All Questions
Tagged with real-analysis pr.probability
388 questions
6
votes
1
answer
239
views
Positivity of $ \int_{-\infty}^{\infty} \left\{{2^{1/\beta-1/2} \over v}\right\}^{it} { \Gamma\{(it+1)/\beta\}\over \Gamma\{(it+1)/2\} }dt$
I have the following function
$$
\int_{-\infty}^{\infty} \left\{{2^{1/\beta-1/2} \over v}\right\}^{it}
{ \Gamma\{(it+1)/\beta\}\over \Gamma\{(it+1)/2\} }dt
$$
where $1<\beta<2$, $v>0$. Need ...
- 93
12
votes
3
answers
2k
views
Looking for sufficient conditions for positive Fourier transforms
I am looking for some sufficient conditions for an even, continuous, nonnegative, non-increasing, non-convex function to be non-negative definite. In other words
$$
\int_0^\infty f(x)\cos(x\omega) \, ...
- 178
3
votes
1
answer
983
views
About the metrizability of the space of Probability measures $\mathcal{P}(S)$
It is often proved in Books that the space of Probability measures $\mathcal{P}(S)$ on a Polish metric space $(S,\rho)$ endowed with the weak/narrow topology induced by declaring it to be be the ...
- 309
2
votes
1
answer
112
views
Static Widom-Rowlinson model
In Elena Pulvirenti's slides she introduced a $\textbf{static Widom-Rowlinson model of one species}$. Consider $\Lambda\subset R^2$ with periodic boundary conditions, $\Lambda$ set of particle ...
- 288
2
votes
2
answers
155
views
Existence of classical solution for a parabolic equation without Hölder continuity in time for its coefficients
Consider equation
$$\partial_t u = \partial_x u + \partial_{xx} u - c u + f, \hbox{ on } (t, x) \in (0, \infty) \times \mathbb R$$
with initial condition $u(0, x) = g(x).$
Suppose that $c(t, x)$ and $...
- 1,399
3
votes
1
answer
404
views
The sign of the tail of Fourier transform of a positive function/ characteristic function
I am interested in a specific density (positive function) and would like to prove that the tail of its characteristic function (Fourier transform) is positive ($>0$). Here is the density $f(x)=c_\...
- 178
1
vote
1
answer
533
views
A probability question from sociology
We know that $\frac{1}{2} \leq a \leq p \leq 1$. And, $n \geq 3$ is a positive odd number, and $t$ is an integer. $a$ satisfies the equation below.
\begin{equation} \small
\begin{aligned}
&\sum_{t=...
- 39
14
votes
0
answers
718
views
Lower bounds on analytic functions connected to Fox H
The question is related to the one I asked before and never got an answer to. Fourier transform of $f_a(x)= a^{-2}\exp(-|x|^a)$, $a \in (0,2)$, is decreasing in $a$ . I need to demonstrate that the ...
- 178
0
votes
1
answer
179
views
How to show $\max_{1\leq i\leq n}(X_i+Y_1)\preceq \max_{1\leq i\leq n}(X_i+Y_i)$?
Let two collections of random variables $\{X_i\}$ and $\{Y_i\}$ be independent and let $\{Y_i\}$ be i.i.d. Then
$$\max_{1\leq i\leq n}(X_i+Y_1)\preceq \max_{1\leq i\leq n}(X_i+Y_i).$$
where $\...
- 288
3
votes
1
answer
334
views
Reversing the order of conditioning in a sum to compare conditional variances
Suppose $Z=X+Y$ where $X$ is independent of $Y$ and $Y\sim N(0,1)$. I would like to compare $\text{var}(E(X|Z))$ to $\text{var}(E(Z|X))$. Obviously, $\text{var}(E(Z|X))=\text{var}(X)$.
In particular, ...
7
votes
1
answer
463
views
Boundedness of total current in electrical network
Consider the following symmetric matrix (adjacency matrix):
$$A=(a_{ij})_{1\leq i,j\leq n}$$
such that $a_{ij}=a_{ji}, a_{ii}=0$ and $a_{ij}=0$ for $|i-j|\geq k$ where $k\geq3$. We also have $1\leq a_{...
- 1,495
12
votes
1
answer
1k
views
Riesz–Markov–Kakutani representation theorem for compact non-Hausdorff spaces
Let $X$ be a compact Hausdorff topological space, and $\mathcal C^0 (X) = \{f:X\to\mathbb{R}; \ f \text{ is continuous }\}$. It is well known that for any bounded linear functional $\phi: \mathcal C^...
- 333
1
vote
1
answer
2k
views
Lipschitz continuity of multivariable function in expected value
Suppose $h:\mathcal{X \times Y \times W}\rightarrow\mathcal{X}$, with $\mathcal{X,Y,W} \subseteq \mathbb R^d$, $d \in \mathbb N$ is Lipschitz in $x,y$, i.e.,
$$ \| h(x,y,w) - h(x',y',w) \|_2 \le L_h (...
- 13
1
vote
0
answers
60
views
Maximum value of $\int (aF^2(x)g(x)+G^2(x)f(x))dx$ over all $f,g$ densities satisfying $\int F(x)g(x)dx=1/2$
I want to maximise $$I(f,g):=\int_{-\infty}^\infty (aF^2(x)g(x)+G^2(x)f(x))dx$$ where $a>0$ is a given constant, over all possible probability densities $f,g$ satisfying $$\int_{-\infty}^\infty F(x)...
- 319
0
votes
1
answer
1k
views
Bounding $L^p$ norms in terms of lower-order $L^q$ norms
Suppose $f,g\in L^q(\Omega)$ ($\Omega\subset \mathbb{R}^n$) for all $1\le q\le p$. Here, $L^p(\Omega)$ is defined with respect to some measure $\mu$ that is absolutely continuous wrt Lebesgue measure. ...
- 710
0
votes
1
answer
96
views
Prove $x\mapsto\frac{E[f(X)1(f(X)\geq x X)]}{1+E[X1(f(X)\geq x X)]}$ is Lipschitz-continuous
Let $X$ be a random variable on $[0,A]$, and $f:[0,A]\to[-B_1,B_2]$ be a continuous function.
Let $$g(x) = \frac{g_1(x)}{g_2(x)}$$ where $g_1(x) = E[f(X)\mathbf{1}_{\{f(X)}]$ and $g_2(x) = 1+E[X\...
- 1
0
votes
2
answers
156
views
Show that if $A_{0}(t)+A_{1}(t)W(t)=0$ for all $t$ with $A_{0}$ and $A_{1}$ differentiable in $t$ and $W(t)$ a Wiener process, then $A_{0}=A_{1}=0$
I am learning the quadratic variation of stochastic process, and I am working on an exercise stating that
If for all $t$, we have $$0=A_{0}(t)+A_{1}(t)W(t),$$ where $(A_{0}(t),\mathcal{F}_{t})$ ...
user157103
0
votes
1
answer
86
views
Integral rising from difference of chi-squared random variables
Let $X,Y$ be independent random variables such that $X\sim\chi_{n-1}^{2}, Y\sim\chi_{1}^{2}$ are chi-squared distributed (where $n\geq2$ is a natural number). I am trying to evaluate $\mathbb{P}[X\leq ...
- 109
11
votes
1
answer
3k
views
A sum of two binomial random variables
Let $p\in(0,1)$, $n$ a positive even integer, $k,l\in\{0,\dots,n\}$, and $X_k\sim \text{Binomial}(k,p)$, $Y_{n-k}\sim \text{Binomial}(n-k,1-p)$ independent random variables. I would like to prove that
...
- 947
2
votes
2
answers
380
views
Convergence of fraction of expectation values
Let $X_1,...,X_n$ be iid normal random variables.
I am looking for a strategy to establish the following limit for fraction of expectation values
$$\lim_{N \rightarrow \infty} \frac{E(\prod_{1\le i ...
- 536
2
votes
0
answers
66
views
Properties of solution to Burger's equation using Cole-Hopf transformation
I am currently looking at a $1$D-Burger's equation defined by
\begin{equation} \label{ex burgers}
\left\{
\begin{array}{ll}
{} & \frac{\partial V_m}{\partial t} (t,x) = \frac{\sigma^2}{2} \...
- 357
2
votes
0
answers
189
views
Point wise convergence of Laplace transform and convergence of functions
Assume that functions $f_n(t), f(t)\in C_b(R_+)$. For every $\lambda >0$, we have
$$
\bigg|\int_0^\infty e^{-\lambda t}f_n(t)d t-\int_0^\infty e^{-\lambda t}f(t)d t\bigg|\leq C_\lambda n^{-1},
$$
...
- 411
2
votes
1
answer
549
views
$2$-Wasserstein distance between mixtures
I am stuck on the following problem. I have a discrete distribution $\mu_0$ (it is actually an empirical distribution). I have some $\mu_i$ (again discrete, some empirical distribution). I have some ...
- 371
0
votes
1
answer
146
views
Extension of subharmonic functions at infinity
Let $W$ be the complement of a compact set $K$ in $\mathbb{R}^{n}$, and $u$ a subharmonic function on $W$. Can we find, under some conditions, a function $\tilde{u}$ that is subharmonic on $W\cup\{\...
- 411
0
votes
1
answer
60
views
A question on the problem of Dirichlet 2
Let $U$ be an open set in $\mathbb{R}^{n}$ with $n\geq2$ and $V$ an open set containing the boundary $\partial U$ of $U$. Suppose $u$ is subharmonic on $V$. We know that the generalized solution of ...
- 411
3
votes
1
answer
299
views
Lipschitz functions that saturate the Lipschitz inequality on the average (part 1)
Consider a 1-Lipschitz function $f: \mathbb R^n \to \mathbb R$ satisfying the inequality
\begin{align*}
|f(x) - f(y)| \le \|x-y\|_2, \;\forall x,y \in \mathbb R^n.
\end{align*}
For $n \ge 2$, can we ...
- 1,731
9
votes
1
answer
556
views
A non-recursive, explicit formula for the Fabius function
The Fabius function $F\colon\mathbb R\to[-1,1]$ may be defined as the unique solution of the functional integral equation
$F(x)=\int_0^{2x}F(t)\,dt$ for all real $x$ such that $F(1)=1$.
The recent ...
- 128k
9
votes
1
answer
652
views
Scaling in Mehta's integral
The following expression is known as Mehta's integral and deeply connected to random matrix theory:
$$\frac{1}{(2\pi)^{n/2}}\int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty} \prod_{i=1}^n e^{-...
- 124
7
votes
1
answer
1k
views
Properties of convolutions
Consider the function
$$f_{n}(x)=e^{-x^2}x^n.$$
and the function
$$h_p(x):=e^{-\vert x \vert^p}.$$
My goal is to analyze
$$ F_p(y):=\frac{(f_2*h_p)(y)}{(f_0*h_p)(y)}- \left(\frac{(f_1*h_p)(y) }{(f_0*...
- 173
2
votes
1
answer
178
views
Non-convergence to a Gaussian
Let $f_n: \mathbb R^2 \rightarrow \mathbb R$ be a family of probability distributions with the property that they vanish on the diagonal $f_n(x,x)=0.$
I would like to know: Can we show that a ...
- 183
1
vote
2
answers
295
views
Monotonicity of maximum of convex combination of two scaled concave functions
Let $f:R\rightarrow R$ be a concave function with a unique and finite maximum. Let $$g(x, \beta) = \beta f(\alpha \cdot x) + (1-\beta) f((1-\alpha) \cdot x), $$ where $\alpha \in [0,1/2]$ and $\beta \...
- 13
1
vote
1
answer
115
views
$K(x,y)\in L^{\infty}(R^n\times R^n, m\times m)$, $K(x,y)=K(y,x)$, so $K(x,y)=\sum_{k=1}^{\infty}\lambda_k \phi_k(x)\phi_k(y)$, are $\phi_k$ bounded?
Consider a symmetric function
$$
K(x,y):R^n \times R^n \to R
$$
satisfying $K(x,y)=K(y,x)$ and
$$
\int_{R^n\times R^n} K^2(x,y)dm(x) dm(y) <\infty.
$$
Let $m$ be a probability measure on $R^n$.
...
0
votes
1
answer
135
views
Bounds on expectation of $X/(X^2 + c)$ with $X$ ~ Gaussian and $c > 0$
I'm trying to compute expectation of $X / (X^2 + c)$ when $X$ is normally distributed with mean $\mu$ and variance $\sigma^2$, and $c$ is some positive constant. I think this cannot be solved ...
- 11
2
votes
2
answers
451
views
If $0 \le \mu(A) < p < 1$, when is it true that there exists a measurable $B \supseteq A$ such that $\mu(B)=p$?
Let $(X,\mu)$ be a probability measure space and $A$ be a measurable subset of $X$ such that $0 \le \mu(A) < p < 1$.
Question
When is it true that there exists a measurable $B \subseteq X$ ...
- 6,853
1
vote
1
answer
860
views
Right continuous filtration
In optimal control theory, we often need a filtration do be right continuous. Consider a filtered probability space $(\Omega, \mathcal F, \mathbb P)$ equipped with a right continuous filtration $\...
- 553
2
votes
1
answer
99
views
Does bounded integral over sequence of subsets of $X$ whose union is $X$ imply bounded integral over X?
I came across the following problem while doing a piece of research on automata theory.
Suppose we have a probability space $(\Omega, \mathcal{F}, \mu)$, where $\Omega$ is a set, $\mathcal{F}$ is a $\...
- 333
1
vote
1
answer
143
views
Lower bound for log-Ratios
Can we find a universal constant $c>0$ such that for all $p,q\in\Delta:=\lbrace x\in (0,1)^{n}\ \colon\ x_{1}+\dots+x_{n}=1\rbrace$ it is true that
\begin{equation}
|p_{i}-q_{i}|\le c\left|\ln\frac{...
- 289
6
votes
2
answers
759
views
How to control Wasserstein distance in terms of characteristic function
Let $\mathcal P(\Omega)$ be the set of probability measures supported on some compact subset $\Omega\subset\mathbb R^d$. For $\mu\in\mathcal P(\Omega)$, denote by $F_{\mu}$ its characteristic function,...
user128095
4
votes
2
answers
145
views
Understanding equiprobable trinomial identity
With $f(x_1,x_2,x_3,x_1+x_2+x_3;\,1/3,1/3,1/3):= \frac{(x_1+x_2+x_3)!}{x_1!\,x_2!\,x_3!\, 3^{x_1+x_2+x_3}}$ denoting the probability mass function of the equiprobable trinomial distribution as in wiki/...
- 284
5
votes
0
answers
205
views
Strange inequality relating Binomial pmf and cdf
I'm encountering a strange inequality I need to prove, relating the Binomial pmf and cdf.
Suppose we have $n$ coin flips, and fix an arbitrary $k \le n/2$ heads. Suppose further that we have some ...
- 101
2
votes
1
answer
636
views
Sufficient condition for function of conditional probability density to be increasing
Let $Y$ and $W$ be two jointly distributed random variables; $Y$ takes values on $(y_1,y_2)$ and $W$ takes values on $(w_1,w_2)$. The conditional probability density of $W$ given $Y$ is given by $f_{W|...
- 143
6
votes
1
answer
433
views
Triangle inequality for Ito integral?
For Lebesgue integrals one has the triangle inequality saying that for continuous functions let's say
$$\left\vert\int_0^t f(s) \ ds\right\vert \le \Vert f \Vert_{\infty} \int_0^t \ ds$$
Now if ...
- 536
2
votes
1
answer
111
views
Sufficient conditions for inequality with integral of reliability functions
Let $Y$ and $W$ be two random variables with support $(y_1,y_2)$ and $(w_1,w_2)$ and distributions $F_Y$ and $F_W$, both twice continuously differentiable (densities $f_Y$ and $f_W$). Assume that both ...
- 143
2
votes
0
answers
208
views
On the difference of conditional differential entropy of two correlated random variables
Problem Definition
Let $\mathbf{G}$ and $\mathbf{S}$ be jointly distributed random variables where
$\mathbf{S}$ is continuous and is related to $\mathbf{G}$ through a conditional pdf $f(s|g)$ defined ...
- 31
1
vote
2
answers
889
views
Simplify Wasserstein distance between Gaussians with binary cost function
Let $\mu_1$ and $\mu_2$ be 1D gaussian distributions with means $m_1$ and $m_2$ respectively and common variance $\sigma$. Let $\Omega$ be a closed subset of $\mathbb R^2$, and consider the cost ...
- 6,853
0
votes
1
answer
133
views
Product of sets with the Radon-Nikodym Property (RNP)
I have read that it is somewhat well-known that if two Banach spaces $X$ and $Y$ have the Radon-Nikodym Property (RNP), then their product $X\times Y$ also has the RNP.
Does the above result ...
- 1,245
11
votes
2
answers
841
views
Computing the sum of an infinite series as a variant of a geometric series
I came across the following series when computing the covariance of a transform of a bivariate Gaussian random vector via Hermite polynomials and Mehler's expansion:
$$
S = \sum_{n=1}^{\infty} \frac{\...
- 984
2
votes
0
answers
109
views
Average number of pieces of a random piecewise-linear function
Let $I$ be a (nonempty) compact interval in $\mathbb R$ and $a_1,b_1,\ldots,a_L,b_L \in \mathbb R$. Let $\varphi$ be a piecewise function with $T \ge 2$ pieces(for example $T=2$ for the choice $\...
- 6,853
11
votes
0
answers
381
views
Concerning Luzin-(N)-property
Definition: a function $f:\mathbb{R}\to \mathbb{R}$ has Luzin-(N)-Property if $f$ maps any null set to a null set.
By https://www.encyclopediaofmath.org/index.php/Luzin-N-property, it is known that ...
- 4,201
5
votes
1
answer
170
views
Ratio of integrals with increasing dimension over Euclidean balls
Let $f_n(x)\geq0$ be any sequence of nonnegative $L^1(\mathbb{R}^n)$ functions such that $\int_{\mathbb{R}^{n}}f_n(x)dx=1$ where $dx$ is the Lebesgue measure on $\mathbb{R}^n$. For any $a>1,\...
- 1,495