# Questions tagged [moments]

The moments tag has no usage guidance.

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### What's the moment of inertia of a swept n-sphere?

I live in $n$ dimensions. I have a uniformly dense $n$-capsule $S$ of radius $r$ and semi-length $x$ - that is, I took an $n$-ball of radius $r$ from $[-x, 0, ..., 0]$ and swept it through to $[x, 0, ....

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118 views

### Are there any conditions on the moments that make a measure a probability measure?

For a positive Borel measure $\mu$ on the real line interval $[-1, 1]$, let $\displaystyle{m_n = \int_{-\infty}^\infty x^n d\mu(x)}$, i.e. the $n$th moments of the measure. Are there any conditions ...

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54 views

### Are Stochastic Process Characterized by Their conditional Moments

Suppose that $X_t$ is a real-valued stochastic process. Then is $X_t$ characterized by it's conditional moments? In the sence that, if $Y_t$ is another process, such that
$$
\mathbb{E}\left[\int_s^T\...

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139 views

### Is the covariance of squares always bounded from below by two times the covariance?

I came across the following inequality in one of my calculations ($X,Y$ are centered random variables):
$$\operatorname{E}(X^2Y^2)-\operatorname{E}(X^2)\operatorname{E}(Y^2) \geq 2 \operatorname{E}(...

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130 views

### Bound for a conditional expectation

Let $a_i, i=1, \ldots, n$ be real numbers. Let $\epsilon_i, \, i=1, \ldots, n$ be a random variables that take values $\pm 1$ with equal probability and $r_i, i=1, \ldots, n$ be random variables ...

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191 views

### Random averages over a Point process - Campbell's Theorem

Let $(N_t)_{t \geq 0}$ be a non-homogeneus Poisson process with intensity function $t \mapsto \lambda(t)$. Let $(T_n)_{n \in \mathbb{N}}$ be the corresponding simple point process (arrival times), ...

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417 views

### Summing moments and Riemann zeta values

Let $d\mu_n(x)=\cos^{2n}x\,dx$ and consider the averages of moments
$$\alpha_n=\frac{\int_0^{\pi/2}x^4d\mu_n(x)}{\int_0^{\pi/2}d\mu_n(x)}.$$
Then, I have encountered a curious evaluation
$$\sum_{n=1}^{...

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619 views

### $\mathbb{E}[X^4]=1$, $X,Y$ iid, what's the best upper bound of $\mathbb{E}[(X-Y)^4]$?

Let $X,Y$ be i.i.d. random variables, $\mathbb{E}[X^4]=1$, what's the best upper bound for $\mathbb{E}[(X-Y)^4]$ ?
A trivial upper bound is $16$, since $(X-Y)^4 \leq 8 (X^4+Y^4)$ then take ...

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64 views

### Convergence rate of $\operatorname E|\langle X,f_n\rangle|^p$

Suppose that $X$ is a random element with values in a separable Hilbert space $\mathbb H$ such that $\operatorname EX=0$ and $\operatorname E\|X\|^2<\infty$. Suppose that $f_1,f_2,\ldots$ form an ...

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84 views

### Existence of moment-constrained maximum entropy distribution with support $[0,1]^n$

Given a finite set of moment values $\{\mu_1,\ldots,\mu_N\}$, for which the multi-dimensional finite Hausdorff moment problem is determined. That is, we know that at least one distribution $\mathcal{D}...

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87 views

### 1D functional equation: solve for function with given expected value w.r.t normal density

Given scalars $c_1, c_2 > 0$, how would one go about solving, for non-expansive (i.e 1-Lipschitz) $\phi: \mathbb R \rightarrow (-\infty,+\infty]$, the following equation
$$
\begin{split}
\mathbb ...

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231 views

### Limit of the logarithm of the $L^p$ norm over the logarithm of $p$ as $p$ goes to infinity

Let $\mu$ be some positive measure on $\mathbb{R}$. For technical reasons, I would like to know if the limit
$$\lim_{p\rightarrow\infty}\frac {\ln \|f\|_{L^p(\mu)}}{\ln p}$$
exists in $[0,\infty]$ for ...

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83 views

### General result for the N-point correlation of the Poisson process (and its derivative)?

I am specifically interested in computing:
$$\mathbb{E}[S_p S_q S_s S_t]$$
where $S_t=\frac{dN_t}{dt}$ and $N_t$ is a Poisson process (so $S_t$ is a "Poisson pulse train"):
$$\mathbb{P}(N_t=n)=\frac{(\...

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46 views

### Using empirical expectation of orthogonal polynomials for density estimation

Estimating the probability density function using empirical moments is quite popular. Is there any advantage to using a different polynomial basis than the usual $1$,$x$, $x^2$ ... etc? For the moment ...

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238 views

### A Minkowski-like inequality

Assume that $X$ and $Y$ are two arbitrary non-negative random variables. Is the following inequality true for $1\leq\alpha\leq 2$?
\begin{align}
\left(\mathbb{E}\left[X^\alpha\right]-\mathbb{E}\left[...

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153 views

### uniquely determining a distribution using moments

Let $A$ be a parametric family of probability distributions that include all distributions in the form of $\phi(X)$ where $X\sim\mathcal{N}(0,\mathbf{I})$ is jointly Gaussian and $\phi:\mathbb{R}^d\to ...

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537 views

### Numerical evaluation/approximation of non-central high-order moments of high-dimensional Gaussian measures?

I need to numerically evaluate/approximate non-central high-order moments of high-dimensional Gaussian measures/distributions with given mathematical expectations and covariance matrices.
The Gaussian ...

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235 views

### Fractional moments of Poisson distribution

I wonder if there is a formula to calculate the fractional (0,1) moments of a Poisson distribution?
Thanks in advance.

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117 views

### Does this moment inequality hold for any probability measure on the positive real line?

Problem statement
Let $P$ be a probability measure on the positive real line and assume all it's raw moments, $\mu_k = \mathbb{E}[x^k]$, $k=1,2,\dots$ exist and $\mu_k < \infty$ for all $k$. Let $...

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118 views

### Moments of a Normal-Wishart distribution

Do known expressions exist for the moments of a gaussian-wishart (aka normal wishart) distribution?
$$NW(\mu,K\mid\mu_0,\lambda_0, v, W) =
\frac{|\lambda_0K|^{1/2}}{(2\pi)^{d/2}}e^{-0.5([\mu - \mu_0]...

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121 views

### Why study the moment problem in one dimensional case( Hamburger moment problem)

I have been reading about moment problem and I have been curious about the following question.
What is the motivation for studying the Hamburger moment problem(one dimensional moment problem?
I ...

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104 views

### Any chance to get the moments of this exotic distribution?

Let us define the following cumulative distribution:
\begin{align}
\Pr (Y(t)\geq a)=\sum_{n=0}^\infty \frac{e^{-kt}(kt)^n }{n!}\int_a^\infty[\circledast_{f}\circ H_a ]^n\delta (x) dx
\end{align}
where ...

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96 views

### Existence of a Laplace transform that takes specific values on the integers

The classical Marcinkiewicz theorem (1939) states that if a random variable $X$ has a Laplace transform/characteristic function of the form $\mathbb{E}(e^{tX})=e^{P(t)} $ with $P$ a polynomial, then ...

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74 views

### Random variables whose expectations are cumulants

In my research I stumbled about the following class of random variables: Let $X_0,X_1,\dots$ be random variables on a common probability space with finite moments of all orders. We then define
\...

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61 views

### Finding analytic expressions for the cumulants of a correlated random variable

I am working with cumulants of a distribution. I have an example of how the second cumulant may be simplified from:
$k_2 = p\alpha^2\left\{\left(\sum a_i\right)^2 - 2\sum_{i<j}a_ia_j\left(1-\rho^{...

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383 views

### Moments of derivatives of $L$-functions

I'd like to know why it is important to know the moments of the derivatives of $L$-functions.
The moments of $L$-functions are related to the Lindelöf Hypothesis, but what about the moments of the ...

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208 views

### Second moment of cos(x,y) for Normal x,y?

I'm trying to figure out second moment of the following quantity
$$y = \frac{\langle x_1, x_2 \rangle}{\left\|x_1\right\|\left\|x_2\right\|}$$
Where $x_1$, $x_2$ are sampled independently from $\...

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109 views

### Sufficient condition for a solution to Hamburger moment problem

Let $\{m_n\}_{n=0}^{\infty}$ be a sequence of real numbers.
It is well known that there exist a positive Borel measure $\mu$ on the real line with moments given by $\{m_n\}_{n=0}^{\infty}$ if and ...

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145 views

### Comparison of tail behaviour of two (bounded) random variables given their moments

Given: two positive scalar (bounded) random variables $X$ and $Y$ with the following conditions to hold:
$$ E(X)=E(Y),\ E(X^k)\ge E(Y^k), \forall k>1$$
How to show (whether it is possible to show) ...

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270 views

### Mathematical links between entropy and moments of a given distribution

Under some conditions, a distribution is determined by all of its moments. Furthermore, there is a certain value of entropy for a given distribution. So my question is:
1.Can I say that its entropy ...

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377 views

### Hankel determinants of binomial coefficients

For $\{h_{n}\}_{n=0}^{\infty}$ a real sequence, detone by $H_{n}$ the $n\times n$ Hankel matrix of the form
$$
H_{n}:=\begin{pmatrix}
h_{0} & h_{1} & \dots & h_{n-1}\\
h_{1} & h_{2} &...

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523 views

### Fourth moments of Gaussian processes

I am working on a topic outside my main research area, so I am afraid I am reproving obvious results, so I would like to ask for a reference. Google didn't help, mostly because I am looking for ...

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292 views

### Moment problem on [-1,1]: necessary and sufficient conditions

Consider a sequence of real numbers $s=(s_0,s_1,\ldots)$. When is there a Borel measure $\mu$ supported on $[-1,1]$ so that
$$
s_k = \int_{[-1,1]} x^k\,\mathrm{d}\mu,\quad \forall k\in\mathbb N\;?
$$
...

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168 views

### Computing skewness derivative in terms of variance

In the Portilla Simoncelli paper (page 18):
http://www.cns.nyu.edu/pub/lcv/portilla99-reprint.pdf
They go about calculating the derivative of the skewness $\eta(x)$ of a distribution (2D matrix in ...

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298 views

### Gaussian and the convex hull of moment curves

Let $c_1,\dots, c_d$ be the first $d$ moments of the standard normal distribution. Does the point $(c_1,\dots, c_d)$ lie in the convex hull of the set $\{(t,t^2,\dots,t^d)\colon t\in[-b,b]\}$, for a ...

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569 views

### Moment matching: construction of a mixture of Gaussian distribution with lower moments identical to Gaussian

This is a question related to the statistical model behind independent component analysis (ICA).
We assume that $Z \sim N(0,1)$. Our goal is to construct a random variable $X$ that follows a ...

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242 views

### Inverse problem for negative moments

Let $D$ be a bounded simply connected domain in the plane, bounded by a smooth closed curve $\partial D$. Moreover$D$ contains the origin. Assume that all negative complex moments w.r.t arc-length ...

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87 views

### Moment Sequence in l²

I have the following problem/question:
For which finite regular complex measures $\mu$ is the moment sequence
$$
\left(\int_{[-1,1]}t^k\,d\mu\right)_{k\in\mathbb N}
$$
a member of $\ell^2(\mathbb N)$?...

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204 views

### Pair of square matrices related by traces formulas

Let $A$ and $B$ be two $n\times n$ matrices over $\mathbb{C}$. Assume that for every $k\geq 1$ it holds $tr(A^k) = tr(B^{2k-1})$. What can we say about the possible eigenvalues of $A$ and of $B$? How ...

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218 views

### Bai and Silverstein's “Lemma on Quadratic Forms” - question about the constant $C_p$

In the book "Spectral Analysis of Large Dimensional Random Matrices" by Bai and Silverstein, there is the following lemma:
Lemma B.26 (pg. 530) Let $A=(a_{ij})$ be an $n\times n$ nonrandom matrix ...

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131 views

### Joint cumulants of $Z_2^n$ characters

Let $f_{c}:Z_2^n \rightarrow \{-1,1\}$ be the character defined as $f_c(x) = (-1)^{<x,c>}$, where $c,x \in Z_2^n$. It is easy to see that since $f_{c_1}\cdot\ldots\cdot f_{c_k} = f_{c_1 \oplus \...

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### Dimension reduction for low-order moments of Rademacher-weighted sums of vectors

Let $x_1,\dots,x_n$ be vectors in a Euclidean space $H$. Let $\varepsilon_1,\dots,\varepsilon_n$ be independent Rademacher random variables (r.v.'s), so that $P(\varepsilon_i=\pm1)=1/2$ for all $i$.
...