Questions tagged [moments]
The moments tag has no usage guidance.
102 questions
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Moments of Dirichlet $L$-functions on the critical line
I'm looking for a reference for some questions related to the moments of Dirichlet characters on the critical line,
$$ M_k(T;\chi) = \int_T^{2T} |L(1/2+it,\chi)|^{2k}\,dt, $$
where $\chi$ is a ...
- 1,354
1
vote
1
answer
66
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Can this be translated to a truncated multivariate moment problem?
Fix $\mathbf t \in \mathbb{R}_+^d$. I am looking for a r-atomic measure $\nu$ on $\mathbb{R}_{+}^{d}$ that solves the following 'moment' conditions.
$$\forall\, \mathbf i \in \mathbb{N}^{d} \, s.t \, ...
- 686
4
votes
1
answer
964
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Fractional moments of multivariate normal distributions
Is there an analytic formula for fractional moments of multivariate normal distribution?
$E(\prod_{i=1}^k x_i^{\nu_i})={?}$ where $X=(x_1,\ldots,x_k) \sim N_k(\mu, \Sigma)$, $\nu_i\in \mathcal{R}$ and ...
- 41
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0
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61
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Cumulant of functions of weakly dependent random variables
Suppose $X_1,\dots,X_4$ are Gaussian random variables with mean and variance
$$\mathbf E X_i = 0,\quad \mathbf E X_i^2=1.$$
Furthermore suppose that the random variables have a certain weak ...
- 623
3
votes
1
answer
1k
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Expected value of absolute value of shifted binomial distribution
Recently my research needs to calculate the close form of $\mathsf{E}[|X-\frac{n}{2}|]$ where $X$ follows binomial distribution with parameter $(n,p)$. When $p=\frac{1}{2}$, this is just the mean ...
- 31
1
vote
2
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165
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A conjecture on 'truncated joint moments' of binomial coefficients under binomial distribution
This is similar in spirit to Sum of squares of middle binomial sums or 'Truncated mean' of binomial coefficients under binomial distribution but gives some total estimates. Though the other ...
- 1,826
2
votes
2
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258
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Is any real function satisfying basic conditions a moment generating function? [closed]
We all know that a mgf of a random variable $m_X(t)$ is positive and $m(0)=1$. My question is: if a positive real function $f(t)$ satisfies $f(0)=1$ and the function is smooth enough (around 0), does ...
- 29
1
vote
2
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112
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Are these moments related to any usual distribution?
Knowing the moments of a random variable, we are wondering if we can express it in terms of some usual distribution (beta, uniform...), maybe as a product of distributions.
$X$ is a $[0,1]$-valued ...
- 11
1
vote
1
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174
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mollifier satisfying moment conditions
I wish to find a mollifier $\psi\in C_0^{d+1}(-1,1)$ such that
$$
\int_{-1}^1 x^k \psi(x)dx = \begin{cases}
1, & k=0;\\
0, & k=1,\dots,d.
\end{cases}
$$
This paper (https://home.cscamm....
- 640
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0
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171
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Estimating integral of product of terms $\cos(t\log p)$
I would like to prove the following proposition from A. Harper's paper "Sharp conditional upper bound for moments of the Riemann Zeta Function"
Proposition.
Let $T$ be large and let $n=p_1^{\...
- 199
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1
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137
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Approximate $\log \mathbb E_P[\exp(th(x)]$ for a function $h$ which is lipschitz and has finite moments of order 1 and 2 w.r.t $P$
Let $P$ be a probability measure on a space $\mathcal X$ and $h: \mathcal X \rightarrow \mathbb R$ is measurable function with finite moments of order 1 and 2. I'm interested in approximating the ...
- 6,853
1
vote
1
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122
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MGF of a RV that is the ratio between a complex Gaussian and a Chi-squared RVs
Given the following p.d.f., which is the p.d.f. of the real and imaginary parts of a random variable that is the ratio between a complex Gaussian and a Chi-squared RVs:
\begin{equation*}
f_U(u)=\exp\...
3
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0
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187
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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 ...
- 210
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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\...
- 5,407
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2
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189
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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):
$$\operatorname{E}(X^2Y^2)-\operatorname{E}(X^2)\operatorname{E}(Y^2) \geq 2 \operatorname{E}(...
- 617
4
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1
answer
1k
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Bound for a conditional expectation
Let $a_i, i=1, \ldots, n$ be real numbers. Let $\epsilon_i, \, i=1, \ldots, n$ be a random variables that take values $\pm 1$ with equal probability and $r_i, i=1, \ldots, n$ be random variables ...
- 343
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0
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454
views
Random averages over a Point process - Campbell's Theorem
Let $(N_t)_{t \geq 0}$ be a non-homogeneus Poisson process with intensity function $t \mapsto \lambda(t)$. Let $(T_n)_{n \in \mathbb{N}}$ be the corresponding simple point process (arrival times), ...
user83658
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Summing moments and Riemann zeta values
Let $d\mu_n(x)=\cos^{2n}x\,dx$ and consider the averages of moments
$$\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}^{...
- 43.1k
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votes
1
answer
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$\mathbb{E}[X^4]=1$, $X,Y$ 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 ...
- 563
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81
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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 ...
- 423
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vote
0
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115
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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}...
- 11
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votes
1
answer
98
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
$$
\begin{split}
\mathbb ...
- 6,853
3
votes
2
answers
1k
views
Limit of the logarithm of the $L^p$ norm over the logarithm of $p$ as $p$ goes to infinity
Let $\mu$ be some positive measure on $\mathbb{R}$. For technical reasons, I would like to know if the limit
$$\lim_{p\rightarrow\infty}\frac {\ln \|f\|_{L^p(\mu)}}{\ln p}$$
exists in $[0,\infty]$ for ...
- 1,429
3
votes
1
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164
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General result for the N-point correlation of the Poisson process (and its derivative)?
I am specifically interested in computing:
$$\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=...
- 634
5
votes
1
answer
348
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[...
- 287
6
votes
1
answer
365
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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 ...
- 181
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votes
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696
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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 ...
- 1,026
6
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2
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1k
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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.
4
votes
1
answer
197
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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 $...
2
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0
answers
247
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]...
- 121
1
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1
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203
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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 ...
- 125
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0
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119
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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 ...
- 634
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0
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115
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 ...
- 593
4
votes
0
answers
95
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
\...
- 623
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0
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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^{...
- 139
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0
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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 ...
- 241
3
votes
1
answer
345
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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\|}$$
Where $x_1$, $x_2$ are sampled independently from $\...
- 2,442
2
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0
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156
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Sufficient condition for a solution to Hamburger moment problem
Let $\{m_n\}_{n=0}^{\infty}$ be a sequence of real numbers.
It is well known that there exist a positive Borel measure $\mu$ on the real line with moments given by $\{m_n\}_{n=0}^{\infty}$ if and ...
- 671
2
votes
1
answer
252
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Comparison of tail behaviour of two (bounded) random variables given their moments
Given: two positive scalar (bounded) random variables $X$ and $Y$ with the following conditions to hold:
$$ E(X)=E(Y),\ E(X^k)\ge E(Y^k), \forall k>1$$
How to show (whether it is possible to show) ...
- 21
4
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1
answer
581
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Mathematical links between entropy and moments of a given distribution
Under some conditions, a distribution is determined by all of its moments. Furthermore, there is a certain value of entropy for a given distribution. So my question is:
1.Can I say that its entropy ...
- 53
16
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1
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895
views
Hankel determinants of binomial coefficients
For $\{h_{n}\}_{n=0}^{\infty}$ a real sequence, denote by $H_{n}$ the $n\times n$ Hankel matrix of the form
$$
H_{n}:=\begin{pmatrix}
h_{0} & h_{1} & \dots & h_{n-1}\\
h_{1} & ...
- 2,188
9
votes
1
answer
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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 ...
- 20.2k
6
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2
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445
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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
$$
s_k = \int_{[-1,1]} x^k\,\mathrm{d}\mu,\quad \forall k\in\mathbb N\;?
$$
...
- 505
1
vote
1
answer
411
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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 ...
- 113
7
votes
2
answers
605
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 ...
- 1,127
8
votes
2
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2k
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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 ...
- 1,127
3
votes
0
answers
246
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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 ...
- 1,583
3
votes
0
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96
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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 $\ell^2(\mathbb N)$?...
8
votes
1
answer
218
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 ...
- 5,039
2
votes
1
answer
364
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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 ...
- 947