All Questions
Tagged with pr.probability inequalities
91 questions with no upvoted or accepted answers
16
votes
1
answer
743
views
Inequalities for marginals of distribution on hyperplane
Let $H = \{ (a,b,c) \in \mathbb{Z}_{\geq 0}^3 : a+b+c=n \}$. If we have a probability distribution on $H$, we can take its marginals onto the $a$, $b$ and $c$ variables and obtain three probability ...
- 156k
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
6
votes
0
answers
337
views
Maximal inequalities for square of partial sums
Let $S_n = \sum_{i \leq n} X_i$ be the partial sums of a nice sequence of random variables $X_i$. In my application, $X_i$ is a functional of a finite-state, irreducible, aperiodic Markov chain, so ...
- 1,069
6
votes
0
answers
342
views
Maximizing Renyi entropy for a certain channel
The channel under consideration is $T = A + B$, where $A$ and $B$ take on values in $\{0, 1\}$ according to a probability mass function. Let (joint) random vector $(A_1, A_2,\ldots, A_n)$ be denoted ...
5
votes
0
answers
159
views
Log Sobolev inequality uniform in parameters
Fix a positive integer $N$. For $\theta \in [0,2\pi]$, set $\sigma_k(\theta) :=(\cos(k\theta),\sin(k\theta)) \in S^1$ for each integer $1\leq k\leq N$. Now for vectors $x_1,\ldots,x_N\in \mathbb{R}^2$,...
- 873
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
5
votes
0
answers
857
views
Anti-concentration inequality for Gaussian random vector
I am trying to obtain an explicit expression for $C$ in terms of $b$ in the following inequality.
Suppose that $Y$ is a centred Gaussian random vector in $\mathbb R^p$ such that $\operatorname EY_j^...
- 423
5
votes
0
answers
177
views
Inequality about moments of a random variable and of its conditional expectation
This is a follow-up to a question I asked earlier: Moments of a random variable and of its conditional expectation
My claim turned out to be false. Here is a new claim.
Let $X$ be a bounded random ...
- 1,269
5
votes
0
answers
295
views
inequality in a shape of inclusion exclusion formula
I have two inequalities to show, both of which describe some probabilities. First I know how to handle, and it follows from applying arithmetic-harmonic mean inequality:
consider 9 numbers $a_1,a_2,...
- 341
4
votes
0
answers
155
views
Comparing the slackness of Jensen's inequality for some coupled random variables
Let $f:\mathbb{R} \to \mathbb{R}$ be convex and $X,Y$ be random variables with a coupling such that $\mathbb{E}[Y\mid X=x] = x$. A straightforward application of Jensen's inequality gives that $\...
- 141
4
votes
0
answers
131
views
Log of a truncated binomial
Let $X$ follow a binomial distribution with $n$ trials and success probability $p$, and let $0\leq k\leq n$. Are there any natural approximations or bounds for the ratio $$\frac{\boldsymbol{E}\log\...
- 4,049
4
votes
0
answers
96
views
Is this conjecture about the binomial and beta distributions true?
Let $X$ follow a binomial distribution with parameters $n$ and $p$, and also fix $k$ such that $1<k<n$. Define
$$a = \mathbb{E}(X-k)^+$$
and
$$b = \mathbb{E}\log\binom{X}{(X-k)^+}$$
where the ...
4
votes
0
answers
146
views
An inequality for three iid random variables with a log-concave density
It was previously shown that
$$H\ge cG,\tag{1}$$
where $c:=1/14334$,
$$G:=E|X-Y|,\quad H:=E|X-Y|-\tfrac12\,E|X+Y-2Z|,$$
and $X,Y,Z$ are independent random variables with the same log-concave density.
...
- 128k
4
votes
0
answers
414
views
Simmons' inequality on binomial random variables
Fix two positive integers $n,m$ such that $n>2m$ and $n\ge 3$, and let $X\sim \text{Bin}(n,\frac{m}{n})$ be a binomial random variable. For each $i\in \{1,\ldots,m\}$, set
$\alpha_i = \mathbb{P}(X\...
- 41
4
votes
0
answers
190
views
Pedestrian proof of Gaussian chaos for order-two polynomial?
Let $\ell \geqslant 1$. Let us consider $(g_n)_{n \in \mathbb{N}}$ identically distributed independent real gaussian variables and real number $(a_{n_1,\dots n_{\ell}})_{(n_1, \dots, n_{\ell}s)\in\...
4
votes
0
answers
321
views
Examples of measures that satisfy FKG, but not the FKG lattice condition
Let a percolation measure be a measure on $\{0,1\}^n$. We have a natural partial order on $\{0,1\}^n$ given by comparing all coordinates. An event $A$ is called increasing if for all $ \omega \in A $ ...
- 1,054
4
votes
0
answers
405
views
A symmetrization-majorization inequality for i.i.d. zero mean random variables
Let $k\geq 2$ and
$$f(x_1,\ldots,x_k)=\Bigl(\prod_{i\leq k}(1+x_i)+\prod_{i\leq k}(1-x_i)\Bigr)\log\Bigl(\prod_{i\leq k}(1+x_i)+\prod_{i\leq k}(1-x_i)\Bigr)-k(1-x_1 x_2)\log(1-x_1 x_2).$$
If random ...
- 175
4
votes
0
answers
141
views
Is there an example that both Berry-Essen bound and DKW bound are attained?
The Berry-Essen bound stated that
$$\sup _{{x\in {\mathbb R}}}\left|\widehat{F_{n}(x)}-\Phi (x)\right|\leq C_{0}\cdot \psi _{0}$$
where $\psi _{0}(n)={\Big (}{\textstyle \sum \limits _{{i=1}}^{n}\...
- 8,071
4
votes
0
answers
205
views
Dimension reduction for low-order moments of Rademacher-weighted sums of vectors
Let $x_1,\dots,x_n$ be vectors in a Euclidean space $H$. Let $\varepsilon_1,\dots,\varepsilon_n$ be independent Rademacher random variables (r.v.'s), so that $P(\varepsilon_i=\pm1)=1/2$ for all $i$.
...
- 128k
4
votes
0
answers
1k
views
Total variation and Hellinger distance inequality between truncated Gaussians
We know that the total variation distance, $d_{TV}(P,Q) = \frac{1}{2}\left|\left|P-Q\right|\right|_1$, between any two distributions $P$ and $Q$ is lower bounded by their squared Hellinger distance, $...
- 41
4
votes
0
answers
213
views
Optimization problem involving Multivariate Normal
I use $\phi(t)$ to describe the standard normal distribution density and $\Phi(t)$ as the normal distribution CDF and would like to prove that for all
$n\geq3$, the function:
$$h(\mu_{1},\ldots,\...
- 41
3
votes
0
answers
81
views
Order of $\mathbb{E}[ \max_i |x_i + z_i| - \max_i |z_i|]$
Let $z_1, \dots, z_n$ be iid standard Normal, and let $x \in \mathbb{R}^n$. Put $\|u\|_\infty = \max_i |u_i|$.
Define
$$
F(x) = \mathbb{E}\Big[\|x + z\|_\infty - \|z\|_\infty\Big]
$$
If $\|x\|_\infty \...
- 420
3
votes
0
answers
92
views
Tighter Freedman's inequality for a special martingale difference sequence
Let $X_{1}, \ldots, X_{T} \in \{0, 1\}$ be a sequence of Boolean random variables with
$$
\mathbb{E}[X_{t} | X_{1}, \dots, X_{t - 1}] = p_{t}.
$$
Consider the sequence $Y_{t} := X_{t} - p_{t}$ (which ...
- 41
3
votes
0
answers
80
views
Seeking strong bounds on KL-divergence and martingales for a hypothesis-testing inequality
Let's say we have a finite set $\mathcal{O}$ of observations, and let $\mathcal{C}(\Delta\mathcal{O})$ denote the space of closed convex sets of probability distributions.
We have two hypotheses which ...
- 353
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
0
answers
169
views
Probabilistic behavior of greedy point selection in the plane
Let $\mathcal{X} = X_1,\dots,X_n$ be a collection of independent, uniform samples in the unit square. Let $\mathcal{S}=\{X_1\}$, and consider the following process: for $i=2,\dots,n$, let $x^*$ be ...
- 4,049
3
votes
0
answers
84
views
Weak sufficient conditions for non-negative correlation between functions of correlated random variables?
Consider real, nonnegative random variables $A$, $B$, and $X$, and define
$Z = \exp(-A X)$ and $W = \exp(-B X)$, and also $U = \exp(-X - A)$ and $V = \exp(-X -B)$.
What sorts of minimal sufficient ...
- 61
3
votes
0
answers
185
views
Measure change bound for function of subgaussian r.v
Let $X$ be a (sub)gaussian r.v. on $\mathbb{R}^d$; say $X\sim\mathcal{N}(\mathbf{0},\mathbb{1}_d)$; and let $a\colon\mathbb{R}^d\to [0,1]$ be a function with $\mathbb{E}[a(X)] > 0$.
It is not hard ...
- 1,372
3
votes
0
answers
77
views
A concentration problem of product of matrices
Let $A$ be an $n \times m$ matrix with non-negative entries and $B \in \mathbb{R^{n\times n}_{\geq 0}}, C \in \mathbb{R^{m\times m}_{\geq 0}}$ be random matrices where B and C are both symmetric and ...
- 195
3
votes
0
answers
125
views
Concentration of sums of random matrices around the mean, in the Loewner order
Recently, I have found myself interested in concentration properties of random matrices.
Specifically I would like to answer questions of the following sort
Let $\{X_i\}_{i=1}^n$ be i.i.d. copies ...
- 393
3
votes
0
answers
451
views
concentration bounds on weighted multinomial sum
Consider i.i.d random vectors $Y_{1},..,Y_{n}$ and they are chosen uniformly at random from $\{e_{1},..,e_{L}\}$ where $e_{i}$ is a $L\times 1$ vector with $i$th component be 1 and the others be 0. ...
- 31
3
votes
0
answers
158
views
Worst-Case Solution to (Stochastic) Matrix Inequality
EDIT: Some specific conjectures added.
This problem comes with an associated stochastic process, but I phrase everything as linear algebra in case somebody from a non-probability community has seen ...
- 31
3
votes
0
answers
494
views
Maximization of a total variation distance subject to another total variation distance in Markov chain
Suppose two dependent random variables $X$ and $V$ from finite alphabets $\mathcal{V}$ and $\mathcal{X}$ with known joint and marginal distributions are given. Let $P_{XV}$ and $P_X$ and $P_V$ are the ...
- 1,109
3
votes
0
answers
104
views
Minimizing/Maximizing the tail of the convex combinations of Chi Squared i.i.d random variables
Consider $N$ i.i.d random variables, $X_{1}, X_{2}, \ldots, X_{N}$ , that are chi-squared of degree $K \geq 2$. Also consider the following 3 vectors:
\begin{eqnarray*}
\bar{a} &=& (\frac{1}{...
- 51
3
votes
1
answer
330
views
Does this probability distance metric have an official name?
Let us define a distance metric between two joint probability math functions $p(x,y)$ and $q(x,y)$ as in the following
\begin{align}
\sum_{y}\sqrt{\sum_{x}p(x)\left(p(y|x)-q(y|x)\right)^2}.
\end{...
- 287
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
0
answers
80
views
Inequality involving random vectors and absolute values
Let $\mathbb{X}, \mathbb{Y} \subset \mathbb{R}^d$ be finite sets. Suppose random vectors $X \in \mathbb{X}$ and $Y \in \mathbb{Y}$ are sampled according to a joint distribution $\mathbb{P}_{XY}$. ...
