# Questions tagged [probability-inequalities]

Probability Inequalities are useful for bounding quantities that might otherwise be hard to compute. A related concept is a concentration inequality, which specifically provides bounds on how far a random variable deviates from some value.

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### Does Cramer's condition imply strong mixing?

In Theorem 1.4 of D. Bosq the Cramer's condition is a prerequisite for the tail bound of sum of dependent variables. The Theorem is as follows: Let $(X_t,t\in\mathbb{Z})$ be a zero-mean real-valued ...
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### How to bound sub-exponential variables?

I am trying to understand bounding sub-exponential variables. Suppose for $t=2,\cdots,n$, we have \begin{equation} u_{t-1}u_t \end{equation} where $u_t$ and $u_{t-1}$ are sub-Gaussian. We know that ...
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### Does Chebyshev's inequality sacrifice a little power for simplicity?

I followed the proof in Chapter 2 of Ross Introduction to probability and statistics for Engineers. As follows; Chebyshev inequality For any K > 1 ( for K< 1 still holds for 0<K_1 but trivial ...
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### Would this extension of Khintchine's inequality be correct?

This seems trivial, yet I have to make sure it is indeed correct. Referring to Roman Vershynin's High-Dimensional Probability book, the Khintchine's inequality (Exercise 2.6.5, page 27) is defined as ...
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### What are the arguments in getting from the theorem of Vapnik & Chervonenkis (1971) to the common form seen in Devroye, Györfi & Lugosi (1996)?

Context. The theorem below is attributed to "Vapnik, V., and Chervonenkis, A. (1971). On the uniform convergence of relative frequencies of events to their probabilities. Theory of Probability ...
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### Markov's Inequality and convergence in probability

Theorem: (Markov's Inequality): Let x be a non-negative random variable. Then, for all $b >0$ $P[x \geq b] \leq \frac{E(x)}{b}$. ################# Suppose that $x_{t}$ is a non-negative random ...
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### Corollary of Hoeffding’s Inequality

Question I am not from a statistics background. I came across the following corollary of Hoeffding’s Inequality and couldn't find the derivation or proof for it. Could anyone please share some ...
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### 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 \...
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### Does maximizing Jensen–Shannon divergence maximize Kullback–Leibler divergence?

Does maximizing the Jensen–Shannon divergence $D_{\mathrm{JS}}(P \parallel Q)$ maximize the Kullback–Leibler divergence $D_{\mathrm{KL}}(P \parallel Q)$? If so, I'd like to be able to show that it ...
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### Upper Bound for 2nd Raw Moment of Positive Random Variable

Let $X$ be a random variable with support $(0,\infty)$. All I know about $X$ is the support, finite higher moments, and $\mathbb{E}(X)=\mu$. I am trying to come up with a more tractable upper bound ...