# 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.

5,175 questions

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**1**answer

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):= \...

**5**

votes

**0**answers

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 } &...

**3**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**1**answer

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**

votes

**2**answers

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**

votes

**1**answer

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**

votes

**1**answer

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 ...

**1**

vote

**1**answer

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 ...

**0**

votes

**1**answer

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**

vote

**1**answer

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 ...

**2**

votes

**0**answers

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{ ...

**4**

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**0**answers

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**

votes

**1**answer

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**

votes

**1**answer

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\...

**2**

votes

**1**answer

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**

votes

**1**answer

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**

votes

**1**answer

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**

votes

**1**answer

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 ...

**2**

votes

**0**answers

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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**0**answers

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 ...

**3**

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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}...

**5**

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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 ...

**3**

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**0**answers

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**

votes

**1**answer

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 ...

**1**

vote

**0**answers

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**

vote

**1**answer

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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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 }...

**2**

votes

**0**answers

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) =...

**13**

votes

**1**answer

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 ...

**0**

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**0**answers

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 ...

**1**

vote

**0**answers

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**

votes

**1**answer

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 $\sum_i n_i$ ...

**14**

votes

**3**answers

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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**0**answers

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**

votes

**1**answer

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**

votes

**1**answer

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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**1**answer

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**

votes

**1**answer

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 ...

**2**

votes

**0**answers

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 $C$...

**4**

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**0**answers

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?

**5**

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**1**answer

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 ...

**7**

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**3**answers

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**

votes

**1**answer

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\...

**1**

vote

**1**answer

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 ...

**1**

vote

**1**answer

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**

votes

**1**answer

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,...

**2**

votes

**2**answers

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_\...

**4**

votes

**0**answers

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

**2**answers

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 ...