36
votes
Accepted
$\mathbb{E}[X^4]=1$, $X,Y$ iid, what's the best upper bound of $\mathbb{E}[(X-Y)^4]$?
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- 128k
15
votes
Do equal integrals of $1/(1+x^a)$ imply equal measure?
Yes, provided that you know a priori that $\int_{(0,1]} x^{-\kappa} d\mu(x) < \infty$ (and the same for $\nu$) for some $\kappa > 0$. By taking the limit $a \to \infty$, you can detect the size ...
- 10.3k
12
votes
Accepted
Finding $\sum_i x_i$ given $\{\sum_i x_i^{2n}\}_{n\in \mathbb{N}}$
If you plot $\log f_n$ versus $n$, with $f_n=\sum_i x_i^{2n}$, then the asymptotic slope for large $n$ will give you the largest of the $x_i$; subtracting that contribution from $f_n$ and repeating ...
- 188k
11
votes
Accepted
Summing moments and Riemann zeta values
After my first failed attempt I now follow the route suggested by Nemo --- which works smoothly.
Starting from Nemo's identities
$$F(b)\equiv\int_0^{\pi/2}\cos^{2n}x\cos bx\,dx=\frac{\pi (2n)!}{2^{2n+...
- 188k
10
votes
Finding $\sum_i x_i$ given $\{\sum_i x_i^{2n}\}_{n\in \mathbb{N}}$
If $\sum x_i^2$ is finite, the sum $f(z)=\sum \frac{x_i^2}{1-x_i^2z}$ is a meromorphic function on the complex plane, and we know its Taylor series at 0. Thus we know $f$, hence the poles of $f$, ...
- 109k
8
votes
Does this moment inequality hold for any probability measure on the positive real line?
It doesn't hold.
Note that if one chooses $P= \frac{1}{n^2} \delta_n + \frac{n^2-1}{n^2} \delta_x$ with $x$ chosen such that $\mu = 1$, i.e. $x = \frac{n^2-n}{n^2-1}$, then $\mu_3 \sim n$ ...
- 1,095
8
votes
Accepted
Fourth moments of Gaussian processes
The first equality you mention is a special case of Wick's formula or diagram formula. Suppose that you have a Gaussian random vector $X=(X_1,\dotsc, X_n)$ that is centered, i.e., $\...
- 34.7k
6
votes
Accepted
Is the covariance of squares always bounded from below by two times the covariance?
An easy counterexample: $X=Y$, $P(X=\pm1)=1/2$. Then the left side of your inequality is $0$, and its right side is $2$.
A much more general, and perhaps more instructive, class of counterexamples ...
- 128k
6
votes
Accepted
Was this proposition on cumulants of compound Poisson distributions known before I put it into a Wikipedia article?
This is a simple fact, below is a short proof. It is certainly very-well known for quite some time, a sample reference is formula (6.6) in
Cacoullos T. (1989) Generating Functions. Characteristic ...
- 17.2k
5
votes
Accepted
Bound for a conditional expectation
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- 128k
5
votes
uniquely determining a distribution using moments
Looks complicated in general, but here's a derivation that for $d=1$ and $k=2$, in the slightly further restricted case where $\phi(X)=aX^2+b$, three moments suffice.
Let $m_i=E(\phi(X)^k)$.
Recall $...
- 24.8k
5
votes
Accepted
Limit of the logarithm of the $L^p$ norm over the logarithm of $p$ as $p$ goes to infinity
The limit doesn't always exist.
For convenience I will use the Lebesgue measure; similar construction can also be done for most $\mu$.
Let $c_n$ be a sequence of increasing positive integers. ...
- 39k
5
votes
Accepted
Differentiability of characteristic functions and moments of the corresponding measure
A good reference regarding this type of results is the book by Eugène Lukacs "Characteristic Function". For example, chapter 2.3 "Characteristic functions and moments" provides results in this ...
- 778
4
votes
Accepted
General result for the N-point correlation of the Poisson process (and its derivative)?
This question (and its generalization) is conventiently addressed by considering the moment-generating functional. Let $N_t$ be a counting process with (possibly stochastic and time-dependent) ...
- 1,675
4
votes
A Minkowski-like inequality
The answer to this question is yes.
Indeed, let $a:=\alpha$, and suppose $1\le a\le2$. Without loss of generality (wlog), $1<a\le2$, $EX^a<\infty$, and $Y$ takes only values in a fixed finite ...
- 128k
4
votes
Fractional moments of Poisson distribution
Fractional moments of any real order $p$ of any real-valued random variable $X$ with $E|X|^p<\infty$ can be expressed in terms of the characteristic function $f$ of $X$ as follows:
\begin{equation}...
- 128k
4
votes
Are these moments related to any usual distribution?
Note that for all real $k\ge0$ and $a>0$ we have
$$\frac1{k+a}=\int_0^1 x^k x^{a-1}\,dx.
$$
Differentiating in $k$, we further get
$$\frac1{(k+a)^2}=-\int_0^1 x^k \ln x\, x^{a-1}\,dx.
$$
So, doing ...
- 128k
4
votes
Accepted
sign of odd central moments of binomial distribution
Let $X$ have the binomial distribution with parameters $n,p$. Then for any natural $d$
$$E(X-np)^d=E\Big(\sum_1^n Y_i\Big)^d,$$
where $Y,Y_1,\dots,Y_n$ are iid random variables such that $P(Y=q)=p=1-P(...
- 128k
4
votes
Radius of convergence of cumulant generating function
Of course, this is not correct. As a simplest example, let
$X$ be a random variable which takes only values $\{0,\ldots, n\}$, then
the moment generating function is a polynomial of $e^t$, of degree $...
- 91.8k
4
votes
Accepted
Ratio of the first squared and the second moment
This is correct.
Denote $G(t)=\sum_{k=0}^\infty p_k t^k$, where $p_i\geqslant 0$ and $\sum p_i=1$. Then we are given that $\sum kp_k=\infty$ and should prove that
$$
\lim_{t\to 1-0} \frac{(\sum_{k=1}^\...
- 109k
4
votes
Uniqueness of the variance
To address your second question, the functional $v_p$ for $p\in(0,2)$ given by the formula
$$v_p(X):=-\int_0^\infty\frac{\ln|f_X(t)|}{t^{p+1}}\,dt$$
will have your properties 1 and 3, and also ...
- 128k
4
votes
Accepted
Decay estimate of moment of an SDE
Since we can apply Burkholder-Davis-Gundy to control the supremum of moments, we start by removing the tail simply using $1\leq \frac{|X_{t}|}{R}$.
$$
\mathbb E [ |X_t|^p 1_{\{|X_t| \ge R\}} ]\leq \...
- 5,474
4
votes
Accepted
Sum of RVs satisfying Bernstein condition on moments
$\newcommand{\be}{\beta}\newcommand{\si}{\sigma}$No, this is not true. Indeed, suppose that $X_1,\dots,X_n$ are i.i.d. zero-mean normal random variables (r.v.'s) with variance $\si^2$ such that
\begin{...
- 128k
3
votes
Expected value of absolute value of shifted binomial distribution
Mathematica can only produce a useless, tautological expression for $E|X-n/2|$ in terms of the hypergeometric function. Using Lemma 1 (Todhunter's Formula) in the paper you linked and the expression ...
- 128k
3
votes
Accepted
Approximate $\log \mathbb E_P[\exp(th(x)]$ for a function $h$ which is lipschitz and has finite moments of order 1 and 2 w.r.t $P$
To simplify notation, let $X$ be a random variable whose probability distribution is $P$, and then let $Y:=h(X)$. Your question is then is whether
$$\ln E e^{tY}=t\,EY+\frac{t^2}2\,Var\, Y+o(t^2),$$
...
- 128k
3
votes
Hankel determinants of binomial coefficients
This is not an answer.
Although there might be no "closed formula" for the case $p=4$, I wish to add the following to the case $p=2$ for which the determinants evaluate to $2^{n-1}$.
Suppose we ...
- 43.2k
3
votes
Accepted
Computing skewness derivative in terms of variance
I just read the same paper. Initially I made the same mistake as you, which is how I came across your question. It's been more than a year since your post but, in case you are still curious, here is ...
- 46
3
votes
Accepted
Why study the moment problem in one dimensional case( Hamburger moment problem)
It has many applications, especially in probability: e.g. you want to know if there is a probability distribution satisfying some condition on its moments.
Or you know the moments of a random variable ...
- 54.2k
3
votes
1D functional equation: solve for function with given expected value w.r.t normal density
Following approach is true when $ c \geq 2$.
Let $f(x) := a x $ be a linear function. Then, $\mathrm {prox}_{\lambda f}(z)= z-\lambda a$, and
$$
\mathbb c=E_{z \sim \mathcal N(0, 1)}[(\operatorname{...
- 1,991
3
votes
Numerical evaluation/approximation of non-central high-order moments of high-dimensional Gaussian measures?
It seems you may want to turn the large $k$ property to your advantage and use a Laplace-type asymptotic method with controlled bounds. For fixed $d$ you may find some useful tools in the book "...
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