Questions tagged [moments]

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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 ...
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  • 213
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0 answers
55 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 $[0,1]$)
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  • 53
6 votes
1 answer
282 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) ...
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  • 599
0 votes
0 answers
210 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)?
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1 vote
1 answer
235 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, $(...
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4 votes
1 answer
167 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\...
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2 votes
1 answer
87 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 ...
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2 votes
1 answer
70 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}^\...
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  • 2,010
3 votes
0 answers
81 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) \...
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21 views

Reference: Good bounds for Variance of a Random Vector with Known Mean Supported on a Compact Set of Low Metric Entropy

Let $\emptyset\neq M\subseteq \mathbb{R}^n$ be a compact set, $X:\Omega\rightarrow M$ be a random vector defined on a complete probability space $(\Omega,\mathcal{F},\mathbb{P})$ and suppose that $\mu:...
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2 votes
1 answer
256 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 $...
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  • 599
22 votes
2 answers
650 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 > ...
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4 votes
0 answers
92 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 ...
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-3 votes
1 answer
92 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,...
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1 vote
1 answer
123 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.
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3 votes
1 answer
143 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)...
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0 votes
0 answers
100 views

How to get the mean, skewness of a 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 known $B_t$ is a martingale, Is $X_t$ also a martingale? And how can i get the formula ...
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2 votes
1 answer
90 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\...
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  • 91
6 votes
4 answers
718 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 ...
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1 vote
1 answer
122 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 ...
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  • 621
0 votes
1 answer
65 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 ...
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  • 621
1 vote
0 answers
61 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\...
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  • 49
3 votes
1 answer
410 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 ...
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  • 273
0 votes
1 answer
79 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}\...
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  • 621
2 votes
1 answer
279 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 ...
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1 vote
1 answer
51 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 \, ...
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  • 621
3 votes
1 answer
380 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 ...
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2 votes
0 answers
42 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 ...
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  • 563
0 votes
0 answers
96 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 ...
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1 vote
1 answer
627 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 ...
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1 vote
2 answers
116 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 ...
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  • 1,698
1 vote
2 answers
140 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 ...
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1 vote
2 answers
89 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 ...
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  • 11
1 vote
1 answer
134 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....
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3 votes
0 answers
161 views

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^{\...
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1 vote
1 answer
72 views

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 ...
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  • 5,580
1 vote
1 answer
85 views

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\...
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3 votes
0 answers
144 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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5 votes
0 answers
67 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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3 votes
2 answers
167 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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4 votes
1 answer
579 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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2 votes
0 answers
405 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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9 votes
1 answer
440 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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31 votes
1 answer
1k 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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  • 543
1 vote
1 answer
74 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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  • 371
1 vote
0 answers
110 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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  • 11
0 votes
1 answer
90 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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  • 5,580
3 votes
2 answers
897 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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  • 1,390
3 votes
1 answer
117 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=...
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  • 358
5 votes
1 answer
296 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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