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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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1answer
135 views

Hausdorff distance is a lower (or upper bound) for what probability metric?

In a metric space $X=(X, d)$, given a probability measure $\mu$ and two subsets $A$ and $B$ of positive measure, it's not hard to prove that $$ d(A, B) \le W(\mu|_A, \mu|_B), $$ where $d(A, B):= \...
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0answers
160 views

Determinant arising in a problem from probability

Consider the determinant: $$\Delta:= \left|\begin{array}{cccc} A_{j_1} & A_{k_1} & A_{j_1}A_{k_1} & 1 \\ A_{j_2} & A_{k_2} & A_{j_2}A_{k_2} & 1 \\ A_{j_3 } & A_{k_3 } &...
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0answers
85 views

Piece rank probability in this Stratego-like game

There's this game in a 9x8 board where 2 players take turns moving pieces. The players have pieces ranked 1-21. Players can't see the opponent's pieces' ranks, just positions. Pieces landing on the ...
11
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1answer
330 views

How nearly abelian are nilpotent groups?

It is not uncommon to read that "nilpotent groups are 'close to abelian'."1,2 Can this sentiment be made precise in the sense of the Turán and Erdős definition of "the probability that two elements of ...
3
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1answer
55 views

Lower bound on the sum of pmf squared of a hypergeometric distribution

I am working on a proof of correctness for an algorithm I came up with. I encountered the following problem en route. I would appreciate if anyone had some idea or could point me to the relevant ...
2
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2answers
66 views

Divergence rate of geometric sum of random variables

Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of strictly positive and identically distributed random variables and let $\beta\le 1$. I am trying to prove that $$ 0<\lim_{\beta\rightarrow 1}(1-\...
3
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1answer
68 views

Transfer-impedance matrix for edge correlations in random spanning tree

Suppose $G$ is a (weighted) connected graph and let $T$ denote a random spanning tree of $G$, chosen uniformly (or respecting the edge weights). It is known that for any distinct edges $e, f$ $$\...
6
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1answer
119 views

Transportation-cost inequality for pushforward measure

Let $X=(X,d_X)$ and $Y=(X,d_Y)$ be metric spaces and $\varphi: X\rightarrow Y$ be an $L$-Lipschitz map, with $0 \le L < \infty$. Suppose $\mu$ is a probability measure on $X$ which satisfies ...
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1answer
82 views

What is the order of the left tail of a mixture of non-central chi-square?

Let $\mu\sim N(0,1)$, $Z\sim N(\mu,1)$. Then $Z$ can be viewed as a mixture of Gaussians. It can also be viewed as a Gaussian but there is a prior for the mean. Let $X\sim\exp(\lambda)$ where the ...
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1answer
91 views

Sampling a uniformly distributed point INSIDE a hypersphere?

There is a simple algorithm to pick a random point ON an $n$-dimensional hypersphere. Is there one to sample a point from inside it? (Sampling points from a hypercube and rejecting them if they are ...
1
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1answer
118 views

Properties of Cameron Martin Space

In the case that I'm working with a separable Hilbert space, $H$, on which I have a trace class operator, $K$, that's coming from a Gaussian (i.e., $K$ is self-adjoint, and for simplicity, has trivial ...
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0answers
56 views

Recovering a rank-one matrix from its eigendecomposition after randomized rounding

Let $A = xy^T$ be a rank-$1$ matrix, and suppose every entry of $A$ is in $[0,1]$. We can create a binary matrix $A_{\rm rounded}$ by setting $$ [A_{\rm rounded}]_{ij} = \begin{cases} 1 & \mbox{ ...
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0answers
76 views

Geometric meaning of the chi-square “measure of association”

In Statistics, there's a standard test of independence of two random variables taking values in finite sets $I,J$. It relies on the computation of $\chi$-square statistics, $$ \chi^2:=\sum_{(i,j)\in ...
3
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1answer
247 views

wasserstein distance between distributions with bounded ratio

Let $p$ and $q$ be probability distributions on a metric space $X=(X, d)$ with densities $dp$ and $dq$, such that there exists $0 < \alpha < \beta < \infty$ satisfying $$ \alpha d p \le dq \...
4
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1answer
123 views

What is the nearest-neighbor distribution in this picture?

Consider the following process: sample $n$ points uniformly at random in the unit square, and for each point $i$, let $d_i$ be the distance from $i$ to its nearest neighbor. Finally, let $z_i = d_i\...
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1answer
105 views

Multiple Wiener-Ito integral distribution

Distribution of standard Ito integral is well known: $$I_1(f) = \int_0^T f(t)dB(t) \sim \mathcal{N}\bigg(0, \int_0^T f^2(t)dt\bigg).$$ Is it possible to find the distribution of multiple Wiener-Ito ...
2
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1answer
84 views

Stochastic domination of Gaussian random vectors

Let $S$ be the class of all $2$ by $2$ matrices of the form $$\begin{bmatrix} 1 & a \\ a & 1 \end{bmatrix},\, |a|\leq 1.$$ Is there a single matrix $M\in S$ such that for ...
4
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1answer
78 views

Distribution of the $pn$ shortest edges out of $n$ uniform points, $p\to 0$

Suppose I sample $n$ points independently and uniformly at random in the unit square, and then I select the $pn$ shortest edges between all pairs of points, for fixed $0<p<1$. For large $n$ and ...
0
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1answer
122 views

Predictability of countably valued accessible stopping times on complete and cadlag filtrations

The following question is motivated by this part of the proof of Lemma 2 on page 107 of the book Stochastic integration and differential equations of Philip Protter. Lemma 2. Let $T$ be a totally ...
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0answers
75 views

Matrix Chernoff sampling with out replacement

I am interested to know if the matrix Chernoff bound (see Theorem 5.1.1 in https://arxiv.org/pdf/1501.01571.pdf) holds if one samples without replacement. For example, the Bernstein inequality is ...
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84 views

Concerning some Tauberian-type asymptotics of Laplace transform involving $e^{-\sqrt{s}}$

There are some well-known Tauberian theorems concerning the asymptotics of the original function (say as $t$ tends to $0$) and that of its Laplace transform (as $s$ tends to infinity). I want to ask a ...
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0answers
144 views

Cardinality of extreme points of finitely additive probabilistic extensions

Let $\Omega = \{0,1\}^\mathbb{N}$, let $\mathcal{A}$ be the algebra generated by the open subsets of $\Omega$, where we use the product of discrete topologies, and let $\mathcal{F} = \sigma(\mathcal{A}...
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0answers
189 views

Examples in which probabilistic heuristic reasoning fails

There are examples of conjectures in which one can use probabilistic heuristic reasoning to show that they are very likely to be true. For instance, Freeman Dyson used probabilistic heuristic ...
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0answers
97 views

Maximal ergodic theorem on some dyadic intervals

What we refer to maximal ergodic theorem in this thread is the following: let $\left(\Omega,\mathcal F,\mu\right)$ be a probability space and let $T\colon\Omega\to \Omega$ be a measurable and measure ...
0
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1answer
85 views

how to derive stationary distribution of maximal entropy random walk

I was reading the paper 0810.4113v2, burda, which analyzed the stationary distribution maximal entropy random walk on the irregular lattice. I am confused on some of the steps. Description: The ...
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0answers
77 views

How does the graph of percolation probability $\Pi$ vs. $p$ vary for different finite values of $L$?

This is a sequel to my previous question. @Carlo's response here (to my comment) prompted me to ask this question: As mentioned in the textbook "Introduction to Percolation Theory" (Chapter 4) by ...
1
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1answer
120 views

Wasserstein interpolation between two probability measures on a metric space

Question 1 Given probability measures $\mu$ and $\nu$ on the same metric space $X=(X,d)$, and $\alpha \in [0, 1]$, is it always possible to find another probability measure $\lambda_\alpha$ on $X$ ...
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0answers
37 views

Reformulate Wasserstein constraint optimization on product space in terms of marginal

Let $X = (X,d_X)$ be a metric space and $Y$ be an abstract set (with at least two elements). Consider the metric on $X \times Y$ defined by $$d((x,y),(x',y')) = \begin{cases}d_X(x,x'),&\mbox{ if }...
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0answers
41 views

Probability Distribution on permutations with factor structure

Say I define a probability measure over the symmetric group $S_n$ as follows: I specify $n$ positive `potential' functions $G_i : \{1, 2, \cdots, N\} \to (0,\infty)$ I then set $$\mathbb{P}(\sigma) =...
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1answer
219 views

Number guessing game with lying oracle

You are probably already familiar with the usual number guessing game. But for concreteness I restate it. The usual game The Oracle chooses a positive integer $n$ between 1 and 1024 (or any power of ...
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0answers
34 views

Probability One Binomial is Greater than Another Dependent One

Let $\mathbf{n}\sim\text{Multinomial}(n,(p,q,r))\in\mathbb{N}^{3}$ where $p<q$. Can we find an upper bound for $$ \mathbb{P}(\mathbf{n}_1 > \mathbf{n}_2) $$ of order $e^{-C n}$ (with explicit ...
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0answers
139 views

Matrix Bernstein for spherical random variables

Theorem 4.1 in Tropp's Matrix Concentration Inequalities provides an exponential concentration inequality for the spectral norm of a matrix $Z = \sum_i \gamma_i B_i $, where $\gamma_i$ are an i.i.d. ...
2
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1answer
127 views

Colored balls and bins — asymptotic behavior

Suppose I have $N$ bins and a set of balls with $m$ different colors, where there are $n_i$ balls of color $i$. I also have values $0 < p_i \leq 1$ for all colors $i$. I throw all $\sum_i n_i$ ...
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3answers
538 views

Union of random intervals with total length equal to infinity

Let $a_1,a_2,\dots$ be a sequence of positive numbers less than $1$, such that $$\sum_{n=1}^\infty a_i= \infty,$$ and $S^1 = \mathbb{R}/\mathbb{Z}$. Suppose $I_1,I_2,\dots$ be random intervals with ...
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0answers
67 views

Is the maximum of independent Poisson random variables log-concave?

Let $X_1,\ldots, X_n$ be independent Poisson random variables with parameter $\lambda > 0$. Define $$M_n=\max \lbrace X_1.\ldots, X_n \rbrace,\,\,p_j=\mathbb{P}(M_n=j).$$ Is it the case that $M_n$ ...
3
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1answer
131 views

about an interesting moment generating function

Let $X_1,\ldots,X_n$ be iid Rademacher variables, i.e., $P(X_1=1)=P(X_1=-1)=1/2$. CLT says that $Y_n\equiv \sqrt{n}\bar{X}$ converges in distribution to $N(0,1)$ as $n\to\infty$. So $Y_n^2$ is ...
4
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1answer
160 views

What is the probability for a Binomial to be greater than other?

Let $X = B(n, p)$ and $Y = B(k, q)$ be two random variables with binomial distribution and, let $s$ be a positive integer. Assume that $n > k+s$ and $np \geq kq+s$. What is the probability for $X$ ...
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1answer
173 views

Is this a random walk? Does it have a name?

By combining two methods I've stumbled into a rather messy random walk situation. I have the typical random walk setup $$\theta_{i+1} = \theta_{i} + \hat{\theta}_{i+1}$$ Where $\hat{\theta}_{i+1} \...
3
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1answer
145 views

Can we transform $\int_\rho^1 (W_t - W_{t-\rho}) \,dW_t$ to make its law $\rho$-invariant?

I just bumped into the stochastic integral $$ \int_\rho^1 (W_t - W_{t-\rho}) \,dW_t $$ where $0 < \rho < 1$ is a constant and $W$ is a standard Wiener process. It would be nice if we have a ...
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0answers
44 views

Continuous Local Martingales under time change under what conditions are they still local martingales?

This question is motivated by reading a section in Continuous Martingales and Brownian Motion by Daniel Revuz, Marc Yor. In Chapter V there is a section on time-change: Definition: A time change $C$...
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0answers
146 views

gaussian upper bound on spherical heat kernel

It is known that the heat kernel on n-sphere satisfies $p_t(x,y)\leq Ct^{-n/2}e^{-d(x,y)^2/5t}$ for all $t\in (0,T)$. Can something be said about how big C needs to be?
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1answer
283 views

KL divergence and mixture of Gaussians

Do we have an exact formula to compute the KL divergence between 2 mixtures of Gaussians (i.e convex combinations of a finite number of Gaussian distributions)? If not exactly known, are there good ...
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3answers
389 views

Coverage of balls on random points in Euclidean space

We have n points randomly distributed in a d-dimensional unit hypercube. We randomly sample k of those points and center a ball with radius r on each of those k points. Does there exist an estimate of ...
5
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1answer
243 views

Approximation of Wasserstein distance between $p_\theta$ and $p_{\theta + d\theta}$

Given a parametric family of distributions $\{p_\theta\mid\theta \in \Theta\}$, one can show that under some regularity conditions, the following approximation is valid $$\operatorname{KL}(p_\theta\...
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1answer
276 views

A simple two variable analytic inequality, inspired by probability

I'm trying to prove the following inequality: $$ bf_1g_1 + (x-b)f_1g_0 + (y-b)f_0g_1 + (1-x-y+b)g_0f_0 \le (|f_1|^p x + |f_0|^p (1-x))^{1/p} (|g_1|^p y + |g_0|^p (1-y))^{1/p} $$ where $0\le xy\le b\le ...
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1answer
63 views

Is the law of $\sup_{l \leq t \leq u} \frac{|B_t|}{\sqrt{t}}$ atomless?

Let $B_t$ denote a standard Brownian motion, and $0 < l < u$. I am wondering if the law of $\sup_{l \leq t \leq u} \frac{|B_t|}{\sqrt{t}}$ is atomless, that is, $\mathbb{P}\left(\sup_{l \leq t \...
2
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1answer
193 views

Schwartz space on $\bigcup_{n=1}^CR^n$

I have an application where I need to work with the following idea. Let the space $\bigcup_{n=1}^C \mathbb{R}^n$ be associated with the metric $d$ such that for $x=(x_1,\cdots,x_n)$ and $y=(y_1,\cdots,...
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2answers
77 views

Draw samples from distribitions in the neighborhood of a fixed distribution

Disclaimer Sorry in advance for vagueness. I'm still trying to get my ideas right on this one. Setup So, let $P$ be a distribution on a Euclidean space $X$ with an $\ell_p$ metric, and let $P_\...
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0answers
93 views

Cadlag and adapted (usual conditions assumed) imply progressively measurable (related to Protter's Stochastic Calculus theorem 6)

Hi maybe someone on here can help me. I have been stuck on showing this fact for several months. I asked this question in the stack exchange and it has floated around for a while but to no avail. ...
5
votes
2answers
167 views

Improving equi-integrability for a family $\mathcal F$ in $L^1(\Omega)$

Let $\mathcal F$ be a weakly compact subset of $L^1(\Omega)$. Dunford–Pettis theorem says that $\mathcal F$ is uniformly integrable. Also, by de la Vallée-Poussin theorem we can find an increasing ...