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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Does this filtration have a name?

In the context of Ethier&Kurtz Markov Processes: Characterization and Convergence (Chapter 4, equation (3.2)) as well as the two papers Martingale problems for conditional distributions of Markov ...
Mushu Nrek's user avatar
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Hunting an invisible target

An invisible target on the integer line starts at $0$. On each round it either stays put, moves to the left or moves to the right by $1$ with probability $\frac{1}{3}$ each. You are then asked to ...
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Deriving the distribution of standardized variables with empirical mean and standard deviation

I'm working with a set of independent and identically distributed random variables $\{ x_i \}_{i=1}^N$, where each $x_i$ follows a Gaussian distribution $P_X(x) = \mathcal{N}(x; \mu, \sigma^2)$. This ...
user1172131's user avatar
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Asymptotic Independence of random walks from increments?

Suppose we have two random walks $(S_n:n\geq 1)$ and $(T_n:n\geq 1)$ building from independent identically distributed increment vectors $\{(X_k,Y_k):k\geq 1\}$, i.e. $S_n=\sum_{k=1}^n X_k, T_n=\sum_{...
MikeG's user avatar
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Multi-scale 3- and 5-arm exponents for critical planar percolation

Consider critical site percolation on the planar triangular lattice. Denote by $A_j(m,n)$ the event that there are $j$ arms (paths from the inner boundary to the outer boundary) of alternating colour ...
Julius's user avatar
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The fluctuations of a random path

Suppose I have a $n \times n$ square grid and for each square, I assign 1 with probability $\frac{1}{2}$ and 0 with probability $\frac{1}{2}$. On the boundary, I put 1s on the lower half and 0s on the ...
Frederik Ravn Klausen's user avatar
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Embedding a Markov chain in a Markov process

Let $X_{t\ge 0}$ be a Markov process with values in a metric space $(\mathcal{X},d)$ defined on a probabiltiy space $(\Omega,\mathcal{F},\mathbb{P})$ and let $(\tau_n)_{n=1}^{\infty}$ be a sequence of ...
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Faster Convergence in CLT for sums and convolutions of Gaussians?

Let $n\in\mathbb{N}$ and $\sigma>0$ be fixed. I have a certain class $\mathcal{C}$ of random variables I am interested in analyzing. This contains $\vec X\sim \mathcal{N}(0, \sigma^2I_n)$ Sums of (...
Mark Schultz-Wu's user avatar
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Sum of arrival times of Chinese Restaurant Process (CRP)

Suppose that a random sample $X_1, X_2, \ldots$ is drawn from a continuous spectrum of colors, or species, following a Chinese Restaurant Process distribution with parameter $|\alpha|$ (or ...
Grandes Jorasses's user avatar
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Existence of the limit of periodic measures

Let $T: X \to X$ be a continuous map over a compact metric space. We say that a measure $\mu$ is $T$-invariant if $T_{\ast} \mu= \mu$. We denote by $M(X, T)$ the space of all $T$-invariant Borel ...
Adam's user avatar
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Distance distribution for Cayley graphs of the fintie Heisenberg groups H3(Z/nZ) approaches Gaussian for large "n"?

I wonder several questions about Cayley graphs of finite Heisenberg groups H3(Z/nZ). Question 1: do we know the diameter dependence on "n", at least for the standard choice of generators ? ...
Alexander Chervov's user avatar
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Can we construct close martingales if their terminal marginal laws are close?

Let $M=(M_t)_{0\le t\le 1}$ be a real-valued continuous martingale. Let $\mu := {\rm Law}(M_1)$ and $\varepsilon \in (0,1)$. For any $\nu$ satisfying $W_2(\mu,\nu)\le \varepsilon$, can we construct ...
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Thinning of (mixed) binomial point process

Let $N= \sum_{i=1}^M \delta_{X_i}$ be a mixed Binomial process over $(\mathbb X, \mathcal X)$. I.e., $M$ is a $\mathbb Z_+$ valued random variable with probability mass function $q_M(m)$, $m=0, 1, \...
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Random walk with same directions and different step sizes

Let $X\sim e^{iU}$, where $U$ is uniformly distributed on $(0, 2\pi]$. Define $\chi_1, \cdots, \chi_t$ as i.i.d. random variables with the same distribution as $X$. Consider the following two random ...
Farzad Aryan's user avatar
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Construct a Bregman divergence from Wasserstein distance

I was wondering whether one has studied the Bregman divergence arising from a squared Wasserstein distance. More precisely, let $\Omega\subset \mathbb{R}^d$ be a compact set and $c\in \Omega\times \...
John's user avatar
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Iterated chaos expansion

Using the notations from Normal Approximations with Malliavin Calculus, Chapter 2 random variables $F$ in the probability space generated by an iso-normal Gaussian family $X(h)$, $$E[X(h)X(g)] = \...
Julian's user avatar
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Distribution of waiting time conditioned on a fixed time length

FYI, this question is a duplicate from math stack exchange I ask here again because I got no response. Suppose, I work in a factory production line. The time for me to finish wrapping product $A$ (or $...
Fellow InstituteOfMathophile's user avatar
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Time reversal of infinite-dimensional SDE

Consider the SDE $${\rm d}X_t=b(t,X_t) \, {\rm d}t+\sigma(t,X_t) \, {\rm d}W_t,\tag1$$ where $b:[0,T]\times V\to H$, $\sigma:[0,T]\times V\to\operatorname{HS}(U_0,H)$, $$V\subseteq H\subseteq V^\ast\...
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Lower & upper bound on the maximal component given the system of power sums

Given a non-negative vector $x=(x_1,x_2,\dots,x_n)\in\mathbb{R_{>0}^n}$ and $m\in\mathbb{N}$, construct a system of power sum symmetric polynomials (or norms, if you like) $$ \begin{cases} x_1+x_2+\...
Polylemma's user avatar
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A question on the convex hull of independent random walks

Consider $m$ independent random walks $X^1_n, \dots, X^m_n$ driven by a probability measure $\mu$ in $ \mathbb{Z}^d$. Assume that the $\mu$ has no drift, that is, the expected value of a $\mu$-...
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A few questions on Feller processes

Update. Most of my questions have been answered in the comments. I am adding these answers to the post. There are at least three definitions of Feller semigroup and the corresponding processes: $C_0 \...
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Defining a metric on $\mathbb Z^n$ using Green's function for the simple random walk

Let $G$ be Green's function for the simple random walk on $\mathbb Z^n$ for $n\ge 3$, i.e., $G(x)$ is the expected number of visits to $x$ when the walk starts at the origin. Define $d(x,y)=G(x-y)^{1/(...
Alexander Pruss's user avatar
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141 views

Fractional Brownian motion covariance with a twist

Let $H \in (0, 1)$, $D \in \mathbb{R}$ and assume that the following function $$ r ( t, s ) = \frac{1}{2} \, \Big[ t^{2H} + s^{2H} - | t - s |^{2H} \Big] + D \, t^H s^H, \quad t, \, s \geq 0 $$ is ...
tsnao's user avatar
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Reference request : A SPDE model

Let $\Omega_0\subset\mathbb R^d$ be open and bounded with sufficiently smooth boundary $\partial\Omega_0$. Let $O\subset \Omega_0$ be a random open subset. Set $\Omega:=\Omega_0\setminus O$. Consider ...
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Joint tail for Brownian motion $P[B_{t_1}>g_1,...,B_{t_n}>g_n]$

Maybe not surprisingly there seems to be a lack of in-depth study of sharp estimates for the joint tail of Brownian motion over different times $$P[B_{t_1}>g_1,...,B_{t_n}>g_n]$$ for strictly ...
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Why is it impossible to create a numerically balanced die with more than 120 sides?

I allow myself to contact you as a mathematics enthusiast. I have recently been intrigued by the concept of balance in dice and the assertion that it would be impossible to create a numerically ...
Matthieu Nauly's user avatar
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222 views

Ball games: How to allocate $N$ balls into $M$ boxes so as to maximize the expected number of taken balls

Consider the following ball games, which looks like very intuitive and simple but I have tried for a long time. Assuming we have $M$ identical boxes and $N$ identical balls, we distribute these $N$ ...
koko's user avatar
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Random matrices may be asymptotically free but never free themselves?

It is well known that independent $N\times N$ unitarily-invariant random matrices (or independent families of random matrices) may be asymptotically free as $N\to \infty$ with respect to the ...
user510187's user avatar
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Dirichlet series solution to Poisson Point Process question (repost from math.SE)

Reposting here because the bounty on the original math.SE post expired, with no solutions or comments received. For any discrete subset $S$ of $\mathbb{R}^d$, consider a digraph formed by placing an ...
Jim Ferry's user avatar
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Equivalence of score function expressions in SDE-based generative modeling

I am studying the paper "Score-Based Generative Modeling through Stochastic Differential Equations" (arXiv:2011.13456) by Yang et al. The authors use the following loss function (Equation 7 ...
Po-Hung Yeh's user avatar
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222 views

Singular values of Kronecker product of random matrices

I'm looking for a way to evaluate $\mathbb{E} \| (\mathbf{X} \mathbf{Q})^+ \|$ for a random matrix $\mathbf{X} \in \mathbb{R}^{r \times n}$ and a (fixed) matrix $\mathbf{Q} \in \mathbb{R}^{n \times \...
Uria Mor's user avatar
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Measure algebra for a family of probability measures

Let $(X,B,P)$ be a probability space, $I_P$ the $\sigma$-ideal of $P$-null sets and \begin{align} B_P = B \ltimes I_P &= \{ A \mathbin{\triangle} N \mid A \in B, N \in I_P \} \end{align} the ...
Packo's user avatar
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What is an efficient non-adaptive group testing scheme if the number of defectives, $d$, grows proportionally to the number of items, $n$?

Suppose that for some $p \in \left(0, 1\right)$ and some $n \in \mathbb{N}$, we have $n$ independent Bernoulli random variables, $X_{1}, X_{2}, \dots, X_{n}$, each with mean $p$. We shall call $X_{1}, ...
Matthew Barber's user avatar
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A question related to the jumps of a Levy process

The Lévy–Khintchine formula says that any Lévy process, $X=(X(t), t \geq 0)$, has a specific form for its characteristic function. More precisely, for all $t \geq 0$, $u \in \mathbb R^d$: $$ \mathbb{E}...
André Goulart's user avatar
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133 views

An attempt to define expected value of a Riemannian manifold valued random variable - what'll go wrong?

Let $X:\Omega\to (M,g)$ be a random variable taking values in a Riemannian manifold $(M,g)$ with the Riemannian volume form denoted by $dvol_g(x).$ We know that there's no standard way to generalize ...
Learning math's user avatar
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45 views

$\sigma$-compactness of probability measures under a refined topology

Denote Polish spaces $(X, \tau_x)$ and $(Y, \tau_y)$, where $X$ and $Y$ are closed subsets of $\mathbb{R}$. Consider a Borel measurable function $f: (X \times Y, \tau_x \times \tau_y) \rightarrow \...
Hans's user avatar
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Linear regression with heavy-tailed gradient noise and generalized CLT

I want to investigate the distribution of the stochastic gradients when applying SGD to linear regression, in very simple but non-trivial situations. For dimension $1$ and when estimating only the ...
Stefan Perko's user avatar
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124 views

Almost sure convergence of a recursively defined sequence

Let $(X_k)_{k\ge 1}$ be a sequence of i.i.d. positive random variables of exponential distribution $\mathcal E(1)$, i.e. $$\mathbb P[X_k>x]=e^{-x},\quad \forall x\ge 0.$$ Fix some $s\ge 0$. For ...
Fawen90's user avatar
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106 views

Log Sobolev inequality for log concave perturbations of uniform measure

Suppose $\Omega$ is a convex bounded open set of $\mathbb{R}^n$ (I would be happy with just $\Omega$ as the $n$-dimensional cube). Let $\mu$ be the uniform measure on $\Omega$ and consider the ...
Matt Rosenzweig's user avatar
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118 views

Can a diffusion process admit an invariant measure with a non-differentiable density?

The precise domain of the generator $A$ of an Itō diffusion on a Hilbert space $H$ (assume $H=\mathbb R^d$, if that's easier for you to work with) can usually not be determined explicitly$^1$. Usually,...
0xbadf00d's user avatar
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1 answer
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Functional integral formulas for the wave equation and other hyperbolic PDEs

The Feynman–Kac formula provides a functional (Wiener) integral representation of the solution $u$ to the heat equation \begin{align*} \partial_t u &= \frac{1}{2}\Delta_x u,\\ u(0,x) &= ...
Emily's user avatar
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Martingale regularization

Consider a submartingale $X,$ then for almost every $\omega \in \Omega,$ for every $v \in \mathbb{R},\lim_{u \in \mathbb{{Q},u \uparrow v}}X_u(\omega)$ exist in $\mathbb{R}.$ I was wondering if there ...
mathex's user avatar
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Limiting value of $\dfrac{1}{m}\mathrm{tr}(FAF^\top (FBF^\top)^{-1})$, where $F$ has iide $N(0,1)$ entries and $A,B$ are deterministic

Let $F=F_{m,d}$ be a random $m \times d$ matrix with iid entries from $N(0,1)$. Let $A=A_d$ and $B=B_d$ be deterministic $d \times d$ positive-definite matrices. In case it helps, it may be assumed ...
dohmatob's user avatar
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Recursive sequence of renewal type : when does one term dominate them all?

Let $(b_n)_{n \geq 0}$ be an increasing sequence of non negative real numbers. Let $(u_n)_{n \geq 0}$ be recursively defined by $u_0 =1$ and $$u_{n} = \sum_{k=0}^{n-1} u_{k} b_{n-k}$$ Find a ...
Olivier's user avatar
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An additive combinatoric probability question

I asked the question in cs-theory stack exchange but was advised a pure math forum would be more apt. Link to the question: https://cstheory.stackexchange.com/questions/52930/an-additive-combinatoric-...
Rishabh Kothary's user avatar
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0 answers
233 views

Correlation decay rate

Let $T$ be a continuous transformation of a probability measure space $(X,\mathcal{B}(X),\mu)$ and $\varphi ,\phi \in L^2(\mu)$ (so-called observable) . The correlation function of $\varphi ,\phi$ (a ...
Mrcrg's user avatar
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Local martingale for a (two-dimensional) diffusion

Let $X$ be a two-dimensional diffusion (a solution of $dX_t=f(X_t)\,dt+dB_t$, with $B$ a standard two-dimensional Brownian motion) living on some open set $\Lambda\subset \mathbb{R}^2$. Let $h:\Lambda ...
Serguei Popov's user avatar
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0 answers
75 views

Random walks on randomly evolving graphs

I am interested in analyzing a random walk on a growing tree with vertices labelled on a tree with following properties. The number of nodes at depth $k$ is a an exponential function of $k$. One can ...
user82261's user avatar
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Trouble understanding a Lemma in Pastur's Paper

I'm having trouble understand Eq 3.51 Lemma 3.3 in https://arxiv.org/pdf/2001.06188.pdf The basic premise is $$\begin{align} &\eta _{j}(t)=t^{1/2}\eta _{j}+(1-t)^{1/2}q_{n}^{1/2}\gamma _{j}, \;t \...
Anvit's user avatar
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Regularity on $\mathbb{T}^3$ of the "functional average" of a map $S : C^\infty(\mathbb{T}^3, \mathbb{R}) \to L^2(\mathbb{T}^3, \mathbb{R})$

For simplicity, let $C^\infty(\mathbb{T}^3, \mathbb{R})$ be the real Frechet space of periodic smooth functions on $\mathbb{R}^3$. Here, $\mathbb{T}^3$ is the $3$-dimensional torus. For a fixed ...
Isaac's user avatar
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