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1
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1answer
60 views

1-wasserstein distance v.s. total variation distance

Suppose that $\mu_1$ and $\mu_2$ are two distributions defined on $\mathbb{R}^n$ and $\gamma$ is a symmetric distribution (around $0$) on $\mathbb{R}^n$ with compact support. Let $\gamma_x$ denote the ...
1
vote
1answer
43 views

Discrete Maximum Entropy Distribution with given mean

For a given mean $\mu$, what is the entropy maximizing probability distribution on the nonnegative integers? Different sources indicated either the geometric or the Poisson distribution for this. As ...
0
votes
1answer
124 views

Book on Convergence Concepts in Probability without Measure Theory [on hold]

I am looking for a comprehensive book on Probability which discusses Convergence of Random Variables in detail, excluding portions of Measure Theory. Allan Gut's "Probability: A Graduate Course" seems ...
0
votes
1answer
76 views

Generating random variables from the Cantor Distribution [on hold]

I am looking for a method (exact, if possible, but at least asymptotically correct) for generating random variates from a Cantor Distribution? It seems like its abstract definition prevents this. In ...
0
votes
1answer
34 views

convergence in distribution and convergence of moments

Suppose that the sequence of r.v $\{X_{n}\}_{n\geq 1}$ has all the moments, and $X_{n}\stackrel{D}{\longrightarrow}X\sim N(0,\sigma)$. Assume that $E\left\{(X_{n})^{K}\right\} ...
2
votes
1answer
30 views

Sampling point uniformly at random satisfying equality constraints

First of all, I apologize in advance if the question has already been asked in some way on this site and/or if there is a widely known solution to this problem. The description of my problem is ...
0
votes
1answer
67 views

Do constrained random walks converge weakly to the Wiener measure on the space of constrained paths (that corresponds to the heat equation)?

Let $U$ be an open subset of $\mathbb{R}^n$ such that $\partial \overline{U}$, the boundary of $\overline{U}$ is ''nice'' (for simplicity you can assume piecewise smooth). I also want to allow the ...
2
votes
0answers
92 views

Probability question involving drawing balls from an urn

Suppose there's an urn containing $r$ red balls and $b$ blue balls. At each trial, I'm drawing a ball at random from the urn, without replacement. Let $R$ denote the event of drawing a red ball, and ...
1
vote
0answers
64 views

Multiple Bipartite graphs and matchings

I've been told recently that it's better i just for help regarding my 'specific' problem rather than lots of little questions around the same topic which appear somewhat unclear. I would first like to ...
2
votes
0answers
77 views

Under what conditions do time averages of ergodic transformations satisfy a central limit theorem?

Let $(X, \mu)$ be a probability space and $T:X\rightarrow X $ an ergodic transformation, i.e. $T$ is measure preserving and the only $T$ invariant subspaces have either measure $0$ or measure $1$ ...
1
vote
0answers
373 views

Can one generate a sequence of natural numbers whose density has a given distribution? [migrated]

Suppose $\{ p_{k} \}$ is a collection of real numbers with the following properties: 1) $p_k \in (0,1)$ $~~~~$(i.e. $0$ and $1$ are not allowed values) 2) $\sum_{k=1}^{\infty} p_k =1$ An ...
3
votes
1answer
99 views

Does bounding moments make distributions close in total variation distance?

Let $W\sim\mathcal{N}(0,\sigma^2)$ be a "reference" Gaussian random variable. Suppose I have a set of distributions, $\mathcal{W}$, where $W_a\in\mathcal{W}$ if it satisfies the following criteria: ...
1
vote
0answers
102 views

Is it possible to improve the order of convergence of averages of random variables if they are not identically distributed?

Let $X_n$ be a sequence of independent random variables (but not necessarily identically distributed) taking values in $[-1,1]$ that have the following property: 1) The average $A_n := \frac{(X_1+ ...
1
vote
1answer
89 views

Deriving Newtonian capacity of sphere from Brownian motion

We have the following result by Spitzer (see (1) or Port) $lim_{t\to \infty}\frac{1}{t}\int_{\mathbb{R}^{n}/B_{r_{0}}}P_{x}(T_{B_{r_{0}}}<t)dx=Cap(B_{r_{0}})=\frac{r_{0}}{4\pi}$ By Chuancun and ...
1
vote
1answer
46 views

PDF of th product of normal and cauchy distributions

I am having trouble in finding out the resulting PDF of the product of normal and cauchy distributions. It turns out that we have a general formula for calculating the PDF of product of two random ...
3
votes
1answer
127 views

distribution discretization

Let $\mu$ be a distribution on $\mathbb{R}^n$. We partition $\mathbb{R}^n$ into small cubes congruent with $[0,\delta)^n$, parallel to the axes. In each cube, pick a point $x$ (for instance, the ...
2
votes
1answer
47 views

Hitting probability of semiball

For fixed x and hemisphere H of radius r and centered at the origin, I wonder what is $P_{x}(T_{H}<\infty)$. Attempt Firstly, I wonder if there is any relation between $P_{x}(T_{H}<N)$ and ...
0
votes
0answers
23 views

Density of $\int_{B}\frac{|1-|B_{T}|^{2}|}{|y-B_{T}|^{3}}dS(y)$

For $B\subset \partial B(0,1)))$ and random variable $B_{T}\in Int(B(0,1))$ with density $p_{T}$, is there a density for $\int_{B}\frac{|1-|B_{T}|^{2}|}{|y-B_{T}|^{3}}dS(y)$? Context The original ...
7
votes
4answers
285 views

Gaussian distributions as fixed points in Some distribution space

I'm taking a course on topology and probabily. Today, the professor remarked something along the lines of: If you look at the space of probability distributions with $0$ mean and variance $1$, ...
0
votes
2answers
111 views

What are some examples of isotrophic sets?

What are some examples of isotrophic sets? and is there a "good" way to describe them? Isotrophic meaning that a random vector X uniformly distributed in the set has the isotrophic property for all ...
3
votes
0answers
91 views

Properties of a finite random walk

Consider the simplest random walk - $X_0 = 0$ and from there on (i.i.d), $X_i=X_{i-1}+1$ with probability $p$ or $X_{i-1}-1$ otherwise. Let $Y_N$ be the highest point $X$ have reached on the first ...
1
vote
0answers
95 views

How to show that two linear combinations of Bernoulli random variables have jointly Gaussian distribution (and more)

Let $X_1,\ldots,X_n$ be independent Bernoulli random variables such that $\mathbb{P}(X_i=\pm 1)=1/2$ and consider two collections of real numbers $a_1,\ldots,a_n, b_1,\ldots, b_n$. For the moment let ...
2
votes
0answers
92 views

Speed of Approach to Invariant Measure

Let $X_t$ represent a continuous-time Markov process on $\mathbb{R}^d$, say a diffusion with locally Lipschitz coefficients. Suppose that there exists a unique invariant measure $\mu$ on the space, ...
1
vote
0answers
82 views

A natural sum over multisets (expectation over multinomial)

I think this is a natural question but am not sure where to find resources. Consider the possible multisets arising from choosing $n$ times an item from one of $k$ categories. We can represent one ...
2
votes
0answers
50 views

Property of relative entropy [closed]

For $X$ a measurable space and $P,Q$ two probability measure on $X$ s.t. $Q$ is absolutely continuous with respect to $P$, the relative entropy is defined as $$D(Q\|P)=\int_X ...
-1
votes
1answer
114 views

Property of relative entropy [closed]

For $X$ a measurable space and $P,Q$ two probability measures on $X$ s.t. $Q$ is absolutely continuous with respect to $P$, the relative entropy is defined as $$D(Q\|P)=\int_X \log(\frac{dQ}{dP})dQ,$$ ...
0
votes
0answers
51 views

Computation on Random Bipartite graphs

I'm looking at a random bipartite graph $K_{\omega(n)}*K_{\omega(n)}$ where $\mathrm{log}(n)\leq \omega(n) \leq n^{1/2}$, in which each of the $\omega(n)^{2}$ edges is placed randomly with probability ...
1
vote
0answers
52 views

A series with long-tailed terms

Let's consider the following series: $$ \zeta = \sum_{k=1}^{\infty} a_k \xi_k, $$ where the sum is understood as the limit in $L_2(\Omega)$, $a_k \in \mathbb{R}$, $\sum_{k=1}^{\infty} a_k^2< ...
6
votes
2answers
109 views

Geometric interpretation of the average of two independent Cauchy distributions

Let me state two facts: (1) It is well known that if one takes a point uniformly distributed on the unit circle, and then takes it stereographic projection, the corresponding measure induced on the ...
2
votes
0answers
83 views

Probability question involving simulations of picking balls from a bag

I’m working on a chemistry problem, which essentially translates to finding the answer to a related probability problem. However, my knowledge in probability is very limited and I'd be grateful if ...
0
votes
1answer
63 views

Distribution of the $\alpha$-parameter of a $2\times 2$ Haar-distributed, unitary matrix

It is well known that any $2\times 2$ unitary matrix $\mathbf{U}$ can be parametrized as $$\mathbf{U}=\begin{pmatrix} 1 & 0 \\ 0 & \mathrm{e}^{\mathrm{j}\beta_1}\end{pmatrix} \begin{pmatrix} ...
5
votes
1answer
344 views

lower-bound for $Pr[X\geq EX]$

Given n random variables, $X_1, ..., X_n$, each takes value 0 or $a_i \in[0, 1]$. $X = \sum_{i=1}^n X_i$ and $EX \geq 1$ is the expected value of $X$. Can we get a lower-bound for $Pr[X \geq EX]$? It ...
5
votes
0answers
157 views

A note on Doob's theorem

I have faced the following problem, regarding to the Martingale Theory. Because this area far from my area I don't know whether this problem is in literature or this can be simple question for ...
8
votes
1answer
260 views

Algorithm to produce random number with a gamma distribution

I'd like to produce pseudo-random numbers with different distributions for a Monte Carlo simulation. I've got the poisson distribution working nicely with an algorithm from Knuth. I'm having trouble ...
0
votes
1answer
89 views

Expectation of exp(-1/(ax^2)) when x is a standard normal variable and a>0 is a parameter [closed]

I would like to know if the mean value of $\exp(-1/(ax^2)) $ when $x \sim N(0,1)$ and $a>0$ is a parameter is known.
0
votes
1answer
185 views

Probability of the maximum of a throw of an infinite number of $n$-sided dice being $k$ [closed]

Let $X$ be the random variable obtained as the maximum of a throw of $m$ dice (each of which is $n$-sided). In other words, $X = \max\{l_1,\cdots, l_m\}$ where $l_i$ can take any value between $1$ and ...
2
votes
0answers
66 views

A 1-D random variable from a random distribution

I have a random variable $X$ that is drawn from the pdf $$ f(x; \mu, \sigma, \sigma_{\mu}, \sigma_{\sigma}) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{1}{|\hat{\sigma}|\sqrt{2\pi }} ...
1
vote
0answers
22 views

Perturbing moments of multivariable distributions

Let $P$ be a multivariate probability distribution on $\mathbb R^n$ which is moment-determinate and let $\{m_k : k \in \mathbb N_0^n\}$ be the sequence of moments $P$. Fix an order $p$ and consider ...
4
votes
0answers
179 views

Total variation and Hellinger distance inequality between truncated Gaussians

We know that the total variation distance, $d_{TV}(P,Q) = \frac{1}{2}\left|\left|P-Q\right|\right|_1$, between any two distributions $P$ and $Q$ is lower bounded by their squared Hellinger distance, ...
0
votes
0answers
40 views

Monotonicity of a function of order statistics with respect to the sample size

There are $n$ ($n \ge 3$) independent random variables $\{ {c_i}\} _{i = 1}^n$ identically drawn on the interval $[\underline c,\bar c]$ ($\underline c>0$), with cdf $F(\cdot)$ and pdf $f(\cdot)$, ...
6
votes
1answer
73 views

Summability of ratios of moments a weight

Recently, I encounter the following problem: Let $w$ be a probability density on $[0,1]$. Let mk be the $k$-th moment, i.e., $$m_k=\int_0^1t^kw(t)dt.$$ Under what condition can we have ...
1
vote
0answers
55 views

Inversion of Fourier transform of a multivariate gamma distribution in polar form?

Let $\mathbb{S}^{N-1}$ be the unit sphere in $\mathbb{R}^N$ under the Euclidean norm $||\cdot||$. Let $\mu$ be an infinitely divisible Borel measure. If there exists a finite measure $\alpha$ on ...
2
votes
0answers
50 views

Mean and variance of a general multivariate skew normal distribution

I have a problem about a general multivariate skew normal distribution. There is a $p\times 1$ vector, $\mathbf{y}=(\mathbf{y}_1',\mathbf{y}_2',\ldots,\mathbf{y}_n')',p>n$, which has the density as ...
0
votes
0answers
37 views

Help in finding the probability density function

This may seem trivial but I will appreciate help in determining the functional form of the probability density function (pdf) for the following case. Will highly appreciate some guidelines on how to ...
3
votes
1answer
166 views

Is it possible to construct any random variable on the Euclidean Probability space?

Let $(\Omega,\mathscr A,P)$ be an arbitrary probability space, and let $X:\Omega\to\mathbb R$ be a random variable. Then, one can generate a random variable $Y$ from the probability space ...
1
vote
1answer
86 views

Entropy on a draw from a random distribution.

Suppose I am attempting to calculate the entropy of a continuous, normally distributed random variable $X$, from the distribution $\mathcal{N}(\mu, \sigma)$. This is easy to to do - I just calculate ...
0
votes
0answers
30 views

Beta distribution - changes in multiple time points

Let's say I have a set of daily data (assume iid) that I know is beta distributed (between 0 and 1). I can estimate the parameters of the distribution and calculate the tails etc. This would tell me ...
3
votes
1answer
124 views

Two matrix Fisher distributions on SO(3)?

After the uniform distribution (normalized Haar measure), the matrix Fisher distribution seems to be the most popular probability distribution on the Lie group SO(3). The density is proportional to ...
1
vote
0answers
29 views

Family of (Cumulative Distribution) Functions

I'm looking for a 2 (or more)-parameter family of functions $F$ with the following properties: For each $f \in F$, $f(0)=0$, $f(1)=1$, and $f$ is (weakly) increasing. $F$ is closed under products. ...
0
votes
1answer
98 views

Behavior of the integral of products of probability densities

Assume $z \in \mathbb{R}^m$ and $x \in \mathbb{R}^n$. Assume we have proper density function $P(z)$ and proper conditional density function $P(x|z)$. We give the definition $$ T(x_1,\ldots,x_n) := ...