Questions tagged [moment-problem]

The moment problem arises as the result of trying to invert the mapping that takes a measure $μ$ to the sequences of moments, and to resolve the problem of determinacy of such measure.

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33 views

Second moment of Silverman kernel

I am learning kernel smoothing method, in the Wiki page , I find that second moment of Silverman kernel is equal to zero (i.e $\int u^{2} K(u) d u$), I am not sure how to derive this result, that ...
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27 views

Find Force from Moment Vector, Given Radius and Unit Vector

Given a known Moment ($\vec{M}$), distance ($\vec{r}$), and unit vector $\hat{f}$ of force ($\vec{F}$), is it possible to find the resultant magnitude of force (using the formulation of the cross ...
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53 views

If $\mathbb{E}[X^n] =\mathbb{E}[Y^n] $ then $\varphi_X = \varphi_Y$

Let $X,Y$ be real random variables. I know that if $\mathbb{E}[X^n] =\mathbb{E}[Y^n] $ and $\mathbb{E}[X^n] \leq \frac{Mn!}{r^n} \hspace{0.1cm} \forall \hspace{0.1cm} n \in \mathbb{N}$, then $\...
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40 views

A specific form of the Hamburger Problem

I'm having a hard time trying to solve a moment-related problem. Let M be a random variable, I denote its k-th moment $m_{k}(X)=E({X^k})$ . How can i prove that the sequence $ u_{k}=\frac{m_{k+2n}(X)}...
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13 views

Positive definite 2-sequence from positive definite 1-subsequences?

If $p = (p_{n,m})_{n,m \geq 0}$ is a 2-sequence for which it is known that all the 1-subsequences $s^{(ab)} = (p_{ka,kb})_{k \geq 0}$ are positive semidefinite, for arbitrary $a,b \in \mathbb{N}_0$, ...
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28 views

Moments of probability distributions

I am looking for a concrete random variable with skew $\gamma_1$ and kurtosis (not excess) $\gamma_2$, where \begin{align} \frac{\gamma_2}{2}-\frac{\gamma_1^2}{3} = 0. \end{align} I alreagy tried ...
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15 views

Uniqueness to moments problem on an interval

The following result is well known. It is a special case of for example the more general Theorem 12.4.4 in [1, p235]. Here I am asking if there exists a more elementary reference or proof for this ...
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24 views

Vertical Variance

Variance is a numeric description of how spread out a probability mass function or probability density function is horizontally. A low variance means that the data is horizontally squished toward the ...
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24 views

how to integrate this by using contour integral and residue

$\int_{x=-\infty}^{\infty} \frac{\beta\alpha^\beta(1-\alpha)}{\pi(1-\alpha^\beta)}e^{itx} \frac{[(0.5+\frac{1}{\pi}\arctan(x))(1-\alpha)+\alpha]^{-(\beta+1)}}{(1+x^{2})}~~-\infty\leq x\leq \infty,\...
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27 views

Determining a measure from a moment sequence?

I am considering the Stieljes moment problem (https://en.wikipedia.org/wiki/Stieltjes_moment_problem), and its solution for a special class of moment sequences derived from quantum mechanics. One is ...
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49 views

Determining a measure based on recurrence relation for orthogonal polynomials

Suppose you were handed a sequence of polynomials $P_n(x)$ such that the $n$th one is degree $n$ and such that they satisfy a constant coefficient recurrence relation whose coefficients are taken from ...
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34 views

Derivation of skewness and kurtosis algebra of random variables

In algebra of random variables, the symbolic rule for computing variance of random variable $X\in\mathbb{R}^{n\times p}$ multiplied by a coefficient vector, $a\in\mathbb{R}^p$, is $$Var(X\cdot a) = a^\...
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73 views

If $\operatorname{supp}\mu\subseteq \{x_1,\dots, x_n\}$, then $\mu=\sum_{i=1}^na_i\delta_{x_i}$ for some $a_1,\dots, a_n\geq 0$

First, we begin with a definition: Let $\mu$ be a Radon measure on a Hausdorff space $X$. Let $\{G_i\}_{i\in I}$ be the family of open $\mu$-null sets, and put $G:=\bigcup_{i\in I}G_i$. Then $G$ is ...
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134 views

Measure on $\mathcal B(\Bbb R)$ having uniformly bounded central moments

Consider $\Bbb R$ with Borel $\sigma$-algebra $\mathcal B(\Bbb R)$. Let $\mathcal \mu:\mathcal B(\Bbb R)\to[0,\infty]$ be a measure such that for all $p\in [1,\infty)$ there exists $M>0\ ($...
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Does a probability distribution exist given a finite number of moments?

Given a finite set of moments $(m_n)$ for $n\in V \subset \mathbb Z^+$, is there a way to determine that there exists a probability distribution $d\mu(x)$ such that $m_n = \int x^n d\mu(x)$ for $n\in ...
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61 views

Uniqueness of measures related to the Stieltjes transforms

Suppose that $\mu$ is a Radon measure on $\mathbb{R}$, and let $I(\mu)$ be the Stieltjes transform of $\mu$, i.e. $I(\mu)(\lambda)=\int_{\mathbb{R}}\frac{1}{x-\lambda}\,\mathrm{d}\mu(x)$ for $\lambda$ ...
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28 views

Polynomials of the second kind of almost “the second kind” again

Let $s=(s_n)_{n\geq 0}$ be a positive definite sequence with orthonormal polynomials $(p_n)_{n\geq 0}$ and polynomials $(q_n)_{n\geq 0}$ of the second kind. Then $p^*_n=q_{n+1}$, $n\geq 0$ are the ...
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87 views

Prove support of function is compact (Urysohn lemma)

I want to prove that a function has compact support. (I'm writing a thesis on the moment problem). I have proven with help from Urysohns lemma, with the normal space $X$, that there exists a $K \...
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58 views

Proof of Stieltjes moment problem

The Stieltjes moment problem consists in finding the conditions on a sequence $m_n$ such that $$m_n = \int_0^\infty x^n d\mu(x)$$ for some positive measure $\mu$. I am looking for a proof of this ...
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47 views

Anybody knows an specific procedure for solving this excercise?

I'm new here. If I break any rule, please, tell me and I will fix it immediately. Well, long story short. Due to COVID-19, I had some financial troubles and I coudln't keep up paying my internet ...
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1answer
75 views

If the moments of a sequence of distributions converge, do they represent a probability?

Assume ${\bf x} \in \mathbb{R}^n$ denotes a real-valued and bounded random variable. Then, the moments of $\bf x$ uniquely define its distribution. Assume we have not only one distribution, but a ...
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29 views

How do I generalize a certain Markov model?

This question is a further attempt to generalize a certain Markov model of limit & market orders arriving in a financial exchange as first proposed in [1]. See Solving another non-trivial ...
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10 views

How to match moments to a log-normal random variable

If i have $E[X]=a$ (first moment of an arbitrary arithmetic average) how to i "match" the moments to a log-normal random variable Z in order to solve for $\mu$ and $\sigma$ I know under the log-...
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117 views

Determining a random variable through the Taylor expansion of its moment generating function

Let $X$ be a random variable defined on a compact set $K\subset \mathbb{R}$. The moment generating function (MGF) of $X$, denoted as $M_X(t), t\in \mathbb{R}$, is defined as $$M_X(t) = \mathrm{E} [e^{...
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7 views

Moment of a force parallel to line passing center of n points

If we have n points in the plane and a force acting parallel to a line passing through their center: 1) Why is the sum of moments for this force about all points n times its moment with respect to ...
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20 views

Beta x Gaussian parametric distribution

I'm currently studying a probabilistic filter for estimating a point depth along an optical ray from a monocular camera. The approach is defined in video-based, real time multi view stereo and comes ...
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31 views

To show that $\mu$ is determinate through a condition from a note

In Proposition 1.7 (p. 3), it says that If $\mu$ is [a positive Borel measure on $\mathbb{R}$] such that $$ \int_{\mathbb{R}}\exp(\epsilon|\lambda|)\,\mathrm{d}\mu(\lambda)<\infty $$ for ...
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Do d “moments of surprisal” determine probability distribution on d events?

Consider a probability distribution on $d$ events, with the probabilities $p_j$ gathered in a vector $\vec p\in \mathbb{R}^d$. For natural numbers $k$, define the $k$-th moment of surprisal as $$m_k(\...
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29 views

Limit order book

This question is related to Steady state of a non-trivial Markov chain. . In a financial exchange customers submit orders which can be roughly divided into three types.Firstly, there are limit ...
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52 views

Steady state of a non-trivial Markov chain.

This is a generalization of the question Solving another non-trivial recurrence relation. Let $\lambda^C \ge 0$, $\lambda^M \ge 0$ and $\Lambda \ge 0$ and $q\in (0,1)$..Without loss of generality we ...
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43 views

Rate for Sum of Random Variables

I Just read a Paper where we have $(X_1, ..., X_n)$ independent RV with zero mean. And there were an Expression like $$ Var = V + v $$ where $V:= \sum_{1=j}^n \sum_{1=i}^nE(X_j^2)E(X_i^2)$ and $v:= ...
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136 views

Proving uniqueness of Hausdorff moment problem

I want to prove that a positive measure $\mu$ on $\mathbb{R}$ with compact support is uniquely determined by its moments $$ m_k = \int_{\mathbb{R}}x^k d\mu(x). $$ I have looked through a lot of ...
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40 views

Method of moments and limiting distribution

Suppose that $X_1, X_2,\dots$ are random variables (taking positive real values, say) so that for all $k\ge 1$ $\lim_{n\to\infty}\mathbb E[X_n^k] = C_k$ for some $k$ where the $C_k$ are positive and ...
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42 views

Uniqueness of conditional distributions

Let $X$ denote a real-valued bounded random variable and $Y$ denote a discrete-valued random variable with $K$ states. Given $E[X^m]= \sum_{k} c_{km} P(Y=k)$ with constants $c_{km} \in\mathbb{R}$ ...
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71 views

Moments of products of independent random variables: $E[X^mY^n]$ Part II

This is a follow up question from here. Let $X$ and $Y$ denote two real-valued bounded random variables. Then all joint moments exist and uniquely define their joint probability $P(X,Y)$. Given for ...
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92 views

What is an admissible solution

In optimization world, what is the definition of an admissible solution? This word admissible solution appears in many papers, but I cannot find any definition of it. For example, in the paper "GLOBAL ...
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Does there exist a probability distribution such that all central moments are equal?

I find here that if the moment generating function of a random variable has positive radius of convergence, then that random variable is determined by its moments. So does there exist a continuous ...
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22 views

Stieltjes transform and distributional solution

let be the Stieltjes transform $$ g(s)= \int_{0}^{\infty} \frac{f(t)}{t+s} $$ then let be $ m_{n} = \int_{0}^{\infty}t^{n}f(t) $ then i get the distributional solution $$ f(t) = \sum_{n=0}^{\...
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34 views

calculating gradient of generalized method of moment of GMM problem

i'm trying to get the gradient of a generalized method of moment that is based on the first and second moments of a Gaussian mixture model problem but i've failed. a few notations: K - amount of ...
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223 views

Two random variables have the same $n^{th}$ moment for all $n\geq 0$. Then they have the same distribution

I am working this question: Let $X$ and $Y$ be $[0,1]-$valued random variables such that $E[X^{n}]=E[Y^{n}]$ for every integer $n\geq 0$. Show that $E[f(X)]=E[f(Y)]$ for every continuous function $...
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Entries of moment matrices are always nonnegative?

I am reading the following paper https://arxiv.org/pdf/1103.0486.pdf . Please see p.4, the part under Theorem 2.2. (just read from the bottom of p.3 to here). To my understanding, if the measure ...
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How to take moment about point of contact in a hemisphere?

How to get horizontal distance between point of contact and point P ?
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Truncated Sieltjes r-atomic moment problem

Given a certain $n \in \mathbb{N}$, can i construct a discrete positive random variable $X$ that fullfill the following conditions : $$\forall k \in \{1,...,n\}, \mathbb{E}(X^k) = \mu_k$$ for some $...
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Existence of Probability Measures With Given Marginals and moment constraints

let $\mu$ and $\nu$ be two probability measures on $\mathbb{R}^d$, my question under which conditions is there a coupling measure between $\mu$ and $\nu$ that satisfies some moment constraints. for ...
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53 views

Bound on the moment of Schatten norm

Let $A$ be a positive semidefinite matrix. Are there any bounds known for the $q$-th moment of the $p$-th Schatten norm of matrix $A$? Here, $1 \leq p,q \leq \infty$.
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How to show these moments are determined?

I'm given a sequence of moments $$ S_k=\int_{1}^{\infty}x^k \exp \left(\frac{-x}{\log(x)}\right)dx $$ and I'm told that this sequence is determined. However, I can't find a way to show this. I tried ...
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Image of the moment operator

Suppose we have a sequence $(a_k)_k$ in $\mathbb{C}$. I am wondering under which conditions it is true that there is a function $f \in L^1 \left[ 0, 1 \right]$ such that $a_k$ is the k-th moment of $f$...
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304 views

If all powers of two random variables are uncorrelated, are they independent?

Let $X$ and $Y$ be random variables on a common probability space. If $$\def\E{\mathbb E}\E[X^nY^m]=\E[X^n]\,\E[Y^m]<\infty $$ for all integers $n,m\ge 0$, does it follow that $X$ and $Y$ ...
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192 views

Moment in parallelogram

If F is a force in the same plane as parallelogram ABCD and the moment of F about A equals -18 moment unit and the moment about B equals the moment about D equals 32, what is the moment of F about C? ...
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146 views

Do orthogonal polynomials determine the moments of their orthogonality measure?

I am currently learning about the inverse problem for orthogonal polynomials for orthogonality measures supported on the real line. My question is not about finding the orthogonality measure from the ...