Questions tagged [moments]
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102 questions
0
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0
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37
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Bounding the error of a truncated moment problem
Let $\{x_{i}\}_{i=1}^{\infty}$ be a non-increasing sequence of non-negative real numbers, and let $\{y_{j}\}_{j=1}^{B}$ be a non-increasing sequence of non-negative real numbers, where $B$ is a finite ...
- 433
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votes
3
answers
782
views
Show there is no positive r.v. $U$ such that $\frac{1}{2} = \frac{\mathbb{E}[U^k 1_{U \ge (k+1)/2 }]}{\mathbb{E}[U^k]}, \, \forall k \in \mathbb{N}_0$
Let $U$ be a non-negative random variable such that for all $k \in \mathbb{N}_0$
\begin{align}
\frac{1}{2} = \frac{\mathbb{E}[U^k 1_{U \ge \frac{k+1}{2} }]}{\mathbb{E}[U^k]}.
\end{align}
In ...
- 671
2
votes
3
answers
338
views
Sum of RVs satisfying Bernstein condition on moments
Let us say that a RV $X$ with mean $\mu$ and variance $\sigma^2$ satisfies Bernstein condition with a parameter $\beta>0$, if for all $k \ge 2$, it holds that
$$
|\mathbb{E}[(X - \mu)^k]| \le \frac ...
- 367
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0
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94
views
Infinite sequence of PSD non-moments in two variables
Define a 2d sequence to be a mapping $a: \mathbb{N}^2 \to \mathbb{R}$ (where $\mathbb{N} = \{0, 1, \dots\}$). Here are two definitions of types of 2d sequences:
We say that a 2d sequence $a$ is a ...
- 161
3
votes
1
answer
219
views
Moment problem, ergodicity and spectral gap on the space of tempered distributions
Let $\{ S_n \}_{n=0}^\infty$ be a collection of tempered distributions where $S_0:=1$ and $S_n$ is a tempered distribution on $\mathbb{R}^n$.
Just below formula [5] in p.122 of the Fröhlich paper, ...
- 3,477
2
votes
1
answer
83
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Azuma-Hoeffding for one-side bounded super-martingale sequence
Suppose we have a real-valued super-martingale difference sequence $\{X_k\}$ w.r.t. some filtration $\mathcal{F_k}$, i.e., $X_k$ is $\mathcal{F_k}$-measurable, and
$$ \mathbb{E}[X_k|\mathcal{F}_{k-1}] ...
- 33
1
vote
1
answer
98
views
Moment generating function of one-side bounded random variable
Suppose we have a scalar random variable $X$ satisfying the one-sided bound $X \leq d$, for some $d > 0$. Additionally, we know that $\mathbb{E}[X] \leq 0$. Can we establish a bound of the ...
- 33
2
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0
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123
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Continuity of entropies, replica trick and Hausdorff moment problem
I could not find a really appropriate title for my question (happy to revise) but let me explain.
Suppose $p(x|c)$ is a probability density function over $x \in [0,1]$ which depends continuously on ...
- 231
1
vote
0
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80
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Moments from characteristic function for matrices
When $x$ is a random variable with the smooth characteristic function $\phi_x(t) = \mathbb{E}e^{itx}$, we can easily compute the moments as $\mathbb{E}[x^k] = i^{-n}\phi_x^{(n)}(0)$. There is no magic ...
- 193
2
votes
0
answers
56
views
Sum of independent Wisharts
Suppose random vectors $y_1,y_2,\ldots,y_m$ are independent and the distribution of each $y_i$ is a $d$-dimensional complex Gaussian with mean $0$ and covariance $\Gamma_i$, that is $y_i \sim \mathcal{...
- 193
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votes
0
answers
32
views
Finding measure representation for rank 2 moment matrices
Assuming the following equation has a solution, I'm interested in finding any concrete values of $x_{1},\dots x_{n},y_{1},\dots y_{n},c_{1},c_{2},R$ that fulfills it.
$$
\begin{bmatrix}
1 & 1 \\
...
- 275
1
vote
1
answer
114
views
Convergence in probability of sample covariance for permutation invariant triangular arrays
Take two triangular arrays $X_{N,i}$ and $Y_{N,i}$ of random variables where $1 \le i \le N$. Suppose that the families $\{X_{N,i}\}$ and $\{Y_{N,i}\}$ are independent, and that the following ...
- 1,084
2
votes
5
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540
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Finding $\sum_i x_i$ given $\{\sum_i x_i^{2n}\}_{n\in \mathbb{N}}$
Can we find $\sum_i x_i$ given $\{\sum_i x_i^{2n}\}_{n\in \mathbb{N}}$, assuming $\{x_i\}_{i\in\mathbb{N}}$ is a set of positive real numbers?
Perhaps an easier question is, can we find $\sum_i x_i$ ...
- 433
2
votes
1
answer
215
views
Decay estimate of moment of an SDE
We consider an SDE
$$
d X_t = b(t, X_t) \, dt + \sigma(t, X_t) \, d B_t,
$$
where $(B_t)$ is a $d$-dimensional Brownian motion on $\mathbb R^d$. We fix $p \in [1, \infty)$. Here $b, \sigma$ are ...
- 835
2
votes
0
answers
162
views
Taylor coefficients of the integral of the ordered exponential
Let $A$ be a continuous $2\times 2$ matrix-valued function on $[0,1]$. Define $X_A$ as the solution of
$$
X_A'(t) = A(t) X_A(t), \qquad X(0) = I.
$$
In other words $X_A$ is the ordered exponential of $...
- 577
1
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0
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75
views
Integral inequality related to the (mixed?) moments of two functions
For $k\in \mathbb{Z}_+$ and $t\in [0,1]$ we set
$$
S_{k}(t) = \{(t_1,\ldots, t_k)\colon 1\ge t\ge t_1\ge\ldots\ge t_k\ge 0\}.
$$
Let $a$ and $b$ be two continuous functions on $[0,1]$. Introduce
$$
...
- 577
1
vote
1
answer
66
views
Upper bound $I_R := \int_{B_R^c} |x| (P_t \ell_\nu) (x) \, \mathrm d x$ in terms of $R, \nu, t$?
Let $(p_t)_{t >0}$ be the Gaussian heat kernel on $\mathbb R^d$ and $(P_t)_{t >0}$ its induced semi-group, i.e.,
$$
\begin{align}
p_t (x) &:= (4\pi t)^{-\frac{d}{2}} e^{-\frac{|x|^2}{4t}},
\...
- 835
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votes
1
answer
184
views
What conditions should be satisfied for a rational function to be a moment generating function?
I have a table of points at which a moment generating function is evaluated (for points $t_0,t_1,t_2,\ldots,t_n$ I know $M(t_0), M(t_1), M(t_2),\ldots,M(t_n)$).
I've approximated these tabular ...
- 49
5
votes
1
answer
203
views
Uniqueness of the variance
The variance assigns a number to each of certain probability distributions on Borel subsets of $\mathbb R$. It has the properties of
(1) shift-invariance, i.e. if $X$ is a random variable with a ...
2
votes
1
answer
136
views
Linear transformation of random vector has bounded moments?
Suppose that the random vector $\mathbf{y}=(Y_1, Y_2,\ldots, Y_p)^{\top}$ satisfies:
$\mathbb{E}(Y_i) = 0$, $\mathbb{E}(Y_i^2)=1$ for any $1\leqslant i \leqslant p$;
$\mathrm{Cov}(Y_i,Y_j)=0$ for $i\...
- 81
2
votes
1
answer
114
views
Eigenvalues of $H_1 H_2 H_1$, where $H_1$, $H_2$ independent $\mathit{GUE}$
Given $H_1$ and $H_2$ i.i.d. $\mathit{GUE}$ matrices, what is the single eigenvalue distribution of $H_1 H_2 H_1$ in the large $N$ limit? This matrix is Hermitian, and so its eigenvalues are still ...
- 524
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0
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72
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A moment-based stochastic order
Given that a Borel probability measure $\mu$ on [0,1] is characterized by its moments, it seems natural to consider the following stochastic order: say that $\mu\le\mu'$ if $$\forall k\in\mathbb N,\...
- 234
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72
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Under what conditions is the least-squares approximation bounded with the same Lipschitz gradient constants?
Let $f(x):\mathbb{R}^K\Longrightarrow \mathbb{R}^L$ denote a multivariate continuously differentiable function. All the partial derivatives of $f$ (all its Jacobian elements) are bounded from above ...
- 21
1
vote
1
answer
118
views
Comparison of Rademacher and Gaussian moments under linear transformations
Let $X$ be an $n$ dimensional standard Gaussian and let $U$ be an $n \times n$ orthogonal matrix. Then, the random vector $Z = U^\top X$ is also distributed as a standard Gaussian in $R^n$ and we have ...
3
votes
1
answer
375
views
Stochastic dominance using moment-generating function
Traditionally, stochastic dominance is defined using the cumulative distribution function(CDF). But sometimes, the CDF is not easily to be obtained. For example, the generalized noncentral Chi-square ...
- 41
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0
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134
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Uniqueness of a moment style problem
This is a leftover from this question (and I've modified slightly to make the question more natural in the new setting). It's maybe not a very fascinating question by itself, but it seems this is what ...
- 24.2k
1
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1
answer
124
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Multiplying a log-concave function to a Gaussian probability density reduces its variance
Let $X$ be a random Gaussian vector with probability density $p_X(x)$. Let $Y$ be the random variable with density proportional to $p_X(x)e^{-g(x)}$ for some convex function $g$. Does it hold that
$$
...
- 11
2
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0
answers
61
views
Approximate logarithmic bound on expected maximum via central limit theorem
If $Z_i$ are standard normal, possibly dependent, one can show that
$$E\left[\max_{i=1,...,M} Z_i^2\right]\leq 3\ln M + 1.$$
I'm looking for a similar (asymptotic) bound for asymptotically normal ...
- 203
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0
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63
views
The moment problem for $m_n=1/n$
What is the p.d.f. for the moments $m_n=1/n$ ?
(They are obtained from $\int_0^1 x^n/x\ dx $, but clearly $1/x$ is not a p.d.f. on $[0,1]$)
- 85
6
votes
1
answer
328
views
Show that $M_X(t) = 2 E \left[ e^{tX} \Phi( aX-t) \right], \forall t \in \mathbb{R}$ iff $X$ is Gaussian
Let $M_X(t)$ denote the moment generating function of a random variable $X$. Now suppose that the following expression holds: for a given $a>0$
\begin{align}
M_X(t) = 2 E \left[ e^{tX} \Phi( aX-t) ...
- 671
0
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0
answers
284
views
Upper bound for moment of the Riemann zeta function
Is it possible to get a good upper bound for the integral $$\int_{0}^{T}\zeta ^{3}(\frac{1}{3}+it)dt$$ (unconditionally)?
- 150
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votes
1
answer
612
views
Was this proposition on cumulants of compound Poisson distributions known before I put it into a Wikipedia article?
The $n$th cumulant $\kappa_n$ of a probability distribution for $n\ge2$ is functional that is a polynomial in the first $n$ moments of the distribution, that has the properties of $(1)$ homogeneity, $(...
4
votes
1
answer
248
views
Ratio of the first squared and the second moment
Let $G(t)$ be a probability generating function of some integer and non-negative r. v. $X$. Suppose that
$$\lim_{t\to1}G'(t)=+\infty.$$
That is
$$
\mathbb{E}X=+\infty.
$$
Can you show that
$$
\lim_{t\...
2
votes
1
answer
365
views
Tail bounds via moments
Suppose X is a discrete random variable with $\mathbb{E}[X] = \mu$, such that $\mathbb{E}[(X - \mu)^k] = \Theta(\mu^{k-1})$ for every $k \geq 2$. What (if anything) can be said about the concentration ...
- 428
2
votes
1
answer
78
views
Asymptotic behavior of the moments of non-negative sequences
We consider a sequence $u = (u_k)_{k\geq 1}$ such that $u_k \geq 0$ for any $k \geq 1$. We assume that there exists a critical $p_c \in \mathbb{R}$ such that, for any $q<p_c <p$,
$$\sum_{k=1}^\...
- 2,306
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0
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111
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Cumulants as coefficients of degree-n polynomial and conditions for real roots
In a $(n-1)$-degree polynomial, $P_{n-1}(z) = c_{n-1} z^{n-1} + c_{n-2} z^{n-2} + \ldots + c_0$, defined by,
\begin{equation}
P_{n-1}(z) = \sum\limits_{i=1}^n \, \prod\limits_{j=1,j\neq i}^n (z-x_j)
\...
- 183
2
votes
1
answer
669
views
Radius of convergence of cumulant generating function
Recall that for a random variable $X$ with a moment generating function $M_X(t)$ the cumulant generating function is defined as
\begin{align}
K_X(t)=\log M_X(t)
\end{align}
The Taylor expansion of $...
- 671
22
votes
2
answers
848
views
Do equal integrals of $1/(1+x^a)$ imply equal measure?
Suppose $\mu$ and $\nu$ are two probability measures on $[0,1]$ with the property that $\int_{0}^{1} \frac{1}{1+x^a} ~d\mu(x) = \int_{0}^{1} \frac{1}{1+x^a} ~d\nu(x)$ holds for every exponent $a > ...
- 765
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votes
0
answers
104
views
Which linear forms are linear combinations of point evaluations?
Let $f_1,\ldots,f_r\in\mathbb{C}[x,y]$ and consider the subalgebras $A_1,\ldots,A_r$ of $\mathbb{C}[x,y]$ that are generated by $f_1,\ldots,f_r$, i.e., $A_i=\mathbb{C}[f_i]$. Using some dimension ...
- 3,031
-3
votes
1
answer
123
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Are the first 4 statistical moments independent? [closed]
Are the first 4 statistical moments independent? Is there a mathematical demonstration that can show independence one from each other? Can the value of one moment influence the value of another? If so,...
- 11
1
vote
1
answer
170
views
Is it always possible to determine the distribution of a random variable given all its moments? [closed]
we we're asked about it, and I know that answer is "NO", and I haven't found an good enough answer yet
and would appreciate an explanation with examples.
- 11
3
votes
1
answer
221
views
Generalized moment problem for discrete distributions
Consider the discrete distribution $\mu = \sum_{i = 0}^{n - 1} \delta(x - x_i)$ with all $a \leq x_0 < \ldots < x_{n - 1} \leq b$ and $[a, b] \in \mathbb{R}$. Suppose that $u_0(x), \ldots, u_n(x)...
- 33
0
votes
1
answer
195
views
How to get the mean, skewness of an Itō integral?
If $B_t$ denotes a standard Brownian motion, and let $X_t = \int f(s)dB_s$, $f(s)$ is a deterministic integrand. I know $B_t$ is a martingale. Is $X_t$ also a martingale? And how can I get the formula ...
- 21
2
votes
1
answer
214
views
sign of odd central moments of binomial distribution
I am interested in the sign of odd, central moments of a binomial distribution. From DOI. 10.1137/070700024 I have the formulae:
$ E\left[\left(X-\mu\right)^d\right]= \sum_{i=0}^n\binom{n}{i}\left(-p\...
- 89
5
votes
4
answers
2k
views
The distance distribution of graphs
The degree distribution of a graph is of main importance, especially for large graphs, and namely random graphs. Its expected value and its higher moments tell a lot about a graph – but of course not ...
- 9,720
1
vote
1
answer
164
views
Is this (somewhat specific) moment problem treated somewhere?
Suppose I have a measure $\mu$ over $\mathbb R_+$ given by its moments $\mu_0,...,\mu_n$, defined as :
$$\mu_k = \int x^{k} \partial\mu(x),\; k \in 1,...,n$$
Using Faà di Bruno's formula, I can ...
- 686
0
votes
1
answer
89
views
Finding a r-atomic solution to the univariate truncated Hausdorff moment problem
Suppose i have a certain $t>0$, and observations of moments of a random variable $X$ given by $\mu_0=1,\mu_1,...,\mu_n$.
How can i:
Check that a measure with support $[0, \frac{1}{t}]$ with thoses ...
- 686
1
vote
0
answers
82
views
Negative moments of Steinhaus random variables
Let $f_i, i=1, \ldots, n$ be independent Steinhaus random variables, i.e. variables which are uniformly distributed on the complex unit circle. Let $a \in R^n$.
1) Find $E\left(\sum_{i=1}^nf_i a_i\...
- 97
4
votes
2
answers
1k
views
Differentiability of characteristic functions and moments of the corresponding measure
Let $f$ be the characteristic function of a real-valued random variable $X$. It is known that if $f$ has a $k$-th order derivative (for some even $k$) then $\mu$ has a finite $k$-th order moment. Is ...
- 283
0
votes
1
answer
123
views
Is there a proper name for those 'shifted moments'?
Suppose that we have a random variable $X$, $i \in \mathbb N$, and a scalar $t$. Is there a proper name for these integrals, that I for the moment call 'shifted moments' ?
$$I_{i}^{t} = \mathbb{E}\...
- 686