# Tagged Questions

**4**

votes

**1**answer

22 views

### Minimum separation between $m$ random points on an $n$-dimensional unit sphere

Consider points $v_1, \ldots, v_m \in R^{n}$, which are uniformly distributed on the $n$-dimensional unit sphere $S^{n-1} = \{v:\|v\|_2 = 1\}$. Let the minimum separation be
$$
\rho = \min_{i,j\in{\{1,...

**4**

votes

**0**answers

123 views

+50

### Hierarchical (Recursive) Random Walk Model

Consider the following hierarchical (recursive) random walk model, which is also known as the hierarchical hidden Markov model in computer science (https://en.wikipedia.org/wiki/...

**-1**

votes

**0**answers

8 views

### Comparing log functions of CDFs and PDFs (related to order statistics) with non-log functions of the same

Let $f$ and $F$ denote the respective pdf and cdf of a probability
distribution on $\mathbb{R}$. Take any natural $n\geq3$ and any real $a$ and
$c$ such that $a\leq c$, and $\rho\geq0$.
We want to ...

**1**

vote

**1**answer

57 views

### Proving that an integral related to order statistics is increasing in a certain parameter

Let $f$ and $F$ denote, respectively, the pdf and cdf of a probability distribution on $\mathbb R$. Take any natural $n\ge3$ and any real $a$ and $c$ such that $a\le c$.
Does it always follow that
$$...

**1**

vote

**0**answers

36 views

### Concentration of functional of Gaussian random variable

Suppose I have two Gaussian distributions
$p(x) = \frac{1}{(2\pi)^{d/2}|\Sigma_p|^{1/2}}\exp(-\frac{1}{2}x^\top \Sigma_p^{-1} x)$ and $q(x) = \frac{1}{(2\pi)^{d/2}|\Sigma_q|^{1/2}}\exp(-\frac{1}{2}x^\...

**1**

vote

**1**answer

43 views

### Trace of the inverse sample covariance as the number of samples and dimension scale to infinity

Let $x_1,\dots,x_n$ be i.i.d. $N(0,I_{p\times p})$, with $n>p$. Let $\hat S=\frac1n\sum_{i=1}^n x_i x_i^T$ be the sample covariance.
Assume the asymptotic setting where $\frac pn\to \alpha<1$.
...

**6**

votes

**2**answers

97 views

### Uniform Concentration Bounds on Weighted Sum of i.i.d. Bernoulli Random Variables

Let $\delta_1,...,\delta_n$ be $n$ independent identically distributed Bernoulli random variables with $\mathbb{P}(\delta_1=1)=p$. We consider a set $\Omega = \{\mathbf{a}:=(a_1,...,a_n)~|~a_i\in [0,c/...

**-2**

votes

**0**answers

18 views

### Drawing graph with some different metric values [closed]

I have some metrics but all of them have different values. Some are decimals, some are integers and some are large numbers.
Assume the metrics are (average values):
- metric1 - 1500
- metric2 - 0....

**14**

votes

**1**answer

712 views

### Distribution of maximum of random walk conditioned to stay positive

I have an $n$ step random walk which starts at zero $X_0 = 0 = S_0$ where the steps $X_i$ are independent uniform random variates in $[-1,1]$, but the walk is conditioned on the hypothesis that it ...

**0**

votes

**0**answers

21 views

### Article Using Kullback Leibler Divergence to Measure Divergence of Observation from Distribution

I am currently attempting to compare an observed distribution to a theoretical distribution, and my current approach is to normalize the two and find the Kullback Leibler Divergence. I am beginning to ...

**0**

votes

**1**answer

56 views

### Discretization of a continuous distribution

For a research project I work with continuous distributions, like the normal distribution. In my use case however the random variable Z generally follows a normal distribution, though it can only take ...

**7**

votes

**1**answer

168 views

### Stochastic Covering Number of a Convex Set

Consider a convex set, say $S = [0,1]^d$. Let $X_1, X_2,\ldots,X_n, \ldots$ be i.i.d. random variables that are uniformly distributed on $S$. Denote the Euclidean ball centered at $x \in \mathbb{R}^2$ ...

**0**

votes

**0**answers

11 views

### Is there any Monte carlo or statistical approach to variational integral problems?

I am just shooting in the dark: From brain data imaging we have integrals of the form
$L(D):=\int_{\Omega}(\left \| A_{tensor}(D)-\widehat{A}\right \|+\sqrt{|\gamma(D)|})d\Omega$,
where we minimize ...

**3**

votes

**1**answer

81 views

### Probability of collision of some family of hash functions

Given $x$ and $y$ in $\mathbb{R}$, and let $\mathcal{H} = \{ h \mid \mathbb{R} \to \mathbb{N} \}$ be a family of hash functions where $ h(x) = \left\lfloor x + \sum^C_{i=1} U_i \right\rfloor$ for some ...

**21**

votes

**2**answers

5k views

### L1 distance between gaussian measures

L1 distance between gaussian measures: Definition
Let $P_1$ and $P_0$ be two gaussian measures on $\mathbb{R}^p$ with respective "mean,Variance" $m_1,C_1$ and $m_0,C_0$ (I assume matrices have full ...

**4**

votes

**1**answer

81 views

### information measure for matrix that is analogous to rank

Is there a measure for matrix that is analogous to rank of the matrix, but it is continuous on matrix elements? Say, we could say the information in identity matrix $I_n$ is $n$, and when the off-...

**1**

vote

**1**answer

107 views

### Convexity of truncated expectation

Let $k, n$ be two positive integers with $k \leq n$, and let $P = \{ (x_1, \dots, x_n) \in [0, 1]^n : \sum_i x_i = k \}$.
Given $x = (x_1, x_2, \dots, x_n) \in P$, let $X_i$ be the random variable ...

**3**

votes

**1**answer

328 views

### What is the mathematics behind the random experiment which produces the data with this strange property?

I have a following scenario. there is a huge collection of data resulting from a random experiment $E$ (I do not say random variable yet, for reasons that you will need to explain in your answer). Let ...

**0**

votes

**0**answers

17 views

### Nonparametric estimation in diffusion

Fan and Wang
In the above paper, the Authors provide estimators for the squared spot volatility process $\left(\sigma^{2}_{t}\right)_{t\geq 0}$.
My question is how to find estimators for the process ...

**1**

vote

**0**answers

22 views

### Generalizing an expected increase in autocorrelation near a bifurcation point to a system of ODE

Near a bifurcation point, a stochastically forced dynamical system should show an increase in autocorrelation and variance. This is due to critical slowing (a loss in resilience to perturbations). ...

**1**

vote

**0**answers

57 views

### Why is the classical secretary problem about ranks?

This relates here: http://math.stackexchange.com/questions/1820997/why-is-the-classical-secretary-problem-about-ranks
You want to stop optimal in a sequence of items presented sequentially, that is ...

**3**

votes

**2**answers

168 views

### The necessary sufficient condition for recurrence of a Markovian random walk

Suppose $\sigma_{1},\sigma_{2},...$are i.i.d random variables.$S_{0}=0$. Define $S_{n}=S_{0}+\sum_{i=1}^{n}\sigma_{i}$, then ${S_{n}}$ is a Markovian random walk.
I want to figure out the necessary ...

**1**

vote

**0**answers

38 views

### System of stochastic equations

I want to know if this system of SDE: $$dX_{t}=b(X_{t})dt+\sigma( X_{t}) dB_{t}$$
$$dY_{t}=b_{0}(Y_{t})dt+\sigma( Y_{t}) dB_{t}$$...

**3**

votes

**0**answers

38 views

### Estimating $E[X ; A]$ where $A$ is, e.g., an inter-quantile range

Estimating $E[X]$ from i.i.d. copies $X_1,X_2,\dotsc$ of a random variable $X$ with unknown distribution $P$ is well studied, obviously. When $X$ has extremely large variance, the Monte Carlo ...

**10**

votes

**1**answer

388 views

### The geometric median of a solid triangle

Let $\Omega\subset \mathbb R^n$ be a compact subset of dimension $n$. Define the geometric median on $\Omega$ as the point $m_{\Omega}\in \mathbb R^n$ such that the integral $\int_{\Omega}|x-m_{\Omega}...

**1**

vote

**0**answers

48 views

### Posterior consistency of non linear model

This is possibly a reference request. Let $G$ : $\mathbb{R}^p \to \mathbb{R}^q$ be a continuous injective/bijective function. Let $\mu$(we may also assume this to be a non degenerate Gaussian) be ...

**7**

votes

**2**answers

909 views

### The James–Stein estimator - counterintuitive estimation of the mean. What means it is better than least squares ? (Understanding Wikipedia)

Background James-Stein estimator and Stein's phenomenon, as described in Wikipedia are rather counterintuitive and amazing.
It is claimed that if one wants to estimate the mean $\Theta$ of
Gaussian ...

**1**

vote

**0**answers

35 views

### Covariance of order statistics [closed]

I'm a researcher in social science and I have encountered the following math formulation of a problem in my field. Note that I have also posted on math.stackoverflow, but given that this seems to be ...

**6**

votes

**1**answer

4k views

### Conjugate prior of the Dirichlet distribution?

What is the conjugate prior distribution of the Dirichlet distribution?

**5**

votes

**3**answers

14k views

### Distance metric between two sample distributions (histograms)

Context: I want to compare the sample probability distributions (PDFs) of two datasets (generated from a dynamical system). These datasets depend on a set of parameters, and I want a concise way to ...

**1**

vote

**0**answers

49 views

### Show this function is strictly concave

Please help me show that $f(w)$ is strictly concave in $w\in[0,\infty)$:
$f(w)=\sum_{j=1}^N P_j (w)\cdot u_j $
where
$P_j (w)=\sqrt{w}\int _{-\infty}^{\infty}\Pi_{k\neq j}\{\Phi[\sqrt{w}(v-u_k)]\}...

**1**

vote

**1**answer

194 views

### connection between the statistical properties of a scalar field and its columns

Consider a scalar field $s:[0,1]^3 \to \mathbb{R}$ and its "column" field
\begin{equation}
c: [0,1]^2 \to \mathbb{R}: (x,y) \mapsto \int_0^1 s(x,y,z) \,\mathrm{d}z.
\end{equation}.
What can be said ...

**0**

votes

**0**answers

32 views

### Product of lognormal random variables

Let $X_1$ and $X_2$ be two normal random variables. Write $X_1\sim N(\mu_1, \sigma^2_1)$ and $X_2\sim N(\mu_2, \sigma^2_2)$, to fix ideas.
Consider the corresponding log-normal random variables: $...

**2**

votes

**1**answer

81 views

### Do product distributions (or graph products) eventually cluster as more products are taken?

Say we have a joint distribution on a finite alphabet $\mathcal{X}\times \mathcal{Y}$. It could be a communication link where we want to send a random message $X$ over a channel, but it gets garbled ...

**4**

votes

**1**answer

56 views

### Concentration of U-statistics for exchangable distributions (and the unbounded case)

Consider the following so-called $U$-statistic of order 2: $$U = \frac1{\binom{m}{2}} \sum_{i < j} h(w_i,w_j)$$ where $w_1,\dots,w_m$ are IID from some distribution and $h$ is symmetric. If $|h(w_1,...

**5**

votes

**1**answer

123 views

### Can samples be compressed?

The Fisher information of a random variable $Y$ about a parameter $\theta$ upon which the probability of $Y$ depends is:
$\mathcal{I}_Y(\theta)= -E\left[\left.\strut \frac{\partial^2}{\partial \theta^...

**9**

votes

**2**answers

11k views

### Coin Pusher Game

While doing laundry at my local laundromat, I saw a coin pusher game. Below is a picture, and here is a video depicting how it works (disregard non-coins).
Essentially, one has a distribution of ...

**0**

votes

**0**answers

33 views

### Experimental Investigations on the Statistics of Infinite, Discrete, Evenly Distributed Pointsets in the Euclidean Plane

I am trying to estimate the distribution of certain planar polygons in the Euclidean plane; to accomplish that, I generate finite set of points, that are evenly distributed in w.l.o.g. the $[0,1)\...

**7**

votes

**1**answer

201 views

### In what sense is the Bayesian posterior mean a “convex combination”?

I asked this on math.stackexchange with no response, I'm hoping someone here might have something.
Suppose I want to estimate $x \in \mathbb{R}^n$ from two signals with zero mean, normally ...

**2**

votes

**0**answers

860 views

### Moments of function of Poisson process

(I'm new to Poisson processes, so please edit if my terminology is incorrect.)
Edit: per comments, here is a (more) general version of the originally posted problem (which is now at the bottom, below ...

**1**

vote

**0**answers

74 views

### Convergence of an rcll process along a random subsequence

I have a process $X_s$, for $s \ge 0$, taking values in a Polish space $T$ with an rcll version where I have shown, for every nonrandom increasing sequence $s_n$, that $X_{s_n} \to c$ in probability, ...

**5**

votes

**1**answer

437 views

### Size of KL-divergence neighbourhoods

I am new here. I was reading another
post
here and this got me wondering what can be said about the size of the following kl divergence neighborhoods.
Consider these two kl-divergence neighbourhood ...

**2**

votes

**1**answer

81 views

### What is the order of the constant $K$ in the multidimensional Dvoretzky-Kiefer-Wolfowitz inequality($Ke^{-c z}$)?

Let $F_n$ be the empirical distribution obtained from an i.i.d. sample
of the distribution $F:R ^d \to [0, 1]$.
Kiefer (1961) shows that the convergence of the empirical distribution is like
$$
P\left(...

**1**

vote

**2**answers

5k views

### Why 1.5IQR whiskers in boxplot? [closed]

Hi math people.
I'm in the process of analyzing some data that I collected through an experiment. The data are (somewhat) normally distributed and I represent the different data-sets using boxplot, ...

**1**

vote

**0**answers

104 views

### Limit theorem : reproduce a proof with an adaption from discrete to continuous time

Im considering Theorem 5.2.2 in M. Sørensen "Exponential Families of stochastic processes".
The setup is as follows:
We have a Levy-Process $X_t$ fullfilling the CLT
\begin{align}
\sqrt{t}(X_t/t-E(...

**0**

votes

**0**answers

64 views

### Basic Monte Carlo Integral Approximation

On the very first page of a well-known book on Monte Carlo techniques, there is the following statement. Let
\begin{equation}
I = \int_D g(\textbf{x})d\textbf{x},
\end{equation}
where $D \subset \...

**2**

votes

**0**answers

47 views

### Literature on transformed Gaussian matrices

I am considering real $n$-by-$m$ matrices of the following type:
$$
M=SM^\prime,\\
M^\prime_{ij}\sim^{iid}N(0,1).
$$
Here, $S$ is a fixed $n$-by-$n$ matrix and the entries of $M^\prime$ (same size ...

**7**

votes

**1**answer

107 views

### Choosing a sample based on where the density function is highest

Is there a name for the following process?
Say I have an absolutely continuous probability density function $f$ with compact support, and I take $k$ independent samples $x_1,\dots,x_k$ from $f$. ...

**3**

votes

**1**answer

111 views

### Is there a closed form expression for $E(X e^{-\mu \sqrt{X}})$, where $X\sim Poisson(\lambda)$ and $\mu >0$?

Is there any closed form expression for $E(X e^{- \mu \sqrt{X}})$, where $X\sim Poisson(\lambda)$ and $\mu >0$? If not, is there any tight upper bound for this quantity? Any idea how to proceed?

**2**

votes

**0**answers

44 views

### A question about probabilistic graphical models

Say one is given a probabilistic graphical model and a cut of the underlying graph. Do we know any statements about when and how can one or many of the marginals (of the sources) or the conditionals (...