# Tagged Questions

**7**

votes

**1**answer

493 views

### Has anyone seen this series?

I come across the following infinite series.
$$
\sum_{n=1}^{\infty} \frac{t^n}{n!\: n^{a}}, \quad\text{for $t>0$ and $a>0$}.
$$
In particular, I am interested in the case where $a=1/4$.
...

**3**

votes

**1**answer

155 views

### Characterization of a set in $\mathbb{R}^d$

Let $X= (X_1,\dots, X_d)$ be a fixed vector of random variables on the space $(\Omega, \mathcal{F}, \mathbb{P})$. Consider the following set.
\begin{equation}\label{main12}
C= \{x\in \mathbb{R}^d ~|~ ...

**4**

votes

**1**answer

80 views

### General version of Skorokhod representation of random variables

Let $F: \mathbb{R} \to [0,1]$ be cumulative distribution function (cdf). The standard way to build a random variable $\tau$ on $([0,1],\mathcal{B},\text{Leb})$ with $F$ as its cdf is using the ...

**0**

votes

**3**answers

193 views

### Lipschitz continuous maps from $\mathbb R^n$ to $\mathbb R^n$ that preserve Gaussian measure?

The only ones I can think of are linear maps like rotations and permutations. Is there a more general characterization?

**1**

vote

**2**answers

140 views

### Smooth but non-analytic kernel functions

Does there exist a (stationary) covariance kernel function which is $C^\infty$-smooth but not real analytic? If so, could you please provide an example?

**4**

votes

**0**answers

190 views

### Inverse of matrix-valued function

Given $c>0$. Let $\gamma_c:{\cal M}_{k \times k}^+\mapsto {\cal M}_{k \times k}^+$ is a function defined by
\begin{equation}
...

**0**

votes

**1**answer

251 views

### Pros and cons of probability model for permutations

I am studying probability model of random permetuation
Let $b(n; k)$ denote the number of permutations of {1,...,n} with precisely k
inversions ($inv(\pi)$). The analytic approach was considered by ...

**0**

votes

**0**answers

99 views

### proof of “supermodular function induces measure”

A function $f:\mathbb{R}^n\longrightarrow\mathbb{R}$ induces a measure by its finite differences, that is
\begin{align}
\mu((\mathbf{a},\mathbf{b}]) := \Delta_{a_1,b_1}\cdots\Delta_{a_n,b_n} f
...

**4**

votes

**1**answer

220 views

### Approximation of an integral over the unit ball of L_1

For every $\varepsilon>0$ find a piecewise continuous function $q:[0,1]\rightarrow \mathbb{R}$ such that $\int_0^1 q(x)dx=1$ and
$$\int_{0}^1 \int_{0}^{s} \left|\frac{q(s)q(t/s)}{s}- ...

**2**

votes

**1**answer

220 views

### Is there a probability density function providing the least expected value?

Fix constant reals $A>1$ and $D>0$. Let $f:\mathbb{R}\to[0,\infty)$ be a probability density function on $\mathbb{R}$, i.e. $\int_{-\infty}^\infty f(x)\, dx=1$, that is continuous almost ...

**13**

votes

**1**answer

676 views

### An Entropy Inequality (generalized)

Let $X,Y$ be probability measures on $\{1,2,\dots,n\}$. For $0\le \alpha \le 1$, set $K=\sum_i X(i)^\alpha Y(i)^{1-\alpha}$ so that $Z:=\frac{1}{K}X^\alpha Y^{1-\alpha}$ is also a probability ...

**28**

votes

**1**answer

2k views

### An Entropy Inequality

Let $X,Y$ be probability measures on $\{1,2,\dots,n\}$, and set $K=\sum_i\sqrt{X(i)Y(i)}$ so that $Z:=\frac{1}{K}\sqrt{XY}$ is also a probability measure on $\{1,2,\dots,n\}$. How can we prove the ...

**1**

vote

**0**answers

143 views

### When does a proper Zariski closed set have measure zero with respect to a conditional measure?

Assume we have a probability measure $\mu$ over $\mathbb{R}^d$ that is absolutely continuous with respect to Lebesgue measure.
Given $m$ polynomials $p_1,\ldots,p_{m}\in \mathbb{R}[x_1,\ldots,x_d]$ ...

**13**

votes

**4**answers

825 views

### A Normal Distribution Inequality

Let $n(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}$, and $N(x) = \int_{-\infty}^x n(t)dt$. I have plotted the curves of the both sides of the following inequality. The graph shows that the following ...

**1**

vote

**0**answers

211 views

### A strange Weakly Compactness in $L^1 ( \Omega, \mathcal{F}, \mathbb{P})$

Hi to everyone,
The ingredients of my problem are the following:
I have a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, a set (continuum cardinality) $\mathcal{Q}$ of probability measures on ...

**5**

votes

**2**answers

294 views

### Local concentration of measure on Erdos-Rényi graph

Let $G_n=(V_n,E_n)$ be an Erdos-Rényi random graph, precisely the vertex set is $V_n=(1,\dots,n)$ and the edge set is $E_n=(ij\in\mathcal{P}_2(V_n)\ |\ \epsilon_{ij}=1)$ where $(\epsilon_{ij})_{ij}$ ...

**5**

votes

**1**answer

212 views

### Extension of measures from the ball sigma-algebra to the borel sigma-algebra

Let $X$ be a metric space, $\Sigma_{1}$ the borel sigma algebra and
$\Sigma_{2}$ the sigma algebra generated by balls (open and closed).
If $\mu$ is a probability measure on $\Sigma_{2}$ can it be ...

**9**

votes

**2**answers

480 views

### The fraction of the sphere a fixed distance from a subspace

The following problem has a beautiful geometric interpretation in terms of the proportion of points on the Euclidean sphere in $\mathbb{R}^d$ that lie at least a certain distance away from a ...

**4**

votes

**2**answers

486 views

### Tails of sums of Weibull random variables

Suppose that $X_1, X_2, \ldots, X_n$ are i.i.d random variables distributed according to Weibull distribution with shape $0 < \epsilon < 1$ (it means that $\mathbf{Pr}[X_i \geq t] = ...

**2**

votes

**1**answer

161 views

### If two probability distributions have the same weak limit and one of them satisfies Large Deviation Principle, what can we say about the other?

If the probability distribution function of two sequences of random variables have the same weak limit and one of the sequences satisfies a Large deviation principle, then does it imply that the other ...

**1**

vote

**2**answers

617 views

### When do maximum and expectation commute?

Hi, I'm looking for conditions on $G(t,x)$ such that
$$
\sup\limits_{t\in [0,1]}E[G(t,X)]=E[\sup\limits_{t\in [0,1]}G(t,X)]
$$
where $X$ is a random variable (it's easy to see that $\sup\limits_{t\in ...

**4**

votes

**3**answers

237 views

### minimum of two probability densities

Consider a smooth probability density $\pi(x)$ on $\mathbb{R}^d$. I am looking for natural for the integral $\iint_{u,v} \ \min\big(\pi(u), \pi(v) \big) \ du \ dv$ to be finite. If $\pi$ is a radially ...

**15**

votes

**4**answers

1k views

### Is a random subset of the real numbers non-measurable? Is the set of measurable sets measurable?

One might say, "a random subset of $\mathbb{R}$ is not Lebesgue measurable" without really thinking about it. But if we unpack the standard definitions of all those terms (and work in ZFC), it's not ...

**3**

votes

**2**answers

557 views

### Reducing system of equations involving Erf, Error Function

I have a system of equations:
$$1/2 + {\rm Erf}(x) - {\rm Erf}(\frac{x+y}{2})=0$$
$$-1/2 + {\rm Erf}(y) - {\rm Erf}(\frac{x+y}{2})=0,$$
Where $x \le y$ and ${\rm Erf}$ is the Error Function.
By ...

**5**

votes

**3**answers

857 views

### Is a semicontinuous real function Borel measurable?

Let $f(x,u): [0,1]^2 \mapsto \mathbb{R}$ be a continuous
function.
[Q] Is $g(x) = \inf_{u\in [0,1]} f(x,u)$ always Borel measurable?
If not, can one find a counter-example?
Note that, for any $c$,
...

**1**

vote

**1**answer

367 views

### Concentration bound for weakly dependent random variables

Hi,
Suppose we observe a sequence $R_1, ..., R_T$ of iid. random variables that equal $0$ with probability $p$ and with probability $1-p$ are sampled from a distribution with expected value $E(R) ...

**3**

votes

**0**answers

190 views

### Elementary analysis: reference request

Given the continuous maps $[0,\infty) \to \mathbb R$ define the following "truncation at level $K$ operator", $T$:
$T(f)(t) = f(\min(t, S_f))$, where $S_f = \inf \{ s : f(s) \ge K \}$
So essentially ...

**-2**

votes

**1**answer

309 views

### Convergence Question [closed]

If $\alpha _{n}\rightarrow \alpha$, then how does one show that for any j=1,2,... and $\epsilon> 0$, if $sup\int \left | x \right |^{j+\epsilon }d\alpha _{n}<\infty$, then $\int x^{j}d\alpha ...

**5**

votes

**1**answer

348 views

### Numerically finding a Mercer expansion for a given covariance kernel

Let $c(r)$ be a nice, continuous function with compact support. For example, $c(r) = \tfrac 1 5 (1-r)^{11} \big( 5 + 55r + 239 r^2 + 429 r^3 \big)$ for $r \in [0,1]$, and $c(r) = 0$ otherwise.
On ...

**5**

votes

**0**answers

296 views

### Independent Events Inducing Probability Measures

Let $\mathcal{F}$ be a sigma algebra over $\Omega$ and $M$ the set of all probability measures on $\mathcal{F}$. Let $\mathcal{C}$ be some collection of pairs $(A,B)$ with $ \ A,B\in\mathcal{F}$. Now ...

**5**

votes

**1**answer

542 views

### Does a log-concave function on a convex set extend continuously to the boundary?

Let $U$ be an open convex set in a locally convex space $X$, and let $f : U \to [0,1]$ be a log-concave function on $U$ (i.e., bounded and real-valued). Under what conditions does $f$ have a ...

**5**

votes

**1**answer

464 views

### Quantitative bounds for multivariate central limit theorem

Hi,
For the univariate central limit theorem, the Berry-Esseen theorem gives a quantitative bound on the rate of convergence of distributions to the Normal distribution under Kolmogorov distance:
...

**6**

votes

**2**answers

356 views

### construction of a random measure with a given mean

Let me first pose a trivial question.
Given a Borel probability measure $\mu$ on the real line, is it possible to construct a purely atomic random measure $M$ whose mean is $\mu$?
The answer is ...

**17**

votes

**3**answers

1k views

### Do convex and decreasing functions preserve the semimartingale property?

Some time ago I spent a lot of effort trying to show that the semimartingale property is preserved by certain functions. Specifically, that a convex function of a semimartingale and decreasing ...

**3**

votes

**1**answer

266 views

### approximately linear functions — more

Suppose $f,g$ are continuous functions from $\mathbb R$ to $\mathbb R$, with the property that
$$f(x)+f(y)=g(x+y)$$
for all $x,y$. Taking $x=y=z/2$ implies that $g(x)=2f(x/2)$ so that the above ...

**6**

votes

**1**answer

778 views

### approximately linear functions

i suppose it's fairly well known that if a (continuous, real-valued) function $f$ on the real line satisfies
$f(x-y)=f(x)-f(y)+const$
then it is necessarily linear.
are there any general ...

**0**

votes

**1**answer

333 views

### a unique solution ? iteration involving conditional distributions

consider the following mappings, G and T,
$y(s) = \[Gx\](s)=\exp\left[\sum_{s'}p(s'|s)\log x(s') \right]$
$z(s) = \[Ty\](s)=\sum_{s'}q(s'|s)y(s')e^{-r(s')}$
where $0< x(s)\leq 1$ ,$r(s)<0$ , ...

**9**

votes

**2**answers

514 views

### Inequality in Gaussian space — possibly provable by rearrangement?

The following problem arose for my collaborators and me when studying the computational complexity of the Maximum-Cut problem.
Let $f : \mathbb{R} \to \mathbb{R}$ be an odd function. Let $\rho \in ...