7

What are the benefits of a unified framework? You are right that we are rapidly going into some much used special cases line logistic regression or Poisson regression, but there is still benefit in having a common framework.
Technology transfer from the general linear model (not generalized!), that is with gaussian errors and identity link. A lot of what ...

5

Even if $X_n\to c$ in probability for some real constant $c$, it is not necessary that $P(Y\le X_n)\to P(Y\le c)$ -- you also need to require that $P(Y=c)=0$.
More generally, if the limit of $X_n$ is not a constant, then you need to assume the convergence, not just of the distribution of $X_n$, but of the joint distribution of $X_n$ and $Y$. In particular, ...

4

For real $u_m>0$, the probability in question is
$$p_m:=P(\|X\|^2\ge u_m|X_1|)=P\Big(\frac{|X_1|}{\|X\|^2/m}\le\frac m{u_m}\Big).$$
Passing to a subsequence, without loss of generality
$$\frac{u_m}m\to c\in[0,\infty]$$
(as $m\to\infty$). By the law of large numbers, $\|X\|^2/m\to1$ in probability. So, by Slutsky's theorem ,
$$p_m\to p_\infty:=P(|X_1|\le1/...

4

There exists a variety of measures of uniformity of a point set. See, for example, On assessing spatial uniformity of particle distributions... for an overview, and a critical comparison when applied to real-world data.
There are two distinct classes of uniformity measures: Quadrat-based measures divide the region into a number of small grids, called ...

answered Sep 23 at 11:31

Carlo Beenakker

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3

Judging from your pictures, it should be sufficient to consider the root-mean-square distance $\rho$ of the points $\vec{x}_k$ from their center of mass $\vec{\mu}$, divided by the radius $R$ of the circle:
\begin{align}
\vec{\mu} &= \frac{1}{N}\sum_{k=1}^N\vec{x}_k, \\
\rho &= \sqrt{\left(\frac{1}{N}\sum_{k=1}^N\left\lVert\vec{x}_k-\vec{\mu}\right\...

2

First, notice that w.l.o.g. you can assume that the matrix $H_n$ is diagonal (from rotational invariance of the isotropic Gaussian).
You thus are interested in
$$
\frac{1}{n} \sum_{i=1}^n \lambda_i(H_n) X_i^2.
$$
So the condition
$$\sum_{i=1}^n \lambda_i^2(H_n)/n^2 \to 0$$
is the key.
Then, one has that
$$
Var\left(\frac{1}{n} \sum_{i=1}^n \lambda_i(H_n) X_i^...

1

Using e.g. the Gauss elimination, we can diagonalize the matrix $(v_{kl})$, that is, write
$$v_{kl}=\sum_{r=1}^d a_r s_{rk}t_{rl}$$
for some real $a_r,s_{rk},t_{rl}$ and all $k,l$; the computational complexity (CC) of this diagonalization is $O(d^3)$; cf. e.g. this source.
Now we can write
$$u_{ij}=\sum_r a_r\,Ex_ix_jX_rY_r,$$
where
$$X_r:=\sum_k s_{rk}x_k,\...

1

The joint distribution of the eigenvalues $\lambda_i$, $i=1,2,\ldots n$ of $A$ is known,
$$P(\lambda_1,\lambda_2,\ldots\lambda_n)=c_{k,n}\prod_{i<j}|\lambda_i-\lambda_j|\prod_m e^{-\lambda_m/2}\lambda_m^{(k-n-1)/2},$$
with $c_{k,n}$ a normalization constant.
The desired expectation value is given by
$$U_{n,k}=\mathbb E[\det(I+(I+A)^{-1})]=\int_0^\infty d\...

answered Sep 22 at 14:00

Carlo Beenakker

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1

$\newcommand\si\sigma\newcommand\Om\Omega$To determine the limit distribution of the process $Z_N$ (and even the distribution of the process $X$), it is not enough to know only $\si^2$; one also has to know the covariances $r_{x,y}:=Cov(X(x),X(y))$ for $x,y$ in $\Om$.
Then, using e.g. the joint moment generating function of $(X(x),X(y))$, one can see that $$...

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