All Questions
Tagged with real-analysis pr.probability
388 questions
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
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
2
votes
0
answers
61
views
Convergence to the probability generating function of a Poisson process
I'm working currently with a Poisson process trying to proove Renyi's Theorem, so far I want to show that
$\prod_{i=1}^{k_n}[z + (1-z)e^{-\mu(A_{n_i})}] \to e^{-(1-z)\mu(A)}$ as $\mu(A_{n_i}) \to 0$, ...
- 21
2
votes
0
answers
198
views
Continuous Local Martingales under time change under what conditions are they still local martingales?
This question is motivated by reading a section in Continuous Martingales and Brownian Motion by Daniel Revuz, Marc Yor.
In Chapter V there is a section on time-change:
Definition:
A time change $C$...
2
votes
0
answers
46
views
increasing inter-class distances results in decreasing linear regression error
Let $\{\mathbf{x}_i, y_i \}$ be a set of binary-labeled samples ($\mathbf{x}_i \in \mathbb{R}^d, y_i \in \{a,b\}, a,b\in\mathbb{R}$). Let $\{ \mathbf{x}'_i, y_i \}$ be also such a set.
Define $\mathbf{...
- 183
2
votes
0
answers
86
views
when is the average of a function with Gaussian inputs bounded away from zero
Define a function $\phi(x):\mathbb{R}\rightarrow\mathbb{R}$. Consider the expected value function defined as follows
\begin{align*}
\mu(\beta)=E[g\phi
(\beta g)]\quad with \quad g\sim\mathcal{N}(0,1)\...
- 363
2
votes
0
answers
63
views
Sensitivity of a function against its random arguments
Let $g:R^{n+m} \to R$ be a deterministic function of some independent random variables $x_1,\ldots,x_n$ with distributions $f_{x_1}(x),\ldots,f_{x_n}(x)$ and some deterministic variables $z_1,\ldots,...
- 482
2
votes
0
answers
254
views
Prove this function is increasing
I'm stuck in showing that the following function is increasing over the domain $\left[0,\hat{b}\right]$:
\begin{eqnarray}
\Pi\left(z\right) & = & \int_{0}^{\phi\left(z\right)}\int_{x}^{\bar{x}...
- 21
2
votes
0
answers
79
views
Compute Mixed Volume with Respect to Some Regular Sets
Let $( \mathbb{R}^n, \mathcal{B}, \gamma)$ be a measure space where $\mathcal{B}$ is the Borel sigma algebra and $\gamma$ is a continuous measure. For $A, B\in \mathcal{B}$ that are convex, the mixed ...
- 1,127
2
votes
0
answers
84
views
limit multiple integral
I want to know if $\lim_{T-> \infty}$ of this integral
$$ \frac{\sigma^{4}C_{H,K}^{2}}{4 T^{4HK}e^{2\theta T }}\\
\times \int\limits_{[0,T]^{4}}e^{\theta(t_{1}-s_{1})}e^{\theta(t_{2}-s_{2})}\left\...
- 31
2
votes
0
answers
160
views
Is it possible to improve the order of convergence of averages of random variables if they are not identically distributed?
Let $X_n$ be a sequence of independent random variables (but not necessarily identically distributed)
taking values in $[-1,1]$ that have the following property:
1) The average $A_n := \frac{(X_1+ \...
- 3,245
2
votes
0
answers
263
views
A strange Weakly Compactness in $L^1 ( \Omega, \mathcal{F}, \mathbb{P})$
Hi to everyone,
The ingredients of my problem are the following:
I have a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, a set (continuum cardinality) $\mathcal{Q}$ of probability measures on $...
- 21
1
vote
2
answers
226
views
Smooth but non-analytic kernel functions
Does there exist a (stationary) covariance kernel function which is $C^\infty$-smooth but not real analytic? If so, could you please provide an example?
- 8,512
1
vote
2
answers
2k
views
Standard Brownian motion, Hölder continuous with exponent $\gamma$ for any $\gamma < 1/2$, not for any $\gamma \ge 1/2$
In some results on Hölder continuity with regards to standard Brownian motion, the following is asserted without proof.
It is not hard to see that for every $k < \infty$, and every $\epsilon >...
user80013
1
vote
2
answers
819
views
Integral formula involving Legendre polynomial
I would like a proof of the following equation below, where $(P_n)$ denotes Legendre polynomials. I guess this formula to hold; I checked the first several values.
\begin{equation}
\int_{-1}^{1}\sqrt{...
- 31
1
vote
2
answers
136
views
Uniform boundedness of integral?
I have perhaps a very simple question where I lack some inutition at the moment: Is the expression
$$\sup_{\alpha < 0, \lambda \in \mathbb N}\int_{-\infty}^{\alpha} e^{-\lambda t^4} \ dt \int_{\...
- 536
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
1
vote
2
answers
169
views
Asymptotic properties of weighted random walks / infinite convolutions of random variables
Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of i.i.d. real-random variables. Let further $0<c<1$. I'm interested in the asymptotic properties of
$$
\sum_{k=1}^n c^k X_k.
$$
I can prove that this ...
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
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
1
vote
1
answer
632
views
Does sequence almost sure convergence imply almost sure convergence?
This is a cross-post of this and this questions from math.stackexchange.com since I have not received any response there. I would like to seek help here.
Suppose $x(t,\omega): [0,T]\times\Omega\...
- 2,239
1
vote
1
answer
203
views
Why study the moment problem in one dimensional case( Hamburger moment problem)
I have been reading about moment problem and I have been curious about the following question.
What is the motivation for studying the Hamburger moment problem(one dimensional moment problem?
I ...
- 125
1
vote
1
answer
50
views
Increasing function of $\theta$ for the Ali-Mikhail-Haq Survival Copula
I have been trying to solve the following function is non-increasing (non-decreasing) with respect $\theta$ where $\theta \in (0,1)$ (resp. $\theta \in (-1,0)$)
\begin{equation}
f(\theta)= \frac{h(t,\...
- 23
1
vote
1
answer
150
views
Is the Boltzmann entropy continuous in the supremum norm?
We define $U : [0, +\infty) \to [0, +\infty)$ by $U(0) := 0$ and $U (s) := s \log s$ for $s >0$. Then $U$ is strictly convex. Let $D$ be the set of all bounded non-negative continuous functions $\...
- 825
1
vote
1
answer
160
views
Existence, uniqueness and regularity of the solution to some integral equation
Let $b: \mathbb R_+\times\mathbb R_+\times \mathcal P\to\mathbb R$ be Lipschitz, where $\mathcal P$ denotes the set of probability measures $\mu$ on $\mathbb R_+$ of finite first moment and is endowed ...
- 1,334
1
vote
1
answer
368
views
Does the almost sure convergence of absolutely continuous r.v.'s imply the weak convergence of the pdf's in $(L^\infty)^*$?
The following question was asked in a comment at Almost sure convergence vs convergence of probability density functions :
Suppose that $(X_n)$ is a sequence of random variables (r.v.'s) converging ...
- 128k
1
vote
1
answer
50
views
Convergence of the sum of a family of real-valued functions
Let $\phi_1,...,\phi_n,...$ be a sequence of real-valued functions so that $\phi_j:[0,1)\to[0,1)$, $\phi_j(0)=0$, and $\phi_j(\delta)$ converges to 0 as $\delta$ approaches 0 from the right for all $j\...
- 13
1
vote
2
answers
194
views
Continuity of the densities of a stochastic process
Let $X=(X_t)_{t\in I}$ ($I\subset\mathbb{R}$ an interval) be a stochastic process with continuous sample paths and such that $X_t$ admits a continuous Lebesgue density $\chi_t\in C(\mathbb{R}^d)$ for ...
- 463
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 $R^n$.
...
1
vote
1
answer
72
views
Independent identical distribution sequence
given a measurable function $\alpha: (\Omega, \mu) \to \mathbb{R}$, and transformation $\sigma : \Omega \to \Omega $.
I found an example such that $\alpha, \alpha\circ \sigma, \alpha\circ \sigma^2, \...
- 553
1
vote
1
answer
196
views
Giving Uniform Bound on Differences of Sums of Converging Polynomials
The title does not quite capture the essence of the difficulty, please allow me to be more explicit here.
I thought of this question when I was trying out an open problem by Ovidiu Furdui(See problem ...
- 99
1
vote
1
answer
200
views
Is regularity closed under products?
Let $G \colon [0,1] \to [0,1]$ be a differentiable cumulative distribution function (monotonically non-decreasing function with $G(0) = 0$ and $G(1) = 1$). We say that $G$ is regular if $$ x - \frac{1-...
- 11.3k
1
vote
3
answers
293
views
Lipschitz continuous maps from $\mathbb R^n$ to $\mathbb R^n$ that preserve Gaussian measure?
The only ones I can think of are linear maps like rotations and permutations. Is there a more general characterization?
- 41
1
vote
1
answer
157
views
Is finding the CDF from the Laplace transform well-posed?
In my study of Dynamic Light Scattering, I came across the following inverse problem. Let $F(s):[0,T]\rightarrow[0,T]$ be the Laplace transform of a probability distribution $f(t)$ on the real line ...
- 654
1
vote
1
answer
187
views
Bound the distance between two vectors on the probability simplex
Let $a,b$ be two vectors with strictly positive elements and $\delta = 1 - \frac{\langle a,b \rangle}{\|a\|\|b\|}$. Bound the following optimization problem as a function of $\delta$
$$\sup_{x>0} \...
- 135
1
vote
1
answer
191
views
Concentration inequality for square roots
Given a sequence of (not-necessarily-iid) real-valued random variables $X_n$ that converge to $a\in\mathbb{R}$ in probability, suppose we have an exponential concentration inequality of the form
$$
P(|...
- 13
1
vote
1
answer
169
views
How to prove that is a consistent estimator?
Let $\hat{\pi}^N$ be an AW-consistent estimator of $\pi$ (i.e., $\hat{\pi}^N$ is a strongly consistent estimator of $\pi$ under adapted (or called nested) Wasserstein distance $AW(\pi, \hat{\pi}^N)\to ...
- 288
1
vote
1
answer
141
views
How to get the estimator?
They introduce a new correlation. For $\pi\in \Pi(\mu,\nu)$ the set of coupling of two probability measures $\mu$ and $\nu$ on a Polish space $(X,d)$. The author introduces a plugin estimator.
...
- 288
1
vote
1
answer
166
views
Discontinuity set of the expected value of a continuous process
Let $X_t$ be a continuous real valued stochastic process on $\mathbb R_+$. Then it is not necessarily true that $E[X_t]$ is continuous in $t$.
Question:
What is known about the discontinuity set of $E[...
- 6,215
1
vote
1
answer
266
views
Decomposition of the sum of nonnegative random variables [closed]
Non-necessarily independent random variables $X_1,~X_2,~\cdots,~X_n$ are supported on $[0,a_1],~[0,a_2],~\cdots,[0,a_n]$ and with mean values $\mu_1,~\cdots,~\mu_n$ respectively, where all $a_i$ and $\...
- 550
1
vote
1
answer
99
views
Mean deviation in $p$-norm for $1 < p < 2$
Let $(X, \mu)$ be a probability space, and let $p \in (1, 2)$ be arbitrary. It is known from Corollary 2.4 of this paper by G. Sinnamon that for any measurable $f : X \to [0, +\infty],$ we have
$$0 \...
- 255
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.,
$$ \| h(x,y,w) - h(x',y',w) \|_2 \le L_h (...
- 13
1
vote
2
answers
295
views
Monotonicity of maximum of convex combination of two scaled concave functions
Let $f:R\rightarrow R$ be a concave function with a unique and finite maximum. Let $$g(x, \beta) = \beta f(\alpha \cdot x) + (1-\beta) f((1-\alpha) \cdot x), $$ where $\alpha \in [0,1/2]$ and $\beta \...
- 13
1
vote
1
answer
143
views
Lower bound for log-Ratios
Can we find a universal constant $c>0$ such that for all $p,q\in\Delta:=\lbrace x\in (0,1)^{n}\ \colon\ x_{1}+\dots+x_{n}=1\rbrace$ it is true that
\begin{equation}
|p_{i}-q_{i}|\le c\left|\ln\frac{...
- 289
1
vote
1
answer
457
views
Taylor's theorem for a composition with $\min:\mathbb R^2\to\mathbb R$ and differentiability Lebesgue almost everywhere
Let
$f\in C^3(\mathbb R)$ with $f>0$ and $$\int f(x)\:{\rm d}x=1$$
$g:=\ln f$ (and assume $g'$ is Lipschitz continuous)
$n\in\mathbb N$, $$s(x,y):=\sum_{i=1}^n\left(g(y_i)-g(x_i)\right)$$ and $$h(...
- 167
1
vote
1
answer
399
views
I want to disprove an equality involving a double integral
I want to show that the following equality does not hold:
\begin{equation}\label{at3}
\frac{\lambda^2-1}{2}x^2-\int_{-\infty}^{\infty}\!\!\int_{-\infty}^{\infty}K(y_1,y_2,x)\ln g(y_1,y_2)dy_2dy_1=...
- 19
1
vote
1
answer
357
views
Does CLT hold for joint distribution of two dependent binomial variables?
Let $S_n$ and $T_m$ be two binomial variables satisfying $S_n\sim B(n,\frac12)$ and $T_m\sim B(m,\frac12)$. Define $\tilde{S}_n=\frac{2S_n-n}{\sqrt{n}}$ and define $\tilde{T}_m$ similarly. For any ...
- 111
1
vote
1
answer
166
views
Question abouth Skorokhod representation of random variables (II)
This is a continuation of
Question abouth Skorokhod representation of random variables
Let $\mu$ and $\nu$ be two probability measures on $\mathbb R$ such that
$$\int_{\mathbb R}|x|^pd\mu(x),~ \...
- 1,835
1
vote
1
answer
159
views
$M_t = f(B_{t \wedge \tau}) + (t \wedge \tau)$ local martingale, $\textbf{E}^x[\tau] = f(x)?$
Suppose $D \subset \mathbb{R}^d$ is a domain and $f: \overline{D} \to \mathbb{R}$ is a continuous function, $C^2$ in $D$, satisfying$$f(x) = 0\text{ for }x\in \partial D,$$$${1\over2} \Delta f(x) = -1 ...
user80013