Questions tagged [tail-bound]
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13
questions
4
votes
2
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Concentration bounds for a mixture of a fixed value and a uniform distribution
Suppose we have a mixed random variable $R= (1-p)X + pU$, where $X$ is just a single value $X \in [-\pi, \pi]$, $0 \leq p \leq 1$, and $U \sim \text{Uniform}[-\pi, \pi]$. I.e. a single trial returns $...
0
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0
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29
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Concentration inequality for sums of independent gamma random variables
I am dealing with the following problem:
Say $X_1, \ldots, X_n$ are independent Gamma random variables, each one having shape and rate parameters $\alpha_i$ and $\beta_i$, respectively. Let $S_n = \...
- 55
7
votes
1
answer
219
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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 $G_i$...
- 173
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0
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32
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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 ...
- 1
6
votes
3
answers
204
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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 ...
- 83
0
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0
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29
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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 ...
- 121
2
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1
answer
70
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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) >...
- 253
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votes
1
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313
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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 ...
- 693
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111
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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
\begin{equation}
p(\theta) = e^{-\phi(\theta)}
\end{equation}
where $\phi(\theta)$ is ...
- 21
3
votes
0
answers
61
views
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 ...
- 1,226
1
vote
0
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Lower bounding the sum of product of two sub-Gaussian variables where one follows an AR(1) process
Suppose we have the sum
\begin{equation}
\sum_{t=2}^{n}\epsilon_{t-1}u_t
\end{equation}
where $\epsilon_t$ and $u_t$ are both sub-Gaussian variables. Further suppose that while $u_2,\cdots,u_n$ are i....
- 1,226
2
votes
1
answer
62
views
A front-loaded Gumbel-like distribution
I'm looking for a distribution that is somewhat like the Gumbel distribution and I was wondering if anyone could help.
The parameters are a positive integer $n$ and real numbers $\mu>0$ and $\sigma&...
- 1,238
2
votes
1
answer
241
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Bounding the tail of sum of discrete distributions (via sub-gaussianity)
I have the following problem: we have a sequence of random variables $Z_1, ..., Z_n$ which are summed up; let's denote $X$ to be their sum. We observe a number $\epsilon$ that is sampled from $X$ and ...
- 512