounded in $L^1$, such that: $$ f_n(t)\to 0 ~~\text{a.e and in } L^1 $$ Can we say that $$ \frac{1}{n}\sum_{i=1}^{n}{...
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1answer
54 views

Minimum mean over all random variables subject to logarithm constraint

Does the following problem have a solution? $$ \min_X \mathbb{E} X \quad\text{subject to}\quad \mathbb{E} \log X = C. $$ Here, the minimization is with respect to all integrable random variables $X$ ...
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1answer
60 views

Distribution of the direction of Gaussian random variable

Let $X$ be a complex normal random variable. (Or, equivalently, a 2D real normal.) Is it possible to say anything useful about the distribution of the phase of $X$? Is it possible to do estimation on ...
4
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1answer
91 views

Rates of convergence to Tracy-Widom?

$\renewcommand{\!}{\mathbf} \renewcommand{\Ai}{\operatorname{Ai}}$ One can define the Tracy-Widom distribution as the Fredholm determinant $F_2(t)=\det(\mathbf I-\mathbf A)$ where $$\mathbf A(x, y)=\...
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1answer
86 views

Sampling i.i.d. variables with restrictions

General Problem: Suppose $X_1,\ldots,X_n \sim \mathbb{P}_X^{\otimes n}$ is a finite sequence of i.i.d. (real- or integer-valued) random variables. Suppose $A\subseteq \mathbb{R}^n$ is a set of "...
2
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1answer
104 views

A question about finitely additive integration

Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space ($\mathbb P$ is countably additive). Let $\{p_\omega: \omega \in \Omega\}$ be a family of (countably additive) probability measures on $(\...
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1answer
36 views

Ensemble averaging in a random graph (or network) in the large $N$ limit [closed]

I have a random graph/network described by the adjacency matrix $(a_{ij})_{N\times N}$ where $a_{ij}=1$ with probability $p$. Each node in the graph is associated with a continuous quantity $\eta_i=\...
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1answer
403 views

Boundedness of total current in electrical network

Consider the following symmetric matrix (adjacency matrix): $$A=(a_{ij})_{1\leq i,j\leq n}$$ such that $a_{ij}=a_{ji}, a_{ii}=0$ and $a_{ij}=0$ for $|i-j|\geq k$ where $k\geq3$. We also have $1\leq a_{...
6
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1answer
430 views

Which books should I read in order to be prepared to study information geometry?

At the moment, I am preparing my master's thesis (in statistics) and I intend to keep studying in order to pursue a doctoral degree. To be precise, I am mainly interested in studying Information ...
2
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1answer
91 views

Eigenspace of Gaussian Markov operator

Consider the (one-dimensional) Gaussian distribution $Q := N(\nu,\tau^2)$ and the (Gaussian) Markov operator \begin{equation*} \begin{array}{rccc} R : & L_1(\mathbb{R},\mathcal{B}(\mathbb{R}),Q) &...
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1answer
87 views

Explicit constant for Carbery–Wright inequality

The Carbery-Wright is a seminal result about the anti-concentration of polynomials of Gaussian random variables. See e.g. https://arxiv.org/pdf/1507.00829.pdf, Theorem 1.4, for the precise statement. ...
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2answers
289 views

Exponential decay of voltage potential difference

Consider the following adjacency matrix of a complete graph: $$A=(e^{-|i-j|})_{1\leq i\neq j\leq n}$$ with 0 on the diagonal. Let $D=diag\{d_1,...,d_n\}$ be the degree matrix where $d_i=\sum_{j\neq i}...
3
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0answers
86 views

Probability that a Voronoi cell contains exactly k random points

Consider two independent point processes in the unit square $[0,1]^2$. The two point processes are identically independent and typically binomial/Poisson. One, say $\Phi^*$, is used to generate a ...
2
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2answers
115 views

is this process a Markov one?

Here is the problem I can't solve. Let $\xi_n$ $(n=1,2,3,\dots)$ be a sequence of i.i.d. random variables on $\mathbb{R}$ with density $p(x)>0$, let $\eta_n=\sum_{i=1}^{n}\xi_i^2$. Define $$\...
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0answers
178 views

Fourier transform of $f_a(x)= a^{-2}\exp(-|x|^a)$, $a \in (0,2)$, is decreasing in $a$

Can one show that Fourier transform of $$ f_a(x) = a^{-2} \exp(-|x|^a), \qquad a \in (0,2)$$ is decreasing in $a$? I have a solution for $a \in (0,1]$ which cannot be used for $a\in (1,2)$.
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31 views

Average number of edges of an induced graph by using the edge-based node selection technique on a graph with arbitrary degree distribution

Let $G(U,V,E)$ be a simple, undirected, bipartite graph and $U=\{u_1,u_2,{\cdots},u_n\}$ and $V=\{ v_{1},v_{2},\cdots,v_{n}\}$. Let $d_k^l$ be the number of vertex with degree $k$ in $l$, where $l \in ...
2
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1answer
60 views

Critical probability for Erdos-Renyi digraphs to be strongly connected

Given $p \in [0,1]$, an Erdos-Renyi graph ${ER}(n,p)$ on $n$ vertices is constructed by defining, for each unordered couple of distinct vertices ${i,j}$ an edge between $i$ and $j$ with probability $p$...
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0answers
149 views

Moments of Plücker coordinates on complex Grassmannian and log-concavity

Consider the Grassmannian $Gr(k,N)\simeq U(N)/(U(k)\times U(N-k))$ which parametrizes $k$-dimensional subspaces of $\mathbb{C}^N$. Let us put on it the $U(N)$-invariant probability measure. Let $\...
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0answers
159 views

Probability that a random multigraph is simple

Question. Consider a given sequence of $n$ integers $d_1$, $d_2$, $\cdots$, $d_n$ with $\sum_i d_i$ even and $d_i\le n$ for all $i$. One may sample a random multi-graph having this degree sequence ...
7
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0answers
251 views

Graph with path of length $\geq n$ along grid diagonals - a known result in graph theory?

Is the following lemma a well known result in graph theory? I am studying a basic existence result that appears to be simple yet powerful. I have not seen it stated as an important result in graph ...
8
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1answer
87 views

Sizes of connected components from a random choice in a grid

This is inspired by the illustration in this recently updated question. So we take a (fairly big) $n$ and an $n \times n$ grid where we draw at random one diagonal in each of the $1 \times 1$ squares. ...
0
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1answer
81 views

Finding a connection between two types of convergence

Please, help me find connections between two types of convergence: Let $\{X_n\}_{n\ge1}: (\Omega,F,P) \rightarrow (\mathbb{R},Bor)$ be a sequence of r.v., there are two convergences: 1) $X_n \...
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1answer
122 views

Is this probability inequality true?

This question may be simple, though I'm not managing to find an answer. Let $X$ and $Y$ be two dependent random vectors in in $\mathbb{R}^d$, with joint probability density $\mu(x,y)$ (with respect to ...
1
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1answer
203 views

Brinksmanship: how to achieve the best outcome by a single statement [closed]

This game is taken from Schelling's Game Theory: How to Make Decisions by R.V. Dodge, in which contenders practice brinksmanship to their own advantages. It goes as follows: Anderson, Barnes, and ...
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1answer
78 views

Can we say that: $ \sum_{n\geq 1}{\frac{1}{n}(f_n(\omega)-g_n(\omega))}<\infty\qquad a.e $

Let $(\Omega,\mathcal{A},\mu)$ be a finite mesure space, and $\{f_n\}$ and $\{g_n\}$ two $L^1$-bounded sequences, such that : $$ \sum_{n\geq 1}{\frac{1}{n}(F_n(f_n)(\omega)-g_n(\omega))}<\infty\...
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0answers
84 views

How to avoid using a probability distribution that doesn't exist?

I have this problem, of which I know the solution, but I'm looking for the mathematically proper way of writing it. Say I have a (infinite) population of people, where each individual is labeled by ...

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