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2
votes
0answers
38 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
257 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
214 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
71 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 ...
0
votes
0answers
11 views

Variance of the maximum likelihood estimator of Rayleigh Distribution [migrated]

I want to calculate the variance of the maximum likelihood estimator of a Raileigh distribution using $N$ observations. The density probability function of this distribution is : $$ f(\sigma,y_i) = ...
5
votes
0answers
113 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
97 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
votes
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 $\mu_x,-1<x<\infty$ ...
0
votes
0answers
87 views

Upper bound involving random orthogonal projection

Let $R$ be an $n\times N$ random matrix with i.i.d. standard Gaussian entries, $n<N$, and let $M:=(RR^T)^{-1/2}R$. Let $u,v\in \mathbb{R}^N$ non-random and s.t. $u^Tv=0$ and $\|u\|>\|v\|$. I ...
1
vote
1answer
166 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
votes
2answers
92 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
161 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
vote
1answer
83 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
170 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
votes
2answers
138 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
vote
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
58 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
101 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
230 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
vote
0answers
54 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
votes
2answers
169 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
vote
0answers
39 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
232 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
90 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
158 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
votes
4answers
168 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
votes
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
147 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
280 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
votes
0answers
71 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
51 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
39 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
182 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
118 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
163 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 ...
1
vote
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
41 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
107 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
112 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
178 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 ...
0
votes
1answer
94 views

Generating random variables from the Cantor Distribution [closed]

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
56 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
60 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
94 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
162 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
99 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
92 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 $1$ ...
3
votes
1answer
122 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: ...