All Questions
Tagged with pr.probability inequalities
346 questions
4
votes
1
answer
168
views
Existence of copula bound pointwise strictly smaller than the Fréchet-Hoeffding upper bound
Consider bivariate copulas $C_1$ and $C_2$ with $\max\{C_1(u,v), C_2(u,v)\}< M_2(u,v)$ for all $u,v \in(0,1)$, where $M_2(u,v) := \min\{u,v\}$ is the Fréchet-Hoeffding upper bound.
Is there a ...
- 143
4
votes
3
answers
161
views
Find distribution that minimises a function of its moments
Imagine a probability density function $f(x)$, defined for positive $x$, and let's note its $n$th non-centred moment $x_{n}$. The mean $x_{1}$ is fixed (and positive).
How can I find $f(x)$ that ...
- 223
4
votes
1
answer
555
views
Conditional Form of Rosenthal's Inequality
Rosenthal's Inequality as stated in the book "Martingale Limit Theory and Its Application" by Hall and Heyde states the following:
If $\{S_i, \mathcal{F}_i, 1\leq i \leq n\}$ is a martingale and $2\...
- 289
4
votes
1
answer
358
views
Bound for type of correlation measure
Assume you have a finite, discrete probability distribution for a joint random variable and such that $P(X=i,Y=j) = p_{i,j}$ for $i \in \{1, \dots, |X|\},j \in \{1, \dots, |Y|\}$. The marginal ...
- 51
4
votes
1
answer
225
views
concentration inequality for $d$-dimensional martingale
Are any concentration inequality available for $d$-dimensional martingale. It is easy to find such inequality using the inequalities for single dimension, but that will contain the dimension $d$ in ...
- 317
4
votes
1
answer
320
views
Sub-Gaussian random variables and convex ordering
Suppose that $X$ is a $1$-sub-Gaussian real-valued random variable, i.e. for all $t \in \mathbf{R}$, it holds that $\log \mathbf{E} \exp \left( t X \right) \leqslant \frac{1}{2} t^2 $.
Does there ...
- 801
4
votes
2
answers
193
views
Minimal coupling
Let $\mu,\nu$ be probability measures defined on a common measure space $(\Omega,\mathcal F)$. A coupling of $\mu,\nu$ is a probability measure $\pi$ on $(\Omega^2,\mathcal F^2)$ with marginals $\mu,\...
- 1,329
4
votes
1
answer
206
views
Inner product of sorted Gaussian vector
Suppose $X_1,\ldots,X_n$ are i.i.d. standard normal. I'm wondering how to analyze the following quantity:
$$\left|\frac{X_{(1)}X_{(n)}+X_{(2)}X_{(n-1)}+\cdots+X_{(n)}X_{(1)}}{n}\right|$$
where $X_{(1)}...
- 1,495
4
votes
1
answer
406
views
Expectation of exponential of a function of independent Rademacher r.v.'s involving the error function
Let $Z,Z'\in\{-1,1\}^n$ be two independent vectors of i.i.d. Rademacher r.v.'s, where $1\leq n \leq d$ are two integers ($d\gg 1$). I am trying to get an upper bound on
$$
\mathbb{E}_{ZZ'}\left[ \exp\...
- 1,372
4
votes
1
answer
347
views
Concentration of functional of Gaussian random variable
Suppose I have two Gaussian distributions
$p(x) = \frac{1}{(2\pi)^{d/2}|\Sigma_p|^{1/2}}\exp(-\frac{1}{2}x^\top \Sigma_p^{-1} x)$ and $q(x) = \frac{1}{(2\pi)^{d/2}|\Sigma_q|^{1/2}}\exp(-\frac{1}{2}x^\...
- 515
4
votes
1
answer
568
views
inequality with exponents
We are given a graph $G$, each vertex $v$ has an assigned value $\gamma_v\in [0,1]$, and it happens that for every $v$ we have $\gamma_v+\sum_{u\in \delta(v)} \gamma_u = 1$. Assume that $\sum_v \...
- 341
4
votes
0
answers
155
views
Comparing the slackness of Jensen's inequality for some coupled random variables
Let $f:\mathbb{R} \to \mathbb{R}$ be convex and $X,Y$ be random variables with a coupling such that $\mathbb{E}[Y\mid X=x] = x$. A straightforward application of Jensen's inequality gives that $\...
- 141
4
votes
0
answers
131
views
Log of a truncated binomial
Let $X$ follow a binomial distribution with $n$ trials and success probability $p$, and let $0\leq k\leq n$. Are there any natural approximations or bounds for the ratio $$\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:
$$h(\mu_{1},\ldots,\...
- 41
3
votes
1
answer
346
views
Simple anticoncentration bound for binomially distributed variable
The following question, which arose during my research, seems deceivingly simple to me, but I could not find any elegant and formal argument.
For a binomially distributed variable $X \sim \text{Bin} \...
- 85
3
votes
3
answers
6k
views
Probability of a random variable greater than its expected value
We have a lot of probabilities lower bounding as (e.g. chernoff bound, reverse markov inequality, Paley–Zygmund inequality)
$$
P( X-E(X) > a) \geq c, a > 0 \quad and \quad P(X > (1-\theta)E[...
- 141
3
votes
2
answers
169
views
On finding an upper bound on the error of a sparse approximation
I posted this question on math.stackexchange earlier, but didn't see any response. So, I am posting it here, in case someone else has an answer.
Original question: https://math.stackexchange.com/...
- 65
3
votes
2
answers
189
views
Is the covariance of squares always bounded from below by two times the covariance?
I came across the following inequality in one of my calculations ($X,Y$ are centered random variables):
$$\operatorname{E}(X^2Y^2)-\operatorname{E}(X^2)\operatorname{E}(Y^2) \geq 2 \operatorname{E}(...
- 617
3
votes
1
answer
1k
views
concentration inequality for averages of dependent random variables
Let $X \in R^n$ be a random vector such that
$$P(|X_i| > \epsilon) > e^{-\epsilon^2}$$
What is a tight bound on
$$P(\sum_{i=1}^n |X_i| > \epsilon)$$
and on
$$P(\max_{1\le i\le n} |X_i| ...
- 461
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
3
votes
1
answer
232
views
Bounds on relative entropy for MLE in Bernoulli coin tosses
In the context of estimating the parameter $p$ from a dataset of $n$ i.i.d Bernoulli coin tosses, we often use the relative entropy $D(p \parallel \hat{p})$ to measure the performance of an estimator $...
- 63
3
votes
1
answer
271
views
Bound on the distribution of a ratio involving Gaussian distributions
Let $U \sim \mathcal{N}(0, I_K)$ be a Gaussian vector of dimension $K$ and $V \sim \mathcal{N}(0,1)$, independent of $U$. Let $\Delta$ be a diagonal matrix with non-negative diagonal elements, $c\in\...
- 197
3
votes
2
answers
636
views
Exponential inequality for the sum of martingale differences $X_1, \dots, X_n$ when $\sum_{i=1}^{n} \operatorname{Var}(X_i) \leq B^2$
Let $X_1, X_2, \dots, X_n$ be a martingale difference sequence such that
$$
X_i \leq y \quad \text{and} \quad \sum_{i=1}^{n} \operatorname{Var}(X_i) \leq B^2.
$$
Question 1: Does the following hold?
$$...
- 33
3
votes
2
answers
402
views
Something between the Chernoff and Hoeffding bounds
Suppose I have $n$ independent 0-1 random variables $X_1, \cdots, X_n$ and I want to show a concentration of $X = \sum_i X_i$.
I can use either the Chernoff bound or the Hoeffding bound.
Suppose $E[...
- 233
3
votes
1
answer
3k
views
Is there a tight lower bound for the expectation of the product of two positive valued random variables?
Let $X,Y$ be two (dependent) random variables with $\mathbb{P}(X\ge 0)=\mathbb{P}(Y\ge 0)=1$.
I want to find a tight lower bound of $\mathbb{E}(XY)$ when $X,Y$ are non-negative, almost surely.
...
3
votes
1
answer
235
views
Inequality for exponential sum in Dvoretzky 1972
I'm currently trying to figure out the following inequality. It looks like an inequality for the exponential sum, but I can't verify it or find a source explaining it any further. Most likely it has ...
- 115
3
votes
1
answer
287
views
Expectation comparison inequality for concave function of symmetric random variables
Suppose that $X_i$, $i\in[n]$ are
independent symmetric
random variables. I think the conjectured result holds in greater generality, but we can additionally assume that each $X_i$ takes the values $\...
- 6,541
3
votes
1
answer
205
views
Bound on an integral representing a difference of two relative entropies
Let $ f : [0,1] \to \mathbb{R} $ be a function satisfying: 1.) $ |f(x)| \leqslant a $ for some $ a < 1 $, and 2.) $ \int_0^1 f(x) {\mathrm d}x = 0 $. I would like to know whether the following ...
- 503
3
votes
1
answer
1k
views
Extension of Bernstein’s Inequality when the random variable is bounded with large probability
Bernstein’s Inequality can be stated as follows : Let $x_1, x_2, \dots, x_n$ be independent bounded random variables such that $\mathbb{E}[x_i] = 0$ and $|x_i| \leq \zeta$ with probability $1$ and let ...
- 33
3
votes
1
answer
229
views
Inequality for difference of consecutive atom probabilities for binomial distribution
Edit: This post was originally two questions, the first of which has been answered, but a reference would still be appreciated if existent. The second question has been removed and migrated to its ...
- 2,720
3
votes
2
answers
1k
views
Expected value of a truncated binomial
Let $X\sim B(n,p)$ be a binomial random variable and fix $0<k<n$. Are there any well-known bounds for $\mathbb{E} (X-k)^+$, where $(X-k)^+ =\max\{0,X-k\}$? I am particularly interested in ...
- 4,049
3
votes
1
answer
171
views
Why does the assumption $|U_t| \le \frac1{p_{\min}}$ work in this paper?
I am reading a 2009 paper right now "Importance Weighted Active Learning" and on page 5, there is a theorem that uses the inequality $|U_t| \leq \frac{1}{p_{\min}}$. I am not sure how the paper found ...
- 33
3
votes
1
answer
105
views
A gaussian based inequality
I came across an inequality in the paper of 'Estimation of a function with discontinuities ...' (AoS, 1998, p.1374) and tried to prove it, but could not get to the result. Some simulations also seemed ...
- 555
3
votes
1
answer
70
views
Multiplicative approximation for a negative moment of the binomial distribution
Let $X$ be a binomial random variable with parameters $n,p$.
Define the function
$f(n, p, t) = E\frac{1}{1 + t X},
$
where $t > 0$.
Question: Can we find an elementary function $F(n, p, t)$ such ...
- 420
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
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 $y>0$,
$$A(s,t,y):=\int_s^t\frac{k(u,y)}{1+a(u)}...
user478492
3
votes
1
answer
197
views
possibility of bounding one functional by another functional
This is a natural follow-up question related to one of my previous questions at here. Assume that $\rho$ is a log-concave probability density function with support $[0,\infty)$ and fixed mean $\mu >...
- 730
3
votes
1
answer
152
views
Question on example 3.0.1 in Yurinsky's book "Sums and Gaussian vectors"
Good day to All.
Let $S_{1,n} = \sum_{i=1}^{n}\xi_{i}$, where $(\xi_{i})_{i \in \mathbb{N}}$ be independent RV with values in some Banach space.
On pages 79-80 in this book author provides an ...
- 65
3
votes
1
answer
1k
views
Chernoff-Hoeffding bound for complex values
Consider the Chernoff-Hoeffding bound, stated as follows: Let $X_1, \dots, X_K$ be i.i.d. real-valued random variables with expectation value
$\mu$ and satisfying $|X_i| \le b$.
Let $\epsilon > 0$. ...
- 298
3
votes
1
answer
553
views
How did the story of Kim-Vu type inequalities continue?
I am interested in the concentration of polynomials of random variables. I have been reading Boucheron, Lugosi, and Massart's "Concentration inequalities" and they give some references. ...
- 233
3
votes
1
answer
431
views
Inequality on the Hellinger distance between Poisson and mixture of Poisson
Let $H$ denote the Hellinger distance; i.e., for two discrete distributions $p,q$ (identified with their pmf) over $\mathbb{N}$,
$$
H(p,q)^2 = \frac{1}{2}\sum_{n=0}^\infty \left(\sqrt{p(n)}-\sqrt{q(n)}...
- 1,372