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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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Moment based characterization of invariance

$X$ is compact, any measure $\mu$ is uniquely determined by its moments \begin{align} y_{\alpha}=\int x^{\alpha} d\mu \end{align} where $x^{\alpha}=x_1^{\alpha_1}x_2^{\alpha_2}\dots x_n^{\alpha}$ is ...
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Convergence of moments implies convergence of the measure in the space of tempered distributions?

Let $\mathcal{S}(\mathbb{R}^d)$ be the Schwartz space and $\mathcal{S}'(\mathbb{R}^d)$ be the space of tempered distributions. Denote by $\{\mu_k\}_{k=1}^\infty$ a sequence of Borel probability ...
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Fourier transform of indeterminate measure

Consider the following probability measure $d\mu(x) = \frac{1}{\sqrt{2\pi}}x^{-1}\exp\Bigl(-(\ln{x})^2/2\Bigl)\chi_{(0,+\infty)}(x)$ whose moments are given by $\mu_n = e^{n^2/2}$. As discovered by ...
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Show that if $\text{plim } A_n = A_0$ and $\lim V(X_n) = B_0$ then $\lim V(A_nX_n) = A_0^2B_0$

I am wondering how the sandwich formula of the variance is derived. Assume $A_n$ and $X_n$ are two random variables in $\mathbb{R}$ such that $A_n$ converges in probability to a constant $A_0$ and $\...
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Proving $m_4m_2\geq m_3^2+m_2^2$

I've been trying to prove: $m_4m_2\geq m_3^2+m_2^2$ How I proceeded: From the Cauchy-Schwarz inequality: $m_3=E[(x-\bar{x})^3]$ $m_2=E[(x-\bar{x})^2]$ $m_4=E[(x-\bar{x})^4]$ $[E(XY)]^2 \leq E[X^2] \...
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Some questions about the number of moments for random variables

I'm reading a paper about econometrics and meeting some questions, and I'm not sure I can describe them perfect. $s_t$ is a scaled score function, and $f_t$ is a stochastic time-varying parameter, and ...
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The central moments of stochastic differential equations

How can one look at the evolution of covariance matrix using Ito lemma for a non-linear stochastic differential equation? when the stochastic term is a Brownian motion? $$ d\mathbf{X(t)} = \boldsymbol{...
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From cumulants of a random variable to its moments

This question is very related to: Infinite sum of Gamma random variables with same shape parameter but different rate parameter In particular I know that a random variable $Q_n'$ has $j$-th cumulant ...
MathRevenge's user avatar
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Example of a random variable that is integrable, but p-th moments for any p > 1 do not exist

I have seen examples of random variables that are integrable, but the second moment does not exist. Is there an example of a random variable that is integrable, but the p-th moment, for any p > 1, ...
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Moment and Cumulant Notation

I recently started working with moments and cumulants for some statistical analysis and among my research I can’t figure out some of the notation. It will be written as: M2,1 and C4,3 I know M ...
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Is the following sequence a moment seqence?

Is the sequence $$\Bigl( \frac{1}{n+1} \Bigr)^\alpha,\quad n \geq 0$$ a moment sequence for any $\alpha \in (0,1]$ for some random variable on [0,1]? We have tried checking whether the corresponding ...
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A closed form expression for spectral moments of arbitrary order in the Wishart ensemble

We are interested in calculating the spectral moments of random matrices sampled from the Wishart distribution. Let $N,T$ be positive integers with $T> N$ . Then the quantities in question are ...
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Moment/cumulant problem complex normal law.

I know that $N = (N_1, N_2)$, where $N_1, N_2$ follow normal distributions on $\mathbb{R}$ is characterized by its moments. I am also aware of Carleman condition for the moments problem for real ...
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Proof of $\mathbb{E}_{\vartheta_0}|\xi_t|^p<Ct^{p/2}$ for $\xi_t$ Gaussian process with mean 0

I currently read paper [2], where, in Chapter 2, one shows that $Y_t-y_t(\vartheta_0)=\varepsilon \xi_t$, where $\xi_t$ is a Gaussian process with $\mathbb{E}_{\vartheta_0}\xi_t=0$. Now, they claim ...
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Is Carleman's condition a sufficient condition for Hausdorff moment problem

We know that Carleman's condition is a sufficient condition for the determinacy of Hamburger moment problem and the Stieltje's moment problem. The first one look at measures on the real line, and the ...
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Convergence of second moment over an expanding set

Suppose $X$ to be a real random variable with $E(X)=0$ and $E(X^2)=1$. How can I prove: $\lim_{n\to\infty}E(X^2\mathbb{I}_{\{X^2\ge n\}})=0$ I am thinking, with no success, about these results: ...
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Bounding sum of RV with high Probability

Let $X_i$ real random variables for $i \in 1, \dots n$ such that $$ \mathbb{P}\left(\|X_i\|\geq \frac{1}{n}\log\left(\frac{1}{\delta}\right) \right)\leq\delta $$ Is there a chance to find a constant $...
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Linear transformation of random vector has bounded moments?

Suppose that the random $p$-vector $\mathbf{y}=(Y_1, Y_2,\ldots, Y_p)'$ with $p\to\infty$ satisfies: $\mathrm{E}Y_i = 0$, $\mathrm{E}Y_i^2=1$ for any $1\leqslant i \leqslant p$; $\mathrm{Cov}(Y_i,Y_j)...
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Carleman's theorem on a compact interval

Let $\mu_n$ be Borel probability measures on $[0, 1]$. I'm trying to prove a special case of Carleman's theorem, i.e., Theorem If the sequence $(\int_0^1 x^k \mathrm d \mu_n (x), n\in \mathbb N)$ ...
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Assume if Odd moment of $h(x)$ is equal to zero, is $h(x)=0$? - Fourier moment problem, $x\in[0,1]$ - $\int _0^1\:h\left(x\right)\cdot x^{2n+1}dx$

Showing that $ \int_0^1 x^{2n}f(x) dx = 0 $ implies $f = 0$ Hi, tried looking at this link. I did not understand how to answer regarding odd function, I receive the answer is that it is not correct, ...
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Is the moment problem a special case of the "functional problem"?

In the Wikipedia page of Hahn-Banach theorem, it mentions, when a normed space $X$ is reflexive, then the following "vector problem" is equivalent to the "functional problem". (...
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Can expected values of the form $E[X^r e^{-sX}]$, for arbitrary $r>0$, uniquely determine the PDF of a random variable $X≥0$?

It is known that the probability distribution of a continuous non-negative random variable, $X$, is uniquely determined by its associated Laplace transform, $$L(s) = E[e^{-sX}] = \int_0^\infty e^{-sx} ...
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Find the moment generating function of a non-canonized gaussian at a specific point t=0

I’ve been trying to solve this problem and hit an integral I’m struggling with. Is my approach the right one? how would you continue? The Question: Let $X \sim N_{[1,4]}$ gaussian random variable. ...
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Determining function from expectation of its powers

I have a problem concerning reconstructing a function from the expectation of its powers. The scalar version of this problem is as follows: Given a probability measure $\mu(x)$ over $\mathbb{R}^n$ ...
Yly's user avatar
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How to prove that for a $p$th moment $m_p$, $((p+1)m_p)^{1/(p+1)}$ is increasing in $p$ over $[0, \infty)$?

Let $f: \mathbb{R}_{+} \rightarrow \mathbb{R}_{+}$be a decreasing, continuous probability density function and let $m_p=\int_0^{\infty} x^p f(x) d x$ be its $p$ th moment. Show that $\left((p+1) m_p\...
Vishnudasa Srinivasan's user avatar
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Finding the first and third moment using MGF

Consider the PDF $$f(x) = 3x^2, 0≤ x ≤ 1$$ I'm trying to get the first and third moment from this PDF. I found the $E(X)$ using the usual formula to be $3/4$. However, I found the MGF to be $$\varphi(...
Owen Alberto Liem's user avatar
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Centre of mass with double integration. What is Moment $Mx$?

First post here. I'm having serious trouble understanding how the Moment $Mx$ is solved for in a typical Centre of Mass problem. So, many people online, are teaching methods to solve for $Mx$ that are ...
Insaan Abdulla's user avatar
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Weak convergence from convergence of integrals of polynomials

Let $\mu_n$ for $n\in \mathbb{N}$ and $\mu$ be probability measures on $\mathbb{C}^d$ with uniformly bounded support. Suppose that $$\int_{\mathbb{C}^d} f(z_1, \ldots, z_d) d\mu_n \to \int_{\mathbb{C}^...
Antoine Labelle's user avatar
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Recurrence relations for even orthogonal polynomials

I have been playing around with the theory of orthogonal polynomials, and it occurred to me that we might be able to build a family of orthogonal polynomials from even powers of a variable $x$. For ...
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Finding the method of moment for Beta given a pdf and n random variable

I have this tough question in my exam prep textbook and I've been struggling with it ...
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A moment problem

What is the p.d.f. for the moments $m_n=1/n$ ? (They are obtained from $\int_0^1 x^n/x\ dx $, but clearly $1/x$ is not a p.d.f. on $[0,1]$)
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Additivity of random variables following normal distribution

Suppose that $X_1,X_2,...,X_n$ are independent and identically distributed random variables. Known that for any unit vector $\mathbf q=[q_1,...,q_n]$ (i.e. $q_1^2+...+q_n^2=1$), $q_1X_1+...+q_nX_n$ ...
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How does the cycle structure of a sub-set of permutations change under a particular mapping?

Let $n \ge 1$ be an integer. Denote by $\Pi^n $ the group of permutations of $\left\{1,\cdots,n\right\}$ and by $\Pi^{(2)}_{2n}$ the subset of all permutations of $\{1,\cdots, 2n \}$ that are composed ...
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Find a distribution given recursive relationship between moments

Let's say I have a continuous random variable $0\le X<\infty$ and I know it has first raw moment $m_1=\alpha$, and subsequent moments are defined recursively as: $$m_{n+1}=(1+\beta n)\alpha m_n$$ ...
Ciaran Harman's user avatar
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Searching sample distribution with given parameteres

I have a very simple question, but it is hard, at least to me, for giving a solution. Given values of mean, variance and median, can you find a method to generate a random sample with these parameters?...
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Reverse Hamburger moment problem

Let $\mu$ be a positive borel measure on $\mathbb{R}$ with $\int_{\mathbb{R}} x^n d\mu (x) = s_n, n \in N_0$. Find a hilbert space $\tilde{\mathcal{H}}$ and a self adjoint extension $A$ of $S$ in $\...
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Can we retrieve $X$ from its odd moments?

The moment problem asks whether there exists a random variable $X$ with given moments. One way to do this: if $X$'s MGF converges about a neighborhood of $0$, we know the MGF uniquely characterizes $X$...
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draw random vector for spherically symmetric exponential distribution

My goal is to draw a random (3-dimensional) vector $X$ from a spherically symmetric exponential distribution $$ X \sim f_X(x) = \frac{\lambda^3}{8\pi}e^{-\lambda |x|} $$ with $x\in\mathbb{R}^3$. My ...
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Differences of moment functionals

For which linear functionals $L:\mathbb{R}[x_1,\ldots,x_n]\to\mathbb{R}$ can we find Radon measures $\mu_1$ and $\mu_2$ such that $L(f)=\int f d\mu_1-\int f d\mu_2$ (both subtrahend and minuend should ...
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About finite moment problem

My question is, given a real random variable $u$, then can we find a discrete random variable $\xi$ with finite possible values, such that $u$ and $\xi$ have the same first $2k$ moments? According to ...
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Asymptotic distribution of maximum likelihood

I got a question: We let $X$ and $Y$ be independent random variables with $X$ Poisson distributed with mean $\lambda$ and $Y$ exponentially distributed with rate $\lambda>0$ and we let $(X_1,Y_1),\...
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Finding moment estimator and its asymptotic distribution

I got a question: We let $X$ and $Y$ be independent random variables with $X$ Poisson distributed with mean $\lambda$ and $Y$ exponentially distributed with rate $\lambda>0$ and we let $(X_1,Y_1),\...
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Conditional second moment normal distribution

Can someone help me find the expression of $$ E(Z^2 | Z \leq 0) \quad when \quad Z\sim N(0,1) $$ I'm aware that $$ E(Z | Z \leq 0) = \frac{2}{\sqrt{2\pi}} \quad , \,\, Z \sim N(0,1) $$ using the ...
nalen's user avatar
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Existence of function with zero positive and negative-integer moments

I am trying to find the existence of a continuous function defined on $(0,\infty)$ such that \begin{align*} \int_0^\infty x^n f(x)\ dx = \begin{cases} 1, &\mbox{ $n = 0$, }\\ 0, &\mbox { $n \...
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Moment of Intertia of hemisphere

Calculate the moment of inertia of a solid uniform hemisphere $x^2+y^2+z^2=a^2$; $z \geq 0$ with mass $m$ about the $z$ – axis. My attempt: We know that MI of a solid hemisphere is $2Ma^2/5$ where $M$ ...
shoyab khan's user avatar
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78 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 ...
PaulWH's user avatar
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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 ...
Krishna Soni's user avatar
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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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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)}...
cocojar's user avatar
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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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