All Questions
Tagged with pr.probability inequalities
346 questions
2
votes
2
answers
161
views
Determine the affine envelope of a random process's MGF
Suppose that a stationary random process $S(t)$ can be characterized as the figure below, which for most of the time is a straight line $S(t)=c\cdot t$, but occasionally would "stall" for a ...
- 265
2
votes
1
answer
404
views
Euclidean distance bound with geometric constraints
Let $S_n$ be a set of $n$ points belonging to $\mathcal{B}_d:=\{\mathbf{x}\in\mathbb{R}^d:\|\mathbf{x}\|_2\le 1\}$, where $d\ll \log(n)$.
Let $s_n$ and $\ell_n$ be respectively defined as follows:
$$...
- 1,677
2
votes
1
answer
406
views
Mean squared absolute value of inner product of unit vectors
Given a nonempty finite subset $S$ of the unit sphere of $d$-dimensional complex Hilbert space, let $\lambda(S) = \frac{1}{\lvert S \rvert^2} \sum_{x,y \in S} \lvert \langle x,y \rangle \rvert^2$ be ...
- 343
2
votes
1
answer
90
views
A probability inequality: $p+(1-p)E[v|v\geq a] \geq E[v|v \geq p+(1-p)a]$
There is a random variable $v \sim F(\cdot)$ with support $[0,1]$. For a parameter $p \in (0,1)$ and $a \in (0,1)$. Define $A$ and $B$ as the following:
$$A=p+(1-p)E[v|v\geq a], B=E[v|v \geq p+(1-p)a]$...
- 121
2
votes
1
answer
599
views
Cantelli's inequality: the original source
Does anyone know where and when Cantelli's inequality was originally published? Strangely enough, I have not been able to find this information online.
- 128k
2
votes
1
answer
246
views
Does small expected value of a random variable show the small probability for its tail?
Do there exist functions $g(\epsilon)$ and $h(\epsilon)$ defined for $\epsilon>0$ such that $g(\epsilon)\to 0$ and $h(\epsilon)\to 0$ as $\epsilon\to 0^+$ with the following property:
If $X$ and $...
- 287
2
votes
1
answer
271
views
How to compute bounding coefficients for McDiarmid's inequality?
I am trying to understand the proof in Sec. A2 of Gretton et al.. To make the question self-contained, I summarize below the key ingredients. At the end of the post, I state my question.
Given a ...
2
votes
1
answer
474
views
How does Azuma's Inequality result from Pinelis Inequality?
According to [1]
Let $(\mathcal{X},||\cdot||)$ be a separable Banach space and let
$S(\mathcal{X})$ denote the class of all sequences
$f=(f_j)=(f_0,f_1,...)$ of Bochner-integrable random ...
- 619
2
votes
1
answer
105
views
Inequality for Gaussian measures
Let $\mu$ denote a centered Gaussian measure on $\mathbb{R}^k$, $K=(-\infty, a] \times \mathbb{R}^{k-1}$ ($a\ge 0$) and $L=\mathbb{R}\times C$ where $C$ is a convex set in $\mathbb{R}^{k-1}$, ...
- 197
2
votes
1
answer
199
views
Gaussian Poincare inequality in $1$ dimensions together with localization issue
Let $d\mu$ be a Gaussian measure on $\mathbb{R}$ with the center $a \in \mathbb{R}$ and variance $1$.
Let $B(a,r) \subset \mathbb{R}$ be the interval $[a-r,a+r]$.
Then, for any smooth mapping $f : \...
- 3,477
2
votes
1
answer
150
views
Normalized concentration inequality for empirical CDF (iid sum)
Consider the empirical and population CDF,
$$
F_n(t) = \frac{1}{n} \sum_{i=1}^n 1\{X_i \leq t\} \quad \mbox{and} \quad
F(t) = \mathbb{E} [F_n(t)],
$$
where above $X_1, \dots, X_n$ are iid, real-...
- 420
2
votes
1
answer
154
views
Random probability following a log concave distribution of order p
In the article "Concentration of the information in data with
Log-concave distributions" of Bobkov and Madiman, it is written that if $X$ is a positive random variable following a log ...
2
votes
1
answer
248
views
Ratio of expectation involving random unit vectors
Let $u=(u_1,...,u_n), v=(v_1,...,v_n)$ be two random vectors independently and uniformly distributed on the unit sphere in $\mathbb{R}^n$. Define two other random variables $X=\sum_{i=1}^nu_i^2v_i^2$, ...
- 1,495
2
votes
1
answer
421
views
A question about Gaussian Processes suprema
Suppose $\{X_t; t \in \mathcal{X}\}$ is a centered Gaussian Process with covariance function $k(\cdot,\cdot)$, and let $d(x,y) = \mathbb{E}[(X_x-X_y)^2]$.
I am trying to find a tail bound for the ...
- 145
2
votes
1
answer
178
views
Maximal inequality over two indices
In Freedman's series of 3 books on Markov processes, I find that I keep on running into terms like:
P[$\max_{0 \leq s \leq 1, s \leq t \leq rs}$ | B(t) - B(s) | > $\epsilon$]
in the background of ...
2
votes
0
answers
97
views
+100
Inequalities for norm of centered Gaussian and uncentered Gaussian
Let $g$ denote a standard Gaussian vector in $\mathbb{R}^n$, and $\|\cdot\|$ a norm.
Let $x \in \mathbb{R}^n$ and define
$$
F(x) = \mathbb{E}[\|x + g\| - \|g\|].
$$
I am wondering if it is possible to ...
- 420
2
votes
0
answers
124
views
Generalization of the triangle inequality to complex exponents: What is $P\left(\left| x^{a+bi} + y^{a+bi} \right| \ge \left|z^{a+bi}\right|\right)$?
Let $x \le y \le z$ be the length of the sides of a triangle whose vertices are uniformly random on the circumference of a circle. In this question, it was proved that if $a \ge 1$, then the ...
- 5,470
2
votes
0
answers
55
views
stochastic process and integral
Let $(X_n(t))_{t\in [1,+\infty], n\geqslant 1}$ be a sequence of nonnegative random variables and $(\mathcal{F}_s)$ a filtration ($\mathcal{F}_s \subset \mathcal{F}_r$ for $s\leqslant r$). We assume ...
- 29
2
votes
0
answers
164
views
Log Sobolev inequality for log concave perturbations of uniform measure
Suppose $\Omega$ is a convex bounded open set of $\mathbb{R}^n$ (I would be happy with just $\Omega$ as the $n$-dimensional cube). Let $\mu$ be the uniform measure on $\Omega$ and consider the ...
- 873
2
votes
0
answers
69
views
A distribution $\pi \propto \exp(-f)$ satisfies log-Sobolev inequality, does $\exp(-af)$ also satisfy LSI?
Assume a distribution $\pi \propto e^{-f}$ satisfies log-Sobolev inequality (LSI)
$$\forall \rho \in P(\mathbb{R}^n), \quad KL(\rho\| \pi) \le \frac{1}{2\lambda} I(\rho \| \pi)$$
with LSI constant $\...
- 21
2
votes
0
answers
337
views
Conditional probability inequality proof
There are two index sets $I= \{1, 2, \dots, m\}$ and $J = \{1, 2, \dots, n\}$. Then, I have independent random variables $X_{ij}, \forall i \in I, j \in J$. Fix $i \in I$; then we have $X_{ij}$ is ...
- 31
2
votes
0
answers
147
views
Is there a way to gain such an estimate?
This problem could be viewed as a polynomial generalization of the Lonely runner conjecture. And $p$, $n$ are taken sufficiently large. Take $n\in \mathbb{N}^*$ fixed, $A_p \subset (\mathbb{Z} / p \...
- 543
2
votes
0
answers
84
views
approximate the square of 2-norm distance between binary distributions with high probability
Suppsose we take $m$ samples from a Bernoulli distribution with probability $p$, and $m$ samples from another probability distribution with probability $q$. We want to calculate a statistic $x$ from ...
- 1,503
2
votes
0
answers
202
views
Probabilistic inequality for sum of squares of zero mean Gaussian random variables
Let $X_1,...,X_n$ be i.i.d. standard normal random variables. How to show that there is constant $c>0$ such that for every $a_k>0$ and for every $n>0$: $P(\sum_{k=1}^{n}a_kX_k^2>\sum_{k=1}^...
2
votes
0
answers
80
views
Bridging between Rosethal Inequalities and log convex tails
Let $X_1,\ldots,X_n$ be independent with $\mathbf{E}[X_i] = 0$ and $\mathbf{E}[|X_i|^t] < \infty$ for some $t \ge 2$. Write $\|X\|_p = (E|X|^p)^{1/p}$.
Then we have the classical "Rosenthal-type ...
2
votes
0
answers
58
views
Bounds for $\sum_{t=1}^Tn_t(s_t)^{-\alpha}\mu(s_t)$ where $n_t(s) = \sum_{1 \le t' \le t} 1_{\{s_{t'}=s\}}$ for $s \in [k]$ and $\mu \in \Delta_k$
Disclaimer: I'm not certain this is the right venue for this post, but I'll give it a try...
So trying prove some bounds in my ongoing work in theoretical reinforcement learning, I encountered the ...
- 6,853
2
votes
0
answers
330
views
Concrete Hanson-Wright inequality?
I'm working on a paper that requires bounding
$$\Pr\left[|\vec x^\top Q \vec y| >= t\right]$$ where $Q$ is a matrix (happens to be symmetric) and $\vec x,\vec y$ are iid real mean-zero subgaussian ...
- 21
2
votes
0
answers
124
views
Intuitive (?) inequality extremal inequality
Consider $N$ pairs of random variables $(X_i, Y_i)$. $X_i$ are iid, with $EX_i=0$ and $EX_i^2=1$. The same conditions hold for $Y_i$. Moreover all $X_i$ are independent of all $Y_j$. It seems very ...
- 461
2
votes
0
answers
687
views
Placing Bounds on Correlation/Covariance Through Correlation with an Intermediate Variable
I am trying to make the most of computations that have already been performed in previous steps of an algorithm. Throughout this problem statement I am only mentioning correlation, but I think it is ...
- 21
1
vote
2
answers
4k
views
Variance of truncated normal distribution
Let $ X \sim \mathcal{N} ( \mu, \sigma^2 ) $, $ - \infty \leqslant a < b \leqslant +\infty $ ($ a, b \ne \infty $ simultaneously) and $ Y $ has a truncated normal distribution on $ (a, b )$, i.e. $...
- 61
1
vote
1
answer
89
views
Inequality for integrals of probability densities
Given three univariate probability densities, $f(x)$, $g(x)$, and $h(x)$, I would like to show that
$\int_{supp(f)\cap supp(g)}\frac{fg}{f+g}dx+\int_{supp(g)\cap supp(h)}\frac{gh}{g+h}dx-\int_{supp(f)\...
- 11
1
vote
1
answer
148
views
An inequality about binomial distribution
Statement
Assume that $\sigma,R\in (1,+\infty)$, $N\in\mathbb{N}^*$, $p\in (0,1)$, $n_1\in\{0,1,2,\cdots,N-1\}$. Prove or disprove that
$$B^\frac{1}{\sigma}(n_1)-B^\frac{1}{\sigma}(n_1+1)<1 .$$
...
- 35
1
vote
1
answer
210
views
Tighter upper bound for $\mathbb{E} [\max_{\sigma \in \{ \pm 1\}^n} \sigma^T W \sigma]$
Following this question I was thinking about ways to improve the upper bound and came up with the following argument. We want to find an upper bound for
\begin{equation}
\mathbb{E} [\max_{\sigma \in \{...
- 237
1
vote
1
answer
125
views
A differential inequality and a special value
Let $G \colon [0,1] \to [0,1]$ be a monotonically decreasing function with $G(0) = 1$ and $G(1) = 0$. Suppose that $G$ is differentiable infinitely many times, and that: $$G(x)G''(X) \leq 2{G'(x)}^2.$$...
- 11.3k
1
vote
1
answer
356
views
Statistical inequality
Let $X$ be a finite discrete variable and $X\ge0$. Is it true that
$$16\operatorname{Var}(X) \le \left[8{\mathbb E}(X) + \operatorname{Range}(X)\right]\operatorname{Range}(X)$$
where $\operatorname{...
- 95
1
vote
1
answer
129
views
Small total variation distance between sums of random variables in finite Abelian group implies close to uniform?
Let $\mathbb{G} = \mathbb{Z}/p\mathbb{Z}$ (where $p$ is a prime). Let $X,Y,Z$ be independent random variables in $\mathbb G$.
For a small $\epsilon$ we have $\operatorname{dist}_{TV}(X+Y,Z+Y)<\...
- 23
1
vote
1
answer
172
views
Ratio of the constants of the Marcinkiewicz–Zygmund inequality for p=1
The Marcinkiewicz–Zygmund inequality states that
$$
{\displaystyle A_{p}E\left(\left(\sum _{i=1}^{n}\left\vert X_{i}\right\vert ^{2}\right)_{}^{p/2}\right)\leq E\left(\left\vert \sum _{i=1}^{n}X_{i}\...
- 13
1
vote
1
answer
113
views
How to upper bound the difference between these two Gaussian-like densities?
$
\DeclareMathOperator*{\argmax}{arg\,max}
\DeclareMathOperator*{\argmin}{arg\,min}
\DeclareMathOperator*{\cov}{cov}
\DeclareMathOperator*{\supp}{supp}
\DeclareMathOperator*{\dom}{dom}
\newcommand{\...
- 657
1
vote
2
answers
2k
views
Upper bound about Gaussian tail bound
From the definition of sub-Gaussian distribution $X$ w.r.t. $\sigma$ i.e.
$$\mathbb{P}(|X-\mathbb{E}(X)|\geq t) \leq 2 \exp(-\frac{t^2}{2\sigma^2}).$$
It's natural that when $X \sim \mathcal{N}(\mu, \...
- 81
1
vote
1
answer
92
views
Lower bound $L_{1}$-metric with $L_{2}$-metric for bounded pdfs, on common support
Setup
To clarify, let constants $0 < a < b < \infty$, and $p \in \mathbb{N}$ be fixed. Further let $B \subset \mathbb{R}^{p}$ be a fixed compact support. We then define the space of bounded (...
- 133
1
vote
1
answer
362
views
An inequality involving the Wasserstein distance and chi-squared distance
$\newcommand{\N}{\mathbb N}$Let $P$ be the set of all probability mass functions on $\N_0:=\{0\}\cup\N$, where $\N:=\{1,2,\dots\}$. Let $P_{>0}$ denote the set of all $q=(q_0,q_1,\dots)\in P$ such ...
- 128k
1
vote
1
answer
144
views
A uniform mixture of order statistics
Let $0<k<n$ be integers, and let $X$ be a random variable obtained as follows: sample $n$ points independently and uniformly at random in the unit interval, and select (uniformly) one of the $k$...
- 4,049
1
vote
2
answers
462
views
lower bound the probability of at least L collisions
Lets say we get a list $M$ containing $|M|=\sqrt{L\cdot N}$ randomly and independtly drawn elements from a set of size $N$. And lets denote the $i$-th element of the list $M$ by $M[i]$.
If we now ask ...
- 123
1
vote
1
answer
499
views
property of iid random variable
Let $ (\xi_i)_{i \ge 1} $ be independent identically distributed random variables, taking values in $ (1,3]$.
Can we show:
$P( \exists N \in \mathbb{N}, \text{ s.t. } \forall k \ge 0, \prod_{i=1}^{...
- 553
1
vote
1
answer
753
views
Probability space analogue of Cauchy-Schwarz inequality
Suppose we have two sets of discrete events, $A$ and $B$. Then I think it is true that:
$$2\sum_{i \in A, j \in B}\Pr[i\ \textrm{AND}\ j] \leq \sum_{i \in A}\Pr[i]+ \sum_{j \in B}\Pr[j] +\sum_{i, j \...
- 13
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
195
views
Reference request: Inequalities involving convex sets and Gaussian variables stated in a paper by Talagrand
I'm looking for references for two facts that are stated without proof in the paper:
Talagrand, M., Are all sets of
positive measure essentially convex?, Lindenstrauss, J. (ed.) et al.,
Geometric ...
- 484
1
vote
1
answer
56
views
Covariance inequality for left skewed distributions
Consider a left skewed random variable $X$ with mean $1$, median $>1$ and support on $[0,2)$. Suppose we have a class of functions $\mathbf{G}$ and each of it's members satisfy $G(x): [0,\infty) ...
- 45
1
vote
1
answer
235
views
Prove inequality on expectation
Random variable $X\geq 0$ and its variance exists. How to prove
$$\mathbb{P}(X\geq(1-t)\mathbb{E}(X))\geq \frac{t^2\mathbb{E}(X)^2}{\mathbb{E}(X^2)}\enspace\text{for}\enspace t\in(0,1]$$
$$\mathbb{E}(\...
- 81
1
vote
1
answer
97
views
A strict inequality for the $L^1$-norm of a symmetrized zero-mean random variable
Suppose that $Y$ is an independent copy of a random variable (r.v.) $X$ with a zero-mean nondegenerate distribution. Is it then always true that $E|X-Y|>E|X|$?
To get the non-strict version of ...
- 128k