# Questions tagged [moments]

The moments tag has no usage guidance.

93
questions

1
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1
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95
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### 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 ...

2
votes

5
answers

523
views

### 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 $\sum_i x_i$ ...

0
votes

0
answers

122
views

### Some improper integral

While studying problems related to fractional Laplacians $(-\Delta)^s$, I keep come across with type of quantities such as
$$
\int_0^R r^{2k}f(r)\,dr+\int_R^\infty r^{2k}\varphi(r)\,dr-\sum_{j=0}^\...

2
votes

1
answer

201
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 ...

2
votes

0
answers

156
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 $...

1
vote

0
answers

70
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
$$
...

1
vote

1
answer

61
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}},
\...

0
votes

1
answer

171
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 ...

5
votes

1
answer

199
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 ...

1
vote

0
answers

76
views

### 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)...

2
votes

1
answer

107
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 ...

0
votes

0
answers

72
views

### 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,\...

2
votes

0
answers

69
views

### 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 ...

1
vote

1
answer

108
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

280
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 ...

2
votes

0
answers

127
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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 ...

1
vote

1
answer

114
views

### 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
$$
...

2
votes

0
answers

59
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 ...

0
votes

0
answers

62
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 $[0,1]$)

6
votes

1
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316
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) ...

0
votes

0
answers

279
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)?

0
votes

1
answer

536
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

231
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

308
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 ...

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}^\...

3
votes

0
answers

108
views

### 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)
\...

2
votes

1
answer

627
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 $...

22
votes

2
answers

810
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 > ...

4
votes

0
answers

101
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
votes

1
answer

122
views

### 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,...

1
vote

1
answer

157
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.

3
votes

1
answer

206
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)...

0
votes

1
answer

190
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 ...

2
votes

1
answer

205
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### 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\...

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 ...

1
vote

1
answer

155
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 ...

0
votes

1
answer

83
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 ...

1
vote

0
answers

77
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\...

3
votes

2
answers

1k
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### 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 ...

0
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1
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117
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### 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}\...

2
votes

1
answer

464
views

### 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
vote

1
answer

60
views

### 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 \, ...

4
votes

1
answer

841
views

### 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 ...

2
votes

0
answers

58
views

### 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 ...

0
votes

0
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117
views

### What are the best unconditional bounds for the discrete moments of $\zeta'$?

I heard this morning through the app Researcher about the following article by Scott Kirila: arXiv:1804.08826v2 in which the author proves under RH that $J_{k}(T)\asymp_{k}(\log T)^{k(k+2)}$, where ...

2
votes

1
answer

1k
views

### 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 ...

1
vote

2
answers

151
views

### 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 ...

2
votes

2
answers

231
views

### 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 ...

1
vote

2
answers

98
views

### 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 ...

1
vote

1
answer

161
views

### 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....