# Questions tagged [inequality]

Use this tag if you question involves the use of an inequality. The inequality may have probabilistic origins or be a purely mathematical inequality. Do not use for measures of inequality, for instance income inequality. For that use the [diversity] tag.

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### Adjustment needed for multivariate Dvoretzky–Kiefer–Wolfowitz inequality on MCMC samples?

I was thinking about studying bounds on the multivariate empirical cumulative distribution function for samples from an MCMC chains. The multivariate Dvoretzky–Kiefer–Wolfowitz inequality would seem ...
18 views

### When a non-subgaussian vector has subgaussian components

The following remark is from Asymptotically Normal and Efficient Estimation of Covariate-Adjusted Gaussian Graphical Model by Chen et al. 2016: Here, $X$ is a design matrix with iid rows and $X^{(1)}$...
154 views

### Expectation of first of moment of symmetric r.v. in terms of variance

Let $X$ be a symmetric random variable with bounded moments and standard deviation $\sigma$. I want to lower-bound $\mathbb E[|X|]$ in terms of $\sigma$. Here is the formal conjecture; I wonder if ...
1 vote
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### Statistical Test for Some Inequality Condition

Suppose that we have three random variables, named $V_1, V_2$ and $V_3$. Here, I want to test the following inequality: $V_1 \geq V_2, V_3$ Is there any reference dealing with this topic?
197 views

### Problem understanding the intuition behind Slepian's inequality

Slepian's inequality is defined as follows: Let $X\in\mathbb{R}^n$ and $Y\in\mathbb{R}^n$ be centered Gaussian random vectors such that \begin{align} \mathbb{E}X_iX_j&\geq \mathbb{E}Y_iY_j,\quad \...
130 views

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### Holder's inequality in the case of $L_1$ and $L_{\infty}$ norm

I am referring to Wainwright's High-Dimensional Statistics book, where at some point it is deduced that \begin{equation} \frac{w'X\Delta}{n}\leq \left\lVert\frac{w'X}{n}\right\rVert_{\infty}\lVert\...
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### Proof that variance is always greater than or equal to zero

It is common knowledge that: $$\begin{equation}\label{3} Var(X) \geq 0 \end{equation}$$ for every random variable $X$. Despite this, I do not remember seeing a formal proof of this. Is there a proof ...
1 vote
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### When is Jensen's Inequality strict?

For a homework problem, I have to prove that for a random sample $X_1, \ldots, X_n$, drawn from a population with finite variance $\sigma^2$, with sample mean $\bar{x}$ and sample variance $s^2$, that ...
105 views

### Question about property of 2-increasing, grounded function with margins

The question is about what may very well be an obvious detail in the proof of a lemma from the book An Introduction to Copulas by Roger Nelsen. I will state all the relevant results and definitions ...
50 views

### Distributive property of probabilistic inequalities involving random variables on both sides

Can I break down $P(h \geq (A + B)$, given all $A,B,h$ are all random variables. Will the following rule works? $$P[h \geq (A + B)] = P(h\geq A) + P(h\geq B)$$ Actually, in one of my mathematical ...
Suppose we have a continuous random variable $X$ and two continuous functions $f$ and $g$ such that $f(X)$ and $g(X)$ are continuous random variables. Let $p_A$ be the probability density function of ...
The Payley-Zigmund inequality states that for a positive random variable $Z$ the following holds \begin{equation} \operatorname{P}( Z > \theta\operatorname{E}[Z] ) \ge (1-\theta)^2 \frac{\...