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### 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 \...
$\newcommand\si\sigma\newcommand\R{\mathbb R}$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 ...
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 ...