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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 ...
Greg Zitelli's user avatar
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 $\sum_i x_i$ ...
CWC's user avatar
  • 423
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}^\...
Jingeon An-Lacroix's user avatar
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 ...
Akira's user avatar
  • 1,063
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 $...
Pavel Gubkin's user avatar
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 $$ ...
Pavel Gubkin's user avatar
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}}, \...
Akira's user avatar
  • 1,063
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 ...
Paul R's user avatar
  • 49
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 ...
Michael Hardy's user avatar
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)...
MHMH's user avatar
  • 71
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 ...
user196574's user avatar
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,\...
DRJ's user avatar
  • 222
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 ...
Yarden Levy's user avatar
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 ...
brownianmotion's user avatar
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 ...
wuhanichina's user avatar
2 votes
0 answers
127 views

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 ...
Christian Remling's user avatar
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 $$ ...
reexpi's user avatar
  • 11
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 ...
Dasherman's user avatar
  • 203
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 $[0,1]$)
Shadumu's user avatar
  • 85
6 votes
1 answer
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) ...
Boby's user avatar
  • 631
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)?
user155294's user avatar
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, $(...
Michael Hardy's user avatar
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\...
Fancier of Mathematica's user avatar
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 ...
Lior Gishboliner's user avatar
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}^\...
Goulifet's user avatar
  • 2,226
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) \...
nGlacTOwnS's user avatar
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 $...
Boby's user avatar
  • 631
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 > ...
Xiaosheng Mu's user avatar
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 ...
Hans's user avatar
  • 2,905
-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,...
Denis's user avatar
  • 11
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.
Matan Cohen's user avatar
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)...
harharkh's user avatar
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 ...
John Doe's user avatar
2 votes
1 answer
205 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\...
qwert's user avatar
  • 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 ...
Hans-Peter Stricker's user avatar
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 ...
lrnv's user avatar
  • 686
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 ...
lrnv's user avatar
  • 686
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\...
volond's user avatar
  • 97
3 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 ...
trisct's user avatar
  • 273
0 votes
1 answer
117 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}\...
lrnv's user avatar
  • 686
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 ...
Anurag Sahay's user avatar
  • 1,278
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 \, ...
lrnv's user avatar
  • 686
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 ...
reynoldking's user avatar
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 ...
Julian's user avatar
  • 613
0 votes
0 answers
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 ...
Sylvain JULIEN's user avatar
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 ...
camel8899's user avatar
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 ...
VS.'s user avatar
  • 1,826
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 ...
Tianxin Zou's user avatar
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 ...
Josue's user avatar
  • 11
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....
user58955's user avatar
  • 630