1
vote
0answers
129 views

bound for $|E[\frac{X}{Y}]−\frac{E[X]}{E[Y]}|$

Is there some bound for $|E[X/Y]−E[X]/E[Y]|$ ? Here $X$ and $Y$ are both summation of a fixed number of Bernoulli random variables and a constant that is >0, which is to guarantee that the denominator ...
2
votes
2answers
85 views

Bounds for the fat tail after trimming the mean?

I am interested in the quantity $$f(X,t) = \int_t^\infty\negthinspace x\ p(x)\ dx,$$ where $p$ is a probability distribution for a positive variable $X$. 1) Does this quantity $f(X,t)$ have a name? ...
4
votes
0answers
114 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: ...
6
votes
1answer
200 views

An inequality for positive definite matrices

Let $K$ and $K^\prime$ positive definite $n \times n$ matrices, such that for all vectors $f \ge 0$ with nonnegative coordinates we have $$\sum_{i,j} K_{ij} f_i f_j \le \sum_{ij} K^\prime_{ij} f_i ...
0
votes
0answers
45 views

Bounds or approximations for the conditional probability of an event involving correlated random variables

Let $\tilde{\gamma_1}, \tilde{\gamma_2}, \ldots, \tilde{\gamma_N}$ be exponential random variables (RVs) that are correlated with each other. Let $\gamma_n$ be another exponential RV that is ...
7
votes
3answers
273 views

Rosenthal like inequality for weak $\mathbb L^p$-norms

Let $p$ be a real number greater than $1$. It is well known (see Hall and Heyde's Martingale limit theory and its applications, Theorem 2.10) that there exists a constant $C_p$ such that if ...
3
votes
0answers
148 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 ...
4
votes
0answers
149 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
1answer
95 views

Deviation bound for the maximum of the norm of Wiener process

Let $W(t)$ be an $n$-dimensional Wiener process. Denote by $\chi_n^2$ a chi-squared random variable with $n$ degrees of freedom. I have recently found the following inequality given without proof: $$ ...
1
vote
1answer
321 views

Prove or disprove $ \int_{0}^{\infty} \int_{-x}^{0} f(x)f(y)dydx > \int_{0}^{\infty} \int_{-\infty}^{-x} f(x)f(y)dydx. $

Consider a symmetric, unimodal distribution $f(x)$ such that $\int_{0}^{\infty} f(x) > 1/2$. I want to prove or disprove the following: $$ \int_{0}^{\infty} \int_{-x}^{0} f(x)f(y)dydx > ...
28
votes
1answer
2k 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 ...
1
vote
1answer
150 views

Distance between the product of marginal distributions and the joint distribution

Given a joint distribution $P(A,B,C)$, we can compute various marginal distributions. Now suppose: \begin{align} P1(A,B,C) &= P(A) P(B) P(C) \\ P2(A,B,C) &= P(A,B) P(C) \\ P3(A,B,C) &= ...
13
votes
4answers
825 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 following ...
33
votes
3answers
2k views

the following inequality is true,but I can't prove it

The inequality is \begin{equation*} \sum_{k=1}^{2d}\left(1-\frac{1}{2d+2-k}\right)\frac{d^k}{k!}>e^d\left(1-\frac{1}{d}\right) \end{equation*} for all integer $d\geq 1$. I use computer to verify ...
1
vote
0answers
105 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 ...
2
votes
0answers
43 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} &=& ...
0
votes
1answer
182 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} ...
3
votes
1answer
220 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 $\epsilon > 0$. ...
9
votes
2answers
480 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 ...
2
votes
1answer
240 views

concentration inequality for averages of dependent random variables

Let $X \in R^n$ be a random vector such that $$P(|X_i| > \epsilon) \le e^{-\epsilon^2}$$ 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| ...
6
votes
2answers
255 views

Inequality involving the weak second moment

I want to ask the following probability inequality: Is it true that for any random variable $X\ge 0$, we have $$ \sup_{t>0}(t\mathbb E(X\mathbf 1_{X\ge t})) \le 2\sup_{t>0}(t^2 \mathbb P(X ...
1
vote
1answer
111 views

Approximating Moment of Sum of RVs

Given $X_i$ are independent random variables. $|X_i| < 1$ $E[X_i] = 0$ $X = \sum_i^n X_i$ $var(X)=\sigma$ Prove: $$ E(X^p)^{1/p} = O(\sqrt{p}\sigma +p)$$ for all even p Things I've tried: ...
9
votes
2answers
449 views

A property of unimodal sequences

It is well-known that $(-1)^j \sum_{i=0}^j (-1)^i\binom{n}{i} \geq 0$. This inequality can be used to prove Bonferroni's inequalities for example. Recently I noticed that a similar inequality applies ...
0
votes
1answer
357 views

Help prove a maximal inequality

Let $X_1,…,X_n$ are exchangeable of random variables, and $n$ is an even number. $S_k=X_1+\dots+X_k$. $M_k=X_{n/2}+\dots+X_{n/2+k}$. I want to prove: $$\Pr(\max_{1 \le k \le n}{|S_k|>\epsilon}) ...
1
vote
1answer
711 views

Probability inequalities

Hi everyone, I am looking for some probability inequalities for sums of unbounded random variables. I would really appreciate it if anyone can provide me some thoughts. My problem is to find an ...
12
votes
1answer
912 views

An $L^0$ Khintchine inequality

Suppose that $\epsilon_1,\epsilon_2,\ldots$ are IID random variables with the Bernoulli distribution $\mathbb{P}(\epsilon_n=\pm1)=1/2$, and $a_1,a_2,\ldots$ is a real sequence with $\sum_na_n^2=1$. ...
3
votes
1answer
874 views

Inequality on probability distributions

I would like to know if the following inequality is satisfied by all probability distributions (or at least some class of probability distributions) for all integer $n \geq 2$. $\int_0^{\infty} ...
1
vote
1answer
319 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 ...
0
votes
1answer
607 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 ...
1
vote
1answer
163 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 ...
3
votes
1answer
396 views

Analogues of the Golden-Thompson inequality

Are there any analogues of the Golden-Thompson Inequality for moment generating functions? The Golden-Thompson Inequality asserts the following: If $A$ and $B$ are two $n \times n$ Hermitian matrices, ...
5
votes
2answers
964 views

Inequality involving probability measures [closed]

I have been working on a problem(alternate minimization) where I want to establish an inequality in which I am stuck. An $\alpha$- parameterized version of the divergence(Kullback-Leibler) takes the ...
1
vote
1answer
635 views

Inequality constraints, probability distributions, and integer partitions

I am interested in the possibility of generating probability distributions using inequality constraints. For instance assume that we have three urns with total of a 10 balls. Thus, $a + b + c = 10$ ...