All Questions
31 questions
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
13
votes
2
answers
1k
views
Probability vector $p$ majorizes its normalized entropy vector $\small \frac{-p\log p}{H(p)}$
I guess the following inequality
$$ \sum_{i=1}^n g \left (\frac{-p_i \log p_i}{H(\boldsymbol{p})} \right ) \le \sum_{i=1}^n g (p_i)$$
holds for any continuous convex function $g$ and any probability ...
- 303
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
11
votes
1
answer
676
views
Entropy arguments used by Jean Bourgain
My question comes from understanding a probabilistic inequality in Bourgain's paper on Erdős simiarilty problem: Construction of sets of positive measure not containing an affine image of a given ...
- 196
3
votes
0
answers
86
views
Finite dimensional distribution of a stochastic process Lipschitz on every relatively compact set
Let $X_t$ be a Markovian Itô diffusion process, defined by an SDE
\begin{equation}
dX_t = \mu(X_t)\,dt + \sigma(X_t)\,dW_t\,.
\end{equation}
Let $f(x,t|x_0,0)$ denote its transition density function. ...
3
votes
1
answer
109
views
Inequality: multivariate normal distribition
Let $p(u,x)=\frac{1}{(4\pi u)^{q/2}}e^{-|x|^2/(4u)},u>0,x \in \mathbb{R}^q.$
Prove that for $r\geq 0,c>1$ there exists $C>0$ (depending on $r,c$) such that $$\forall x \in \mathbb{R}^q,u>...
- 573
5
votes
2
answers
2k
views
Relationship between KL, chi-squared, and Hellinger
There are many well-known relationships between the KL divergence, chi-squared ($\chi^2$) divergence, and the Hellinger metric. In the paper "Assouad, Fano, and Le Cam" by Bin Yu, the author ...
- 63
3
votes
1
answer
142
views
How does the integral of pseudo Gaussian kernel on $(0,\infty)$ depend on its variance?
Let $a, b: \mathbb R_+ \to [0,1]$ be continuous functions. Let $k: \mathbb R_+\times\mathbb R \to [1,2]$ be $1-$Lipschitz. Set, for $0<s<t$ and $y>0$,
$$A(s,t,y):=\int_s^t\frac{k(u,y)}{1+a(u)}...
user478492
-1
votes
1
answer
551
views
Lower bound of an expectation
Suppose a random variable $X$ has unit variance i.e. $\sigma^{2} = 1$. Is there a positive constant $c > 0$ such that
$$\mathbb{E}[\ | X - \mathbb{E}[X] | \ ] \ge c $$
My attempt of a solution is ...
- 643
1
vote
0
answers
77
views
Divergence between random variables after transformation
Let $X$ and $Y$ be random variables with laws $\mu_X$, $\mu_Y$ and $d$ be some $f$-divergence (e.g. KL, total variation, Hellinger). Writing $d(X,Y)$ for the divergence between $\mu_X$ and $\mu_Y$, ...
- 19
5
votes
2
answers
202
views
Monotonicity of a parametric integral
For real $x>0$, let
$$f(x):=\frac1{\sqrt x}\,\int_0^\infty\frac{1-\exp\{-x\, (1-\cos t)\}}{t^2}\,dt.$$
How to prove that $f$ is increasing on $(0,\infty)$?
Here is the graph $\{(x,f(x))\colon0<...
- 128k
12
votes
1
answer
525
views
An inequality about unit vector orthogonal to $(1,1,...,1)$
Does there exist a constant $\alpha>0$ such that the following holds?
$$\liminf_{n\to\infty}\inf_{x\in\mathbb{R}^n, \sum_{i=1}^nx_i^2=1, \sum_{i=1}^nx_i=0}\frac{\sum_{i<j, |i-j|\leq\frac{n}{4}}(...
- 1,495
0
votes
1
answer
159
views
Best bounds on the integral of an increasing function
The following question, somewhat edited here, was asked and then closed at The best bound of the integral of a nondecreasing real function in a closed interval.
Let $F\colon[0,1]\to[0,1]$ be a ...
- 128k
5
votes
2
answers
243
views
Is $\Gamma(s, x=s-1)/\Gamma(s)$ decreasing for real $s>1$? Is $\Gamma(s, x=s)/\Gamma(s)$ increasing?
This has received no full solution at StackExchange.
As per https://dlmf.nist.gov/8.10#E13 we have
$$\frac{\Gamma\left(n,n\right)}{\Gamma\left(n\right)}<\frac{1}{2}<\frac{\Gamma%
\left(n,n-1\...
- 804
15
votes
0
answers
749
views
Prove $\int_{0}^{\infty} \cos(\omega x) \exp(-x^{\alpha}) \, {\rm d} x \ge {\alpha^2 \sqrt{\pi} \over 8} \exp \left( -\frac{\omega^2}{4} \right)$
I would like to prove that
$$\int_{0}^{\infty} \cos(\omega x) \exp(-x^{\alpha}) \, {\rm d} x \ge
{\alpha^2 \sqrt{\pi} \over 8} \exp \left( -\frac{\omega^2}{4} \right)$$
for any $\omega > 0$ and $...
- 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
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 $$\int_{-\infty}^\infty F(x)...
- 319
0
votes
1
answer
1k
views
Bounding $L^p$ norms in terms of lower-order $L^q$ norms
Suppose $f,g\in L^q(\Omega)$ ($\Omega\subset \mathbb{R}^n$) for all $1\le q\le p$. Here, $L^p(\Omega)$ is defined with respect to some measure $\mu$ that is absolutely continuous wrt Lebesgue measure. ...
- 710
1
vote
2
answers
295
views
Monotonicity of maximum of convex combination of two scaled concave functions
Let $f:R\rightarrow R$ be a concave function with a unique and finite maximum. Let $$g(x, \beta) = \beta f(\alpha \cdot x) + (1-\beta) f((1-\alpha) \cdot x), $$ where $\alpha \in [0,1/2]$ and $\beta \...
- 13
1
vote
1
answer
143
views
Lower bound for log-Ratios
Can we find a universal constant $c>0$ such that for all $p,q\in\Delta:=\lbrace x\in (0,1)^{n}\ \colon\ x_{1}+\dots+x_{n}=1\rbrace$ it is true that
\begin{equation}
|p_{i}-q_{i}|\le c\left|\ln\frac{...
- 289
5
votes
0
answers
205
views
Strange inequality relating Binomial pmf and cdf
I'm encountering a strange inequality I need to prove, relating the Binomial pmf and cdf.
Suppose we have $n$ coin flips, and fix an arbitrary $k \le n/2$ heads. Suppose further that we have some ...
- 101
4
votes
1
answer
336
views
Binomial Distributions and Inequality
Suppose $c>2$ is a real number, and $x$ solves equation $f(\frac{1}{2}+x,m,2m)=(2m+1)cx$ for some $m>0$ integer number, where $f$ is probability mass function for binomial distribution, with ...
- 43
3
votes
1
answer
209
views
Log concavity of the maximum of dependent Gaussians
Let $Z_1,\dots,Z_n$ be dependent Gaussian random variables. Is it true that $X=\max\{Z_1,\dots,Z_n\}$ has a log-concave distribution function? This is true for the independent case, but is it true in ...
- 2,288
4
votes
2
answers
654
views
Hypercontractivity of two simple random variables, $E[XY]^s \le E[X^s]E[Y^s].$
For $\alpha,\beta\ge 0$, let $X\in\{1,\alpha\}$ and $Y\in\{1,\beta\}$ be two random variables such that $$XY = \begin{cases}
\alpha\beta \quad & \text{with probability} \quad p_{11}\\
\alpha \quad ...
5
votes
1
answer
279
views
Mixed norm inequality
Suppose we have a product space $(X_1\times X_2,\mu_1\otimes\mu_2)$, with finite measures $\mu_1,\mu_2$ and $p>1$.
Is there a possibility that an inequality of this form holds on the product space?
...
- 156
0
votes
0
answers
112
views
On certain integrals of exponential functions with respect to Gaussian measures
I have questions about the integral
$$F(a,b,c)=\sqrt{\frac{a}{\pi}}\int_{0}^{\infty}e^{-bx^4+cx^3-ax^2}dx$$
for $a,b,c>0$.
What is the asymptotic behavior of $F(a,b,c)$ for small $a,b,c$? In ...
- 505
4
votes
1
answer
388
views
Proofs of inequalities used by Erdos-Renyi in their Random Graphs Paper 1
Please refer to this, it is Erdos-Renyi 1959 paper 1 on Random Graphs. I am currently working on this, but I am stuck on the fifth page, where they use two estimates. More specifically, here's the ...
- 319
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
37
votes
3
answers
3k
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 ...
- 11.4k
16
votes
6
answers
3k
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 ...
- 2,239
9
votes
2
answers
519
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 $k$-...
- 91