5
votes

Accepted

### Limit of distributions

$\newcommand{\R}{\mathbb R}$
Proposition 1: For
\begin{equation*}
s(x)\sim T(x)=\ln P(X>x) \tag{00}\label{00}
\end{equation*}
to hold (as $x\to\infty$), it is necessary and sufficient that
\...

3
votes

Accepted

### Limit of a integral whose integrand diverges under the limit

$\newcommand\si\sigma\newcommand\R{\mathbb R}$We have to find
$$\lim_{\si\downarrow0}p_\si(y)$$
for all real $y$ such that the limit exists, where
$$p_\si(y):=\int_\R\frac{|x|\,dx}{\si\sqrt{2\pi}}e^{-(...

3
votes

Accepted

### Form of minimax estimator

$\newcommand\P{\mathcal P}\newcommand\N{\mathbb N}\newcommand\de{\delta}$You wrote:
Hence I additionally assume that
$\mathcal{P}$ is permutation-invariant, in which case I conjecture that all of the ...

2
votes

### Some identities from graph theory and probability

These inequalities are not too bad. Note that the $t_{ij}$ are independent and mean zero by the fact hyperbolic tangent is an odd function and symmetry of the Gaussians. This implies the first ...

2
votes

Accepted

### Concentration inequalities for random sampling without replacement

$\newcommand\E{\operatorname{E}}\newcommand\var{\operatorname{Var}}\newcommand\si{\sigma}$This will not work. E.g., if $N=10$, $\{c_1,\dots,c_{10}\}=\{-1, -1, -1, -1, -1, 1, 1, 1, 1, 1\}$, $n=5$, and $...

2
votes

### Weak convergence of measures on continuous function spaces

$\newcommand{\sgn}{\operatorname{sgn}}\newcommand{\ep}{\varepsilon}$Here is an elementary proof that $\mu_r$ converges weakly (as $r\to\infty$) the measure $\mu$ that is the distribution of the ...

2
votes

### Weak convergence of measures on continuous function spaces

These measures do converge weakly to a measure with two atoms (of equal probabilities $1/2$) at the two paths $\phi_{\pm}:t\mapsto \pm t$. One way to see it is to consider $\frac{1}{r}B_t$, so that ...

2
votes

Accepted

### Extreme confusion with the exact meaning of Gaussian measure with "translation-invariant" covariance

Here's my guess at what is meant by translation invariance in your question. Let $\mathbb{S}^1 = \mathbb{R} \ / \ \mathbb{Z}$ be the circle. Define $T_x \colon L^2 ( \mathbb{S}^1 ) \to L^2 ( \mathbb{S}...

2
votes

### Does strong stochastic ordering exist?

First, for the discrete topology, $U := \{\mu\}$ and $V := \{\nu\}$ are open and have the property wanted. Surely this is excluded. For the weak convergence topology, the Wasserstein metric and the ...

2
votes

Accepted

### Does strong stochastic ordering exist?

You could define a distance on probability measures by the smallest $c$ such that there exists a coupling giving mass $1$ to a $c$-neighbourhood of the diagonal. Many pairs would be infinite distance ...

1
vote

### What's the lower bound of the correlation coefficient?

$\newcommand\P{\operatorname P}\newcommand\E{\operatorname E}\newcommand\Var{\operatorname{Var}}\newcommand\Cov{\operatorname{Cov}}$As you noted, necessarily $\rho\ge0$, so that $0$ is a lower bound ...

1
vote

Accepted

### Small deviations of real log-concave random variable

We have $f=e^g$, $g$ is concave, $\int f=1$, $\int x f(x)\,dx=0$, and $\int x^2 f(x)\,dx=1$. As you noted, then $f(0)\ge 1/8$ and hence
$$g(0)\ge-a,\tag{0}\label{0}$$
where $a:=\ln8$.
We have to show ...

1
vote

### Why MLEs are asymptotically efficient whereas method of moment estimators are not?

A ``down-to-earth'' observation to see what goes wrong with method of moments is this:
When considering applying the method of moment to $(X_1,...,X_n)$, you may as well apply the method of moments to ...

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