# Questions tagged [tail-bound]

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### Conditions for a Random Variable to Satisfy a Probability Bound on Boundary Points

Let $X$ be a random variable supported on $\mathcal{X}\subset\mathbb{R}^{d}$, and let $\mathcal{X}$ be compact. Consider $f$ as the probability density of $X$. My question is: What conditions ...
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### Bound Product of Independent Gaussians

I'm interested in obtaining upper bounds on $$\Pr[\prod_{i\in[n]}|G_i| > x]$$ where $G_i\sim\mathcal{N}(0,1)$ i.i.d, and $[n] := \{0,1,\dots,n-1\}$. The most naive bound is to note that each $G_i$...
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### Concentration bound for weighted sum of Bernoullis

$\{X_i\}_{i=1,\ldots,n}$ are i.i.d. Bernoulli random variables with parameter $p$. Define $$Y = \sum_{i=1}^n a_iX_i$$ where $a_i>0$ are known(non-random) constants. I want an upper bound on the ...
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### Upper Bound on $\mathbb{E}[\frac{1}{1 + X}]$ where $\mathbb{E}[X] = a$ and $0<𝑎<1$

$𝑋$ is a positive random variable (potentially unbounded) with $0 \le \mathbb{E}[X] = a < 1$. Since $\phi(x) = \frac{1}{x}$ is a convex function, we can use Jensen's inequality to derive a lower ...
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### Probability bound on Maxima under random sampling

I have a set $S$ = {$e_1,e_2,..e_{400}$} of 400 elements and a non-linear function $f:2^{(S)}\to[0,1]$ that takes a subset of $S$ and returns a real number in $[0,1]$. I want to compute the subset for ...
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### For random variable $Z=\max_i X_i$, can we bound $\mathbb{E}(Z|Z>\tau)$ with $\mathbb{E}(Z)$

Let $X_1,…,X_n$ be independent, but not necessarily identical, non-negative random variables. Let $Z=\max_i(X_i)$. Fix a real $\tau > 0$. Is there a way to lower bound \mathbb{E}(Z|Z>\tau) >...
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### How to construct a confidence interval for the coefficients of a multivariate regression with dependence between dependent variables?

Suppose we have two linear regression models $y_1=a+bx+\epsilon_1$ where $\mathbb[\epsilon_1]=\sigma_1$ and $y_2=c+dx+\epsilon_2$ where $\mathbb[\epsilon_2]=\sigma_2$. In other words, I am using the ...
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### Bound on the expectation of a function of random variable having a strictly log-concave probability density

let $\theta \in \mathbb{R}^d$ be a random variable having a strictly log-concave probability density function, i.e $$p(\theta) = e^{-\phi(\theta)}$$ where $\phi(\theta)$ is ...
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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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1 vote
Suppose we have the sum $$\sum_{t=2}^{n}\epsilon_{t-1}u_t$$ where $\epsilon_t$ and $u_t$ are both sub-Gaussian variables. Further suppose that while $u_2,\cdots,u_n$ are i....