All Questions
Tagged with pr.probability inequalities
346 questions
0
votes
1
answer
147
views
Trying to prove an inequality (looks similar to entropy)
I'm trying to prove the following inequality (or something similar, up to a constant factor in either side of the inequality):
$$k\cdot\sum_{i=1}^{k}x_{i}\cdot\ln\left(x_{i}\right)\geq\sum_{i=1}^{k}x_{...
0
votes
1
answer
66
views
Bounding parameter satisfying a collection of inequalities
I have a set of equations with some inequality constraints that I expect generally does not have a unique solution.
The equations take the form below:
$$\alpha/N+(1-\alpha)x_1=a_1$$
$$\alpha/N+(1-\...
- 135
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
0
votes
1
answer
199
views
Is this probability inequality true?
This question may be simple, though I'm not managing to find an answer. Let $X$ and $Y$ be two dependent random vectors in in $\mathbb{R}^d$, with joint probability density $\mu(x,y)$ (with respect to ...
- 195
0
votes
1
answer
143
views
Right tail decay of F distribution [closed]
Suppose $X\sim F(a,b)$. Is there any sharp upper bound of the following probability with large $x$?
$$\mathbb{P}(X\geq x)$$
what is the order of the above probability as $x\to+\infty$?
- 1,495
0
votes
2
answers
280
views
Bounds tighter than the additive Chernoff
Additive Chernoff
Suppose $X_1, \ldots, X_n$ are i.i.d. random variables, taking values in $\{0,1\}$. Let $p=\mathrm{E}\left[X_i\right]$ and $\varepsilon>0$.
\begin{gather*}
\operatorname{Pr}\left(\...
- 105
0
votes
1
answer
182
views
Deducing norm concentration from MGF bounds
Suppose that $X$ is a centered, $\mathbf{R}^d$-valued random variable such that for all $t \in \mathbf{R}^d$, there holds the bound $$\log \mathbf{E} \left[ \exp \langle t, X \rangle \right] \leqslant ...
- 801
0
votes
1
answer
84
views
Can we find the following $k$ so that the following inequality holds for asymptotic normal?
Following this question:Can we find such $k$ so that the following inequality holds?.
Consider a sequence of independent $n-$dimensional random vectors $u, v_1, v_2,\dots, v_k$ uniformly distributed ...
- 288
0
votes
1
answer
133
views
How to demonstrate a correlation inequality? [closed]
If there are 3 vectors X, Y, Z of the same length, for any $x_i \in X,y_i \in Y,z_i \in Z$, we have $0<x_i<1,0<y_i<1,0<z_i<1$.
The correlation between Z, Y is greater than between X, ...
- 11
0
votes
1
answer
1k
views
Bounding $L^p$ norms in terms of lower-order $L^q$ norms
Suppose $f,g\in L^q(\Omega)$ ($\Omega\subset \mathbb{R}^n$) for all $1\le q\le p$. Here, $L^p(\Omega)$ is defined with respect to some measure $\mu$ that is absolutely continuous wrt Lebesgue measure. ...
- 710
0
votes
1
answer
126
views
Berg-Kesten-Reimer inequality on infinite spaces?
See this link for a description of the van den Berg-Kesten-Reimer inequality. How important is the assumption that $\Omega_i$ are finite spaces?
When Berg-Kesten state the inequality in their 1985 ...
- 941
0
votes
1
answer
290
views
Inequality of Partial Taylor Series
Hi,
For a given $\theta < 1$, and $N$ a positive integer, I am trying to find an $x > 0$ (preferably the smallest such $x$) such that the following inequality holds:
$$\sum_{k=0}^{N} \frac{x^k}...
- 51
0
votes
1
answer
115
views
Approximation for an expectation expression
Let $\mathbf{x} \in \mathbb{C}^M$ is an unknown distributed random vector (certainly not gaussian), and matrix $\mathbf{A}\in \mathbb{C}^{M \times M}$ which is fix (known). Also, assume we know the ...
- 25
0
votes
1
answer
101
views
Can we show that for every $\delta>0$, there exist constants $\alpha>0, \beta>0$ so that the following inequality holds with high probability?
Consider two $n-$dimensional random vectors $u$ and $v$ uniformly distributed on the sphere. Define $X_n :=u\cdot v$. Note that as $n\to \infty$, $\sqrt{n}X_n \to N(0,1)$ as $n\to \infty$. Fix $\...
- 288
0
votes
1
answer
188
views
Equality cases in a certain case of Jensen's inequality
Suppose that $Y$ is an independent copy of a random variable (r.v.) $X$ with a zero-mean nondegenerate distribution. Is there a non-tautological, preferably simple characterization of the cases when
$$...
- 128k
0
votes
1
answer
447
views
Properties of $l_q$-balls
For a given $q\in (0,1]$, define the $l_q$-ball as
$$\mathbb{B}_q(R_q)\mathrel{:=}\left\{\theta\in\mathbb{R}^d\,\middle\vert\,\sum_{j=1}^d \lvert\theta_j\rvert^q\leq R_q \right\}. $$
For a given ...
- 100
0
votes
1
answer
966
views
Bound the norm of sum of random vector that generated from standard basis
I have a question like this:
Consider $N$ samples $X_1, X_2, ..., X_N$ that uniformly random generated from standard basis $\{e_i, i=1,2,...,d\}$, i.e. $(1,0,0,\cdots,0),(0,1,0,\cdots,0),(0,0,1,0,\...
- 25
0
votes
1
answer
159
views
Best bounds on the integral of an increasing function
The following question, somewhat edited here, was asked and then closed at The best bound of the integral of a nondecreasing real function in a closed interval.
Let $F\colon[0,1]\to[0,1]$ be a ...
- 128k
0
votes
1
answer
133
views
Projection onto manifold of Gaussian measures by "trunction" of moments
Let $\mathcal{P}_2(\mathbb{R}^n)$ be the set of Borel probability measures on $\mathbb{R}^n$ with finite mean and variance; in the sense that
$$
\int_{x \in \mathbb{R}^n} \|x\|^p d\mathbb{P}(x) < \...
- 5,405
0
votes
1
answer
340
views
Expectation of the ratio of two discrete random variables with combinatorial constraints
We are given a set $S=\{1, 2, \ldots, n\}$ where $n\gg 1$, and for all indices $1\le i \le n$, $i$ is associated with a real value $\alpha_i\!\cdot\! v_i$, where $\alpha_i\in[0,1]$ and $v_i\in(0,1]$.
...
- 1,677
0
votes
2
answers
124
views
Bounding $E[\|\Sigma^{-1/2}(X-\mu)\|_2^3]$ for 2-dimensional Bernoulli
Let $X\in\{0,1\}^2$ have mean $\mu=\left[\begin{smallmatrix}p_1\\p_2\end{smallmatrix}\right]$ and $\Pr[X_1 = X_2 = 1] = p\le \min\{p_1,p_2\}$.
(Note we must have $1-p_1-p_2+p\ge 0$ for the ...
0
votes
1
answer
339
views
Expectations, double integrals and Jensen's inequality
$\def\anonfunc#1{#1(\cdot)}$Consider two random variables distributed $v\sim \anonfunc G$ and
$c \sim \anonfunc F$ with pdfs $\anonfunc g$ and $\anonfunc f$. Let the supports of $c$ and
$v$ be $[x,y]$....
- 13
0
votes
1
answer
503
views
Asymptotics of a 1D integral, or the orthant probability of an equicorrelated random Gaussian vector
Problem: Let $\phi(x)$ be the normal probability density function (pdf), and $\Phi(x)$ the normal cumulative distribution (cdf). I'm interested in the asymptotic behavior of the following integral
$I(...
- 968
0
votes
0
answers
38
views
Bounding the error of a truncated moment problem
Let $\{x_{i}\}_{i=1}^{\infty}$ be a non-increasing sequence of non-negative real numbers, and let $\{y_{j}\}_{j=1}^{B}$ be a non-increasing sequence of non-negative real numbers, where $B$ is a finite ...
- 433
0
votes
0
answers
57
views
Class of covariance matrices invariant under permutations
I am reading a paper on covariance matrix estimation, and in this paper is introduced a class of covariance matrices:
\begin{equation}
U(q, c_0(p),M)=\{\Sigma: \sigma_{ii}\leq M,\quad \max_j\sum_{j=1}^...
- 1
0
votes
1
answer
158
views
Techniques for bounding the operator norm of the expectation of random matrix?
Let $\mu$ be a distribution on the unit sphere in $\mathbb{R}^n$. Let $u \sim \mu$ and consider the random matrix
$$
A = I_n - uu^T.
$$
Question: What techniques are available to provide (reasonably ...
- 420
0
votes
0
answers
153
views
Inequalities on the distribution of the maximum of the normalized sum $\max_{k = 1,\dots,n} \frac{|S_k|}{\sqrt{k}}$
Let $\{X_i\}_{i\in\mathbb{N}}$ be i.i.d. random variables with $\mathbb{E}(X) = 0$,$\mathbb{E}(X^2) = \sigma^2$ and finite moments. Let $S_k = \sum_{i = 1}^{k} X_i$ and consider the normalized ...
- 63
0
votes
0
answers
87
views
Comparison between the expected values of the inverse of the CDF of binomial-distributed random variables
Let us denote with $F(x;j,\mu)$ the cdf of a Binomial distributed random variable with $j$ trial with success probability $\mu$ considered in $x$, and let $f(x;j,\mu)$ be the pmf. Defining $0\leq \...
0
votes
1
answer
92
views
Does point process ordering ever imply conditional intensity ordering?
Let $N$ and $N'$ be regular/non-explosive point processes on $[0,\infty)$. I will take the view that these are collections of random arrival times: $N=(t_n)_{n\in\mathbb N}$ and $N'=(t_n')_{n\in\...
- 215
0
votes
0
answers
91
views
Some new questions on Rademacher complexity
For $A\subset R^n$,$A=(a_1,a_2,\dots, a_n)$, $\sigma_i$ are Rademacher random variable.
Is $|\mathbb{E}_\sigma \inf_{a\in A}\sum_{i=1}^n\sigma_ia_i| \le |\mathbb{E}_\sigma \sup_{a\in A}\sum_{i=1}^n\...
- 185
0
votes
0
answers
78
views
Kernel density estimation is sub-gaussian
Let $X_1, ..., X_n$ be i.i.d. samples drawn from a pdf $f(x)$ on the real line. The kernel density estimator is defined as follows,
$$\hat{f_n}(x) = \frac{1}{nh}\sum_1^n K(\frac{x-X_k}{h})$$
where $K:\...
- 81
0
votes
0
answers
34
views
Inequalities for generalized variance
Let $(X, \mu)$ be a measured space with $\mu(X) = 1$.
Given $\phi \in L^\infty(X, \mu)$, $\phi > 0$, let me define, for $\alpha \geq 1$, $\beta > 0$, the quantity
$$
I(\alpha, \beta) = \left(\...
- 1,263
0
votes
0
answers
293
views
Can we prove the following anti-concentration inequality of polynomials of square Gaussian variables?
Following this question Anti-concentration of Gaussian quadratic form. We have the following inequality:
Let $X_1,\dots,X_n$ denote i.i.d. standard Gaussian random variables. For every $\epsilon>0$...
- 288
0
votes
0
answers
173
views
Lemma 3.10 of paper 'Periodic nonlinear Schrodinger Equation and Invariant measure' by J.Bourgain
I am reading a paper 'Periodic nonlinear Schrodinger Equation and Invariant measure' by J.Bourgain.
And I have a questions in the proof of lemma 3.10.
Please click the paper title for the link.
The ...
- 239
0
votes
0
answers
106
views
Upper bounding the sum with hypergeometric and binomial probabilities
Could you please help me upper bound this tricky expression:
$$P(A)=\sum_{i=0}^n{\left( 1 - \dfrac{\binom kq \binom {n-k}{i-q}}{\binom {n}{i}} \right)}^I \binom ni p^i {(1-p)}^{n-i}$$.
So far I only ...
- 1
0
votes
0
answers
166
views
Lower bound on probability of sum of independent random variables
I'm interested in estimating the following quantity
\begin{equation}\lim_{k \to \infty} \sum \limits_{i=1}^l \sum \limits_{l_1 = 0}^{i} {{nk-k}\choose{l_1}} \Big( \frac{1}{k}\Big)^{l_1} \Big( \frac{...
- 237
0
votes
0
answers
112
views
On certain integrals of exponential functions with respect to Gaussian measures
I have questions about the integral
$$F(a,b,c)=\sqrt{\frac{a}{\pi}}\int_{0}^{\infty}e^{-bx^4+cx^3-ax^2}dx$$
for $a,b,c>0$.
What is the asymptotic behavior of $F(a,b,c)$ for small $a,b,c$? In ...
- 505
0
votes
0
answers
250
views
Can we make two random variables independent at a low cost?
Let $X$ and $Y$ be two discrete random variables with joint probability mass function $p(x,y)$ such that
$$\|p(x,y)-p(x)p(y)\|_1=\sum_{x\in\mathcal{X},y\in\mathcal{Y}}|p(x,y)-p(x)p(y)|\leq\epsilon$$
...
- 287
0
votes
0
answers
141
views
Effect of partitioning the realizations of random variables on the total variation distance?
Let $X$ and $Y$ be two random variables with joint pmf $p(x,y)=p(x)\cdot p(y|x)$ and $X$ has uniform distribution. Also assume that the following relation is satisfied:
\begin{align}
\lVert p(y|x)-p(y)...
- 287
0
votes
0
answers
124
views
Maximal inequality for Markov process
For a Markov process $\{X_n\}$ is there any inequality available for
$$ E[\sup_{0 \leq n \leq k} X_{n}]$$
in terms of moments of $E[X_n], 0 \leq n \leq k$
- 317
0
votes
0
answers
107
views
Conditional version of martingale difference concentration inequality
Let $M_n$ be a $\mathscr{F}_n=\sigma(\eta_m,\theta_m, m\leq n)$ measurable martingale difference sequence. Then is it possible to find a exponential tail bound for the following
$$P(|M_{n+1}| > u|\...
- 317
-1
votes
1
answer
159
views
Bounding difference of two mutual information with different distributions on random variables
Let $A$, $B$ and $C$ be three random variables and $p_{A,B,C}=p_Ap_Bp_{C|A,B}$ and $q_{A,B,C}=p_Aq_{B|A}p_{C|A,B}$ be two distributions on them. Then, we can conclude that?
\begin{align*}
\lvert I_p(A,...
- 287
-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
votes
1
answer
283
views
Lowerbounding expectation value of binomial tail
I'm trying to find a lower bound for the following expression for $q\ge p$:
$$f(q,p,n) := \sum_{v=0}^n \sum_{k=v}^n \binom{n}{v} \binom{n}{k}q^v(1-q)^{n-v}p^k(1-p)^{n-k}.$$
It can be thought of as the ...
- 702
-1
votes
1
answer
76
views
Transforming random variables for having good property?
For arbitrary functions $A$ and $B$ and independent random variables $X$ and $Y$, assume that
\begin{align}
\Omega&\triangleq \{(x,y): A(x,y)=1\},\\
\Lambda&\triangleq \{x: B(x)=1\}.
\end{...
- 287
-2
votes
1
answer
224
views
using jensen's inequality
Suppose we have an expression
f(x, h(x,y)), for some function f and h, and x, y are random variables,
now we know that the function f(a, b) is concave w.r.t. a for given b. Can we use Jensen's ...