All Questions
Tagged with real-analysis pr.probability
388 questions
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
2
votes
1
answer
207
views
Expectation of Truncated Bivariate Gaussian Random Variables
Suppose $Z , \epsilon \sim N(0, 1)$ are independent Gaussian random variables. Let $a \ll 1$ be a small positive number. Let $W = aZ + \epsilon$. It can be show that
\begin{align}
\mathbb{E} [ W^2 (Z^...
- 1,127
2
votes
1
answer
106
views
Lower bounds for the expectation of log ratio between the posterior and prior Beta densities
The quantity I'm interested in is expressed as follows:
$$
I = \mathbb{E}_{k\sim \text{Binom}(n,p)} \left[\ln \frac{\text{Beta}(p;a+k,b+n-k)}{\text{Beta}(p;a,b)}\right]
$$
The term inside the ...
- 63
2
votes
2
answers
667
views
Power series of ratio of Gamma functions
Let $a>1$ and define $G_a(x)=\sum\limits_{n=0}^{+\infty} \frac{\Gamma(\frac{2n+1}{a})}{\Gamma(2n+1)\Gamma(\frac{1}{a})}x^n$ where $\Gamma$ is the Gamma function. This series is convergent on $\...
- 39
2
votes
1
answer
119
views
Exact formula or non-trivial upper bound on p-norm of $f(x)=\|x\|_2$ in $[0,1)^d$
I wonder whether one can exactly calculate the following integral in terms of $d$ and $p\geq 1$ or not, or a better bound(than the trivial one I am going to give) in terms of $d,p$:
$$\left(\int_{[0,1)...
- 715
2
votes
1
answer
99
views
Does bounded integral over sequence of subsets of $X$ whose union is $X$ imply bounded integral over X?
I came across the following problem while doing a piece of research on automata theory.
Suppose we have a probability space $(\Omega, \mathcal{F}, \mu)$, where $\Omega$ is a set, $\mathcal{F}$ is a $\...
- 333
2
votes
1
answer
216
views
Ask for a special function related to the error function
I am wondering whether anyone knows the following integration has a named special function or a reference
$$
F_{a,b}(z) :=\frac{2}{\sqrt{\pi}} \int_0^z \text{erf}(a+b y)\: e^{-y^2} \text{d}y
$$
for ...
- 1,649
2
votes
1
answer
383
views
Is this a log-concave function?
Let $(a_k)$ be a log-concave positive decreasing sequence. Is $\sum\limits_{k=1}^n a_k(1-e^x)^{k-1}$ log-concave in $x<0$, for each natural $n$?
- 79
2
votes
1
answer
3k
views
Inequality for the tail of normal distribution function
Let $ Ф(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{x} e^{-t^2/2} \, dt $ be the cumulative distribution function of the standard normal distribution.
Numerical calculations suggest the following ...
- 61
2
votes
1
answer
469
views
If two probability distributions have the same weak limit and one of them satisfies Large Deviation Principle, what can we say about the other?
If the probability distribution function of two sequences of random variables have the same weak limit and one of the sequences satisfies a Large deviation principle, then does it imply that the other ...
- 3,245
2
votes
1
answer
141
views
Injectivity of two sided Laplace transform
Let $\mu,\nu$ be finite Borel measures on $\mathbb R$.
Assume that there is an open interval $(a,b)$ on which the Laplace transforms exist and coincide:
$$
\int_{-\infty}^\infty e^{-tx}\,d\mu(x) = \...
- 769
2
votes
1
answer
122
views
Can we say that there exists a measurable function $f$ such that $ \nu=f_{\#}\mu$?
Define a coupling $\pi\in \Pi(\mu,\nu)$ on the product space $(X\times X,\mathcal{F}\times\mathcal{F})$. let $\pi_x$ be the disintegration of $\pi$ with respect to the $\mu$, i.e. there exists a Borel ...
- 288
2
votes
1
answer
231
views
Inequality with slowly varying functions
Note: I am reposting this question from Math Stack Exchange, which failed to receive an answer after several weeks and a bounty. Also, I believe it fits the requirements for this website, as it ...
- 127
2
votes
1
answer
268
views
Existence of the derivative of functionals of Brownian motion
Let $v(x, t) = \mathbb E [f(x + W_t)]$ with a Brownian motion $W$. Then, Malliavin calculus leads to the sensitivity in $x$:
$$\partial_x v(x, t) = \frac{1}{t} \mathbb E [ f(x + W_t) W_t ].$$
I am ...
- 1,399
2
votes
1
answer
180
views
Random sequence with positive Lyapunov exponent?
Consider the following self-adjoint matrix
$A_X = \begin{pmatrix} 0 & -i \\ i & X \end{pmatrix},$ where $i$ is the imaginary unit and $X$ is a uniformly distributed random variable on some ...
- 192
2
votes
1
answer
242
views
Modify a random variable to make its range Borel?
Let $X: \Omega\to{\mathbb R}$ be a random variable. Is it always possible to modify it (i.e. change the value of $X$ on a subset of $\Omega$ of zero measure) so that the range of $X$ is a Borel set?
...
- 637
2
votes
1
answer
265
views
A random variable whose characteristic function decreases the fastest
A random variable $X$ is "good" for $(a_0, b_0) \in (0,1)^2$ if its characteristic function $\varphi_X(t)$ satisfies the following constraints:
$\forall t : \varphi_X(t) \geq 0$.
$\varphi_X$...
- 53
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
2
votes
2
answers
152
views
Divergence rate of geometric sum of random variables
Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of strictly positive and identically distributed random variables and let $\beta\le 1$. I am trying to prove that
$$
0<\lim_{\beta\rightarrow 1}(1-\...
- 479
2
votes
1
answer
263
views
Schwartz space on $\bigcup_{n=1}^CR^n$
I have an application where I need to work with the following idea.
Let the space $\bigcup_{n=1}^C \mathbb{R}^n$ be associated with the metric $d$ such that for $x=(x_1,\cdots,x_n)$ and $y=(y_1,\cdots,...
- 21
2
votes
1
answer
165
views
If $Z$ is standard normal and $f$ is analytic. Is $g(t)= E[ f(Z-t)]$ analytic?
Let $Z$ be a standard normal.
Now define
\begin{align}
g(t)= E[ f(Z-t)]
\end{align}
where $f(x)$ is a real-analytic function and $|f(x)| \le x^4$.
Question:
Is it true that $g(t)$ is also a real ...
- 671
2
votes
1
answer
251
views
Automorphism on the unit interval compatible with a measure preserving set function
Cross-posting from math stack-exchange since it's not getting any visibility there.
I am given a function $F: \{[0, y]: y \in I\} \to \Sigma(I)$, such that $\lambda(F([0, y])) = y$, and $F([0, y]) \...
- 4,466
2
votes
1
answer
363
views
Integration against Borel measures on compact Hausdorff spaces
I am studying the properties of integration against Borel measures and Baire measures. And I am not sure whether the following proposition is correct and I tried to give a proof.
Suppose that $X$ ...
- 165
2
votes
1
answer
268
views
Monotonicity of the Hellinger integral/distance
Let $p$ and $q$ be probability densities on $\mathbb R$, with respect to the Lebesgue measure $dx$. The corresponding Hellinger integral and distance are
$H(p,q):=\int_{\mathbb R}\sqrt{pq}\,dx$ and $\...
- 128k
2
votes
1
answer
571
views
Extension of Dynkin's formula, conclude that process is a martingale
This question was asked here, but it did not get enough attention, so I'm crossposting it to MO.
Let $u: \mathbb{R}_+ \times \mathbb{R}^d$ be a bounded $C^2$ function whose first and second partial ...
user61522
2
votes
1
answer
157
views
Is the set of multiple points of the Brownian path $W[0, \infty)$ dense in the plane almost surely?
Let $d = 2$. With probability $1$, is the set of multiple points of the Brownian path $W[0, \infty)$ dense in the plane?
user75246
2
votes
1
answer
186
views
Does $\int \Phi \left( \frac{u}{\xi} \right) f_t(\xi) \mathrm{d} \xi \to \Phi(u)$ imply that $f_t \to \delta_1$?
I'm looking at a family $(f_t)$ of densities of some continuous random variables and know that
$$\int_{-\infty}^{\infty} \Phi \left( \frac{u}{\xi} \right) f_t(\xi) \mathrm{d} \xi \xrightarrow{t \to \...
- 235
2
votes
1
answer
154
views
Smooth conditional expectation with nonsmooth "reverse"
I am looking for a concrete example of the following: $(X,Y)$ are real-valued random variables such that:
$E[Y|X]$ is smooth
$E[X|Y]$ is discontinuous
Even better, I'd like to see an example where ...
- 23
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
2
votes
1
answer
450
views
Show that the absolute value of this function is twice differentiable except on a set of Lebesgue measure $0$
Let
$f\in C^3(\mathbb R)$ with $f>0$ and $$\int f(x)\:{\rm d}x=1\tag1$$
$g:=\ln f$ and assume that $g'=\frac{f'}f$ is Lipschitz continuous (note that this implies that $f'(x)\xrightarrow{|x|\to\...
- 167
2
votes
2
answers
128
views
Spectral decomposition of a combinatorial matrix/Random walks on $s$-sets
$\newcommand{\Z}{\mathbb{Z}}
\newcommand{\J}{\mathcal{J}}
\newcommand{\la}{\lambda}
\newcommand{\1}{\mathbf{1}}
\newcommand{\R}{\mathbb{R}}$
Take any $n\in[3;\infty]$. Here and in what follows, $[k;\...
- 128k
2
votes
1
answer
289
views
On semi-discrete Wasserstein distance
Let $\mu(dx)=\sum_{i=1}^np_i\delta_{x_i}(dx)$ and $\nu(dy)=\rho(y)dy$ be two probability measures on $\mathbb R^d$, where $\nu$ has a bounded support. Consider the $2-$Wasserstein distance below:
$$...
- 345
2
votes
1
answer
103
views
Matuszewska Index and finite variance
Suppose there is a random variable, $X$, with finite variance, and c.d.f. $F(x)$. Does this imply that the upper Matuszewska index of $\bar F(x)$ exists and is strictly smaller than $-2$?
The upper ...
- 294
2
votes
1
answer
71
views
Distances between probability distributions by the variance of the test functions
Let $P$ and $Q$ be two probability distributions on $\mathbb{R}$. The goal is to obtain a notion of ``distance'' between $P$ and $Q$, e.g., total variation distance, K-L divergence.
Let $f\colon \...
- 1,127
2
votes
1
answer
5k
views
Smooth Approximation of Indicator Function of Convex Sets in $\mathbb{R}^n$
Let $( \mathbb{R}^n, \| \cdot \|_P)$ be the $n$-dimensional Euclidean space equipped with $\ell_p$-norm $\| \cdot \|_p$ for some $p\in [1, + \infty]$. Let $A$ be a convex set in $\mathbb{R}^n$ and ...
- 1,127
2
votes
1
answer
144
views
Do we have independence if we let the indices of the events increase?
Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space.
Consider events indexed by $m, n \in \mathbb N$:
$ \ \ \ \ \ \ \ \ \ \ \ A_{1,n}, A_{2,n}, A_{3,n} ...$ are n-wise independent.
$A_{m,1}...
- 247
2
votes
1
answer
169
views
Approximation of the cumulative normal distribution
As is well known, there is no explicit formula for $\int_{-\infty}^\infty step(t−x)\cdot e^{−t^2/2}dt=\int_x^\infty e^{−t^2/2} dt$ for generic $x,$ where $step(z)$ is the step function, $step(z)=1$ ...
- 2,390
2
votes
0
answers
58
views
$L^2$ approximation of delta functions on real algebraic varieties and asymptotic bounds
Let $X$ be a smooth projective variety over $\mathbb{C}$ of dimension $n$. Consider a probability measure $\mu$ on $X(\mathbb{R})$, absolutely continuous with respect to the Lebesgue measure induced ...
2
votes
0
answers
29
views
Steiner symmetrization of smooth function on non-simply connected regions
Given a smooth function $u$ defined on $\mathbb{R}^2$, restrict $u$ to a subset $\Omega \subset \mathbb{R}^2$ (possibly not simply connected) foliated by level sets of a smooth function $\psi: \Omega \...
- 434
2
votes
0
answers
88
views
Dependence and $L^2$ projections of functions
tl;dr: Is it possible that the best approximation to a nonnegative function of three variables with a bivariate function is no better than the best univariate function?
Let $w$ be a density on $\...
- 21
2
votes
0
answers
65
views
Recursive sequence of renewal type : when does one term dominate them all?
Let $(b_n)_{n \geq 0}$ be an increasing sequence of non negative real numbers.
Let $(u_n)_{n \geq 0}$ be recursively defined by $u_0 =1$ and
$$u_{n} = \sum_{k=0}^{n-1} u_{k} b_{n-k}$$
Find a ...
- 468
2
votes
0
answers
136
views
Multiple integral with diagonal constraint (short-range)
I am looking for an upper bound on the following integral:
$$\int_{X_{\delta}}\prod_{j\neq i=1}^{n}\left ( \frac{\delta}{\min (\max(\epsilon, |a_i-a_j|),\delta)}\right )^{b} \prod_{i=1}^{n} da_{i},$$
...
- 5,474
2
votes
0
answers
98
views
Has this "optimal constrained transport" notion of convergence of measures been named and/or studied?
Let $(X,d)$ be a compact metric space, and let $\{\mu_n\}_{n \in \mathbb{N} \cup \{\infty\}}$ be a family of Borel probability measures on $X$.
Fix $L \geq 1$. I will say that $\mu_n$ converges in ...
- 708
2
votes
0
answers
192
views
Convergence of Gibbs distribution to Dirac measure [closed]
Consider the probability density function on $R^d$ for a continuous function $F: R^d \to R$:
$$
q_{\varepsilon}(x) = \frac{1}{Z} \exp\left(-\frac{1}{\varepsilon} F(x)\right).
$$
Denote $x^* = \arg \...
- 31
2
votes
0
answers
115
views
Least positive value of a random polynomial
Fix a positive even integer $d$ and consider the polynomial $f(x)=c_d x^d+\ldots+c_1x+c_0$, where the $c_i$ are independent random variables that follow the uniform distribution in the interval $[-1,1]...
- 3,062
2
votes
0
answers
104
views
Weak convergence rates for integral operators
Suppose $q=\sum_{i=1}^m\pi_i\delta_{x_i}$ is a discrete measure on $\mathbb{R}^n$ and let $q\ast \varphi_\epsilon$ denote the convolution of $q$ with some mollifier $\varphi_\epsilon$, so that $q\ast\...
- 75
2
votes
0
answers
52
views
A certain expectation of a function of independent gammas
Suppose that $Y_1...,Y_n$ are independant gamma random variables: $Y_i \sim \Gamma(\alpha_i,\beta_i)$, with density $f_i(t) = \frac{t^{\alpha_i-1} e^{-\frac{x}{\beta}}}{\Gamma(\alpha) \beta^{\alpha}}$....
- 686
2
votes
1
answer
112
views
Static Widom-Rowlinson model
In Elena Pulvirenti's slides she introduced a $\textbf{static Widom-Rowlinson model of one species}$. Consider $\Lambda\subset R^2$ with periodic boundary conditions, $\Lambda$ set of particle ...
- 288
2
votes
0
answers
66
views
Properties of solution to Burger's equation using Cole-Hopf transformation
I am currently looking at a $1$D-Burger's equation defined by
\begin{equation} \label{ex burgers}
\left\{
\begin{array}{ll}
{} & \frac{\partial V_m}{\partial t} (t,x) = \frac{\sigma^2}{2} \...
- 357
2
votes
0
answers
189
views
Point wise convergence of Laplace transform and convergence of functions
Assume that functions $f_n(t), f(t)\in C_b(R_+)$. For every $\lambda >0$, we have
$$
\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