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1
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0answers
22 views

Writing a function as a sum of functions of bounded diameter

This problem is distilled from one arising in a study of complex random variables, but I've removed as much baggage as I can without (I hope) making it trivial. Fix $D>0$. A function $f:\mathbb ...
0
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0answers
11 views

Copula theory on discrete random variables [on hold]

How can I find the joint pmf on two discrete random variables using the copula theory
1
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0answers
21 views

Is there an equivalent form for Wishart to a power times a normal?

Lin described two equivalent characterizations of the multivariate t-distribution, viz. As a normal vector divided by an independent chi. That is, $t = Z / \sqrt{\chi^2/v}$, where $Z$ is ...
0
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0answers
24 views

Bounds on the moments of truncated sub-gaussian random variables

If $X$ is a centered sub-gaussian random variable, then there exists a constant $c$ such that $$ \mathbb{P}[|X|>t] \leq \exp(1-ct^2) $$ for all $t\geq 0$. Moreover, we know that the normalized ...
0
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1answer
43 views

How can two random variables are continuous infers that their jointly random variable is continuous [on hold]

We assume that $\forall a,b$ suchthat $a^2+b^2>0$, $aX+bY$ is continuous random variable. But we don't assume that $X$ and $Y$ are independent. My question is the following: Is it true that the ...
0
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0answers
27 views

How to compute selection probability of balls in a range [closed]

I have a question about probability that need your help. I assume that I have balls that are numbered from 1 to 100. The probability selection each ball is followed uniform distribution. I divide ...
0
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0answers
94 views

A hypothesis test question [migrated]

Let $X_i$ (for all integer $i$)be Bernoulli random variables (which takes either value -1 or 1, with equal probability). Define a random variable $Y$ to be $Y=\sum_{i=1}^d{X_i}$, where $d$ is a hidden ...
2
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1answer
93 views

What is the distribution of the maximum nearest-neighbor distance of a point cloud sampled from a solid body like?

Let $\mathcal{B} \subseteq \mathbb{R}^n$ be an $n$-dimensional solid body. Assume that we sample $N$ points, say $S = \{ x_1, ..., x_N \}$, from $\mathcal{B}$ uniformly at random. Consider the ...
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0answers
18 views

Order statistics: does distribution of sum of two of them uniquely determine parent distribution? [migrated]

Let $X_1, X_2, \ldots, X_n$ be a sequence of i.i.d. r.v. with bounded range (say, the interval [0,1]), with cdf $F$. Let $Y_1 \geq Y_2, \ldots, \geq Y_n$ be the corresponding order statistics. My ...
2
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0answers
35 views

expectation involving normal pdf and Rayleigh distribution

I need to calculate following definite integral \begin{equation*} \frac{1}{2\pi }\int_0^\infty \frac{x^2 e^{-x^2/\sigma^2 } }{\sigma} \frac{e^{-\frac{\lambda}{{ax^2+b}}}}{\sqrt{ax^2+b}} ~~dx. ...
2
votes
0answers
45 views

Compute the smoothing of functions

Given a function $g:R^d\rightarrow R$, which is not necessarily continuous, I want to compute the "smoothing" of $g$, i.e., $G(\vec{y})=\int_{R^n} g(\vec{x}) f_{\vec{y}, \sigma}(\vec{x}) d\vec{x} $ ...
4
votes
2answers
270 views

Expectation of Mahalanobis norm

Let $(g_i)_{i=1,...,d}$ sampled i.i.d. from a standard Gaussian, and $(\lambda_i)_{i=1,...,d}$ non-random s.t. $\max_i(\lambda_i)=1$ and $\lambda_i>0, \forall i$. I am looking for the expectation ...
7
votes
2answers
219 views

A moment problem

Suppose $X, Y$ are two positive random variables such that $\mathbb{E}[X^\alpha] = \mathbb{E}[Y^\alpha]$ for all $\alpha \in (0, 1/2)$. It is also known that the first moment exists for each of them, ...
2
votes
0answers
75 views

Implication of MGF inequality

Let X and Y be two random variables. Denote by $F_X(x)$ and $F_Y(y)$ their CDFs and by $M_X(t)$ and $M_Y(t)$ their MGFs. It is known that X and Y have the same CDF iff they have the same MGF. My ...
5
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0answers
115 views

Elementary function relative to erf

The modified Bessel function of the 1st kind $I_0$ is defined by $$ I_0(z)=\frac1\pi\int_0^{2\pi}e^{z\cos\theta}\,d\theta $$ and arises, among other places, in the probability density function of a ...
2
votes
1answer
101 views

Variant of Skorokhod's theorem

Consider the following situation: $S, T$ are standard Borel spaces (say $S = [0,1]^k$, $T = [0,1]$ if it is helpful). There is a a random variable $\zeta: \Omega \to S$. $f_n(\zeta) \to^d \eta$, ...
2
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0answers
34 views

logconcave distribution $f(t)$ leads to concave moments $\mu(x)$. logconvex distribution $f(t)$ leads to convex moments $\nu(x)$?

Let $$\mu_x=\frac{1}{\Gamma(x+1)}\int_0^{\infty}u^x f(u) du \tag{1}$$ Suppose that $f(u)>0$ when $u>0$ and $f(u)\to 0$ fast enough when $u\to\infty$ so that $\mu_x,-1<x<\infty$ ...
1
vote
1answer
173 views

About expectation norms on graphs

Let $S \subseteq V$ of a $d-$regular graph $G$ such that $\mu = \frac{\vert S \vert }{\vert V \vert } $. Let $A$ be the adjacency matrix of the graph. Then define the quantity $\phi(S)= ...
4
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2answers
96 views

Finding joint probability from double marginals

Consider three probability distributions in the form $p_1(y,z),p_2(x,z),p_3(x,y)$. When does a global joint probability $p(x,y,z)$ (possibly not unique) exist? The first compatibility condition to ...
4
votes
2answers
165 views

Probable direction of deviations from the expected value in binomial and hypergeometric cases

Suppose I have an urn with N marbles, with frequencies p and q for red and black marbles, and with p > 0,5. I take a sample of r marbles. It sounds intuitive to say that deviations from the mean ...
1
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1answer
84 views

A differential inequality and a special value

Let $G \colon [0,1] \to [0,1]$ be a monotonically decreasing function with $G(0) = 1$ and $G(1) = 0$. Suppose that $G$ is differentiable infinitely many times, and that: $$G(x)G''(X) \leq ...
1
vote
2answers
189 views

Variance of truncated normal distribution

Let $ X \sim \mathcal{N} ( \mu, \sigma^2 ) $, $ - \infty \leqslant a < b \leqslant +\infty $ ($ a, b \ne \infty $ simultaneously) and $ Y $ has a truncated normal distribution on $ (a, b )$, i.e. ...
2
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2answers
140 views

Is this a sufficient condition for joint normal distribution?

Suppose I have a random vector $\boldsymbol{Z}$, if I can prove that for $\forall \boldsymbol{\lambda} \neq \boldsymbol{0}$ where $\boldsymbol{\lambda}$ is a fixed vector, not a random vector, ...
1
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0answers
27 views

Bound on the total variation distance for multiple samples $d_{tv}(P^n,Q^n)$

Given two discrete distributions $P$ and $Q$, with computable total variation distance $d_{TV}(P,Q)=||P - Q||_1$, is there a precise bound for $d_{TV}(P^n,Q^n)=||P^n - Q^n||_1$, as need to estimate ...
0
votes
1answer
60 views

Name of distribution [closed]

I am searching for the name of the following distribution on the set of positive integers (including zero). Let $C\in \mathbb{Z}_+$ and $n\in \mathbb{N}$ are fixed. Vector $p = (p_1,\ldots,p_n)$ is ...
0
votes
0answers
107 views

Log-concavity of convolution log-concave and not log concave density functions

Let $Z = X +Y $ be a random variable (r.v.) where $X$ and $Y$ are independent r.vs. If the density function of $X$ and $Y$, $f(x)$ and $g(x)$ are log-concave in the support of $X$ and $Y$, ...
4
votes
1answer
259 views

Square root of normal distribution

Let $X$ and $Y$ be independent random variates with the same probability distribution, $P(x)$. Assuming that the product $Z=XY$ is a random variate with normal distribution, say $$f_Z(x) = ...
1
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0answers
57 views

Approximate determinantal point process

Consider a random process defined on $2^{\mathcal{X}}$, i.e. all subsets of a set $\mathcal{X}$. It's well known that this process is determinantal if one can find a positive semidefinite matrix K, ...
2
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2answers
173 views

Does $X_n \xrightarrow{d} N(0,1)$ and $X_n/Y_n \xrightarrow{d} N(0,1)$ imply that $Y_n \xrightarrow{d} 1$?

I'm thinking about the following question: If $X_n$ and $X_n/Y_n$ both converge in distribution towards a standard Gaussian random variable and $Y_n \geq 0$ for all $n$, does then $Y_n$ necessarily ...
1
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0answers
40 views

Maximum likelihood estimation with several distributions

My question concerns using Maximum likelihood to estimate unknown parameters used by several (poisson) distributions. The parameters are the pairs $(a_1,b_1),\dots,(a_N,b_N)$, and for each pair ...
6
votes
1answer
237 views

Reference on (discrete) log-concave probability distributions

A discrete distribution $p$ over $\mathbb{N}$ is said to be log-concave if it satisfies the following conditions: The support of $p$ is a contiguous interval, i.e. $\exists a \leq b$ s.t. $p_i > ...
3
votes
1answer
95 views

Lower bound on the tail of the hypergeometric distribution

Suppose there is a bag with $M$ white marbles and $N - M$ black marbles. Let $H(n, N, M)$ be a random variable which is number of white marbles in a draw, without replacement, of $n$ marbles from a ...
0
votes
1answer
159 views

Volume of randomly changing sphere follows beta distribution

We are given $X,X_1,\ldots,X_N$ independent and identically distributed $k$-dimensional vectors. For a given query point $X_q\in\mathbb{R}^k$ assume without loss of generality that $X_1,\ldots,X_m$ ...
3
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4answers
175 views

Central limit theorem with degenerate covariance matrix

Are there known generalisations of the central limit theorem for several random variables when the covariance matrix is degenerate? The usual proof of CLT based on characteristic functions (see e.g. ...
1
vote
0answers
42 views

Shift invariance for the distribution of quadratic polynomials

For a probability distribution $X$, supported on integers, define the shift-invariance of $X$, denoted by $shift(X)$ = total variation distance between the random variable $X$ and $X+1$. Let ...
0
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0answers
43 views

Taking power of the integrand in a Riemann-Stieltjie Integral

This is a problem I am trying to solve as part of a calculation for Value-at-Risk. Given that $P(X<x)=F(x)=\int_{\theta}F(x|\theta)dG(\theta)=1-\alpha$, where $F$ and $G$ are CDF's, is there a ...
5
votes
1answer
148 views

Asymptotic behavior of $X_n$ in a Dirichlet vector $(X_1, …, X_n)$

Let $(\alpha_k)$ be a sequence of positive numbers and let $(Y_k)$ be a sequence of independent random variables $Y_k \sim \text{Gamma}(\alpha_k,1)$. Set $X_n=\dfrac{Y_n}{\sum_{i=1}^nY_i}$. (edit) ...
2
votes
0answers
183 views

Inequality with CDF of order statistics

here is a problem I have been struggling with for a while now. This is for a paper I am working on. Any help would be appreciated! Here we go: Each bidder's valuation $\theta _{i},$ $i=1,...,N$, is ...
3
votes
2answers
295 views

Weak convergence of random measures

Let $\mu_n,n\in \mathbb N$ be a random probability measures and let $\mu$ be a deterministic probability measure on $\mathbb R$. That is to say, that the $\mu_n$ are measurable maps from a probability ...
6
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0answers
77 views

Rate of Convergence of Compound Poisson Laws to Infinitely Divisible Laws

It is known that every infinitely divisible random variable is the limit in law of a sequence of compound Poisson random variables (see for instance Theorem 1.2.18 of Lévy Processes and Stochastic ...
4
votes
1answer
52 views

Numerical approximation to the Wasserstein metric?

Are there numerical methods for approximating/calculating the Wasserstein metric in particular cases? Suppose that $f$ and $g$ are two density functions with the same support. How can I calculate the ...
3
votes
1answer
41 views

Error for the convergence by distribution

A sequence of random variables $X_n$ converges in distribution to $X$, if there is pointwise convergence of its characteristic functions, i.e. $\lim_{n\rightarrow\infty}\phi_{X_n}(\lambda) = ...
0
votes
3answers
190 views

An inequality based on expectation of continuous random variables

I am trying to prove the following statement: $$ E[g(X)] E[X^2g(X)]\ge E[Xg(X)] E[Xg(X)] $$ where $X$ is a random variable, $E[\cdot]$ denotes the expectation operator with respect to ...
5
votes
3answers
120 views

Random partitions with prescribed pairwise membership probabilities

Let $(p_{ij}) \in [0,1]^{n \times n}$ be a given symmetric matrix, with $1$ on the diagonal. Suppose $\pi$ is a partition of $[n]=\{1,\dots,n\}$ and let us write $i \stackrel{\pi}{\sim} j$ if $i$ and ...
0
votes
1answer
189 views

Convergence in the Wasserstein metric and the square root function

Let $f$ be a smooth probability distribution on the unit square $S$ such that $f(x)>0$ on $S$. Let $\{g_i\}$ be a sequence of smooth probability distributions such that $g_i(x)>0$ on $S$ as ...
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0answers
40 views

A curious example envolving moment's convergence

Let $\{X_n\}$ be a random variable sequence and $X\sim N(0,\sigma)$. In general, the convergence $E(X_n^k) \stackrel{n}{\longrightarrow}E(X^k)$ doesn't implie that $E(X_n^{k+1}) ...
1
vote
2answers
42 views

Sensitivity of inverse normal cdf

Let $Q^{-1}$ be the inverse function of a standard normal CDF. For $0 < \epsilon < p,p' < 1 - \epsilon$, how much does the function $Q^{-1}$ change as a function of $|p - p'|$? Any useful ...
1
vote
1answer
112 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
121 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 ...
1
vote
1answer
204 views

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

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 ...