Questions tagged [distribution-tails]

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27 views

How do we proof that Gaussian Tails is interesting area

I have a programming problem that i solve using Gaussian Distribution. The problem is outlier detection. I use the uncertainty of the data, calculated from the classifier confidence, based on the ...
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1answer
34 views

For what kind of random vectors do we have $\sup_{p \ge 1}\|X\|_p < \infty$?

Let $X$ be a random vector on $\mathbb R^m$ (assumed to have zero mean, for simplicity). For $p \in [1,\infty)$, define $e_p(X):=\mathbb E\sum_{j=1}^m|X_j|^p \in [0,\infty]$. Finally, define $\|X\|_p \...
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1answer
14 views

Upper bound lower tail of binomial tail

I have a random variable $Y\sim Bin(n,p)$ and I know that $E[Y]<20$. Now I want to compute $P[Y\leq60]$. I try to compute it by firstly compute $P[Y > 60]$ using Chernoff bound, i.e. $P[Y\geq (1+...
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0answers
36 views

If $X$ is a nonnegative $\sigma$-subGaussian random variable with $P(X=0)\ge p$, what is a good upper bound for $P(X \ge h)$?

Let $X$ be a nonnegative random variable and let $\sigma \in [0,\infty)$ and $p \in (0,1)$ such that (1) $P(X=0) \ge p$ (2) $Var(X) \le \sigma^2$ For $h \ge 0$, define $c_X(h):=P(X \ge h)$. The ...
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8 views

On non-centered subGaussian random variables

Let's say that random variable $X$ is $\sigma$-subGaussian about a point $c \in \mathbb R$ if $\mathbb E[\Psi_2(\sigma |X-c|)] \le 1$, where $\Psi_2(t):=e^{t^2}-1$. Now, suppose the random variable $...
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1answer
25 views

Simple upper bound for $t(2\Phi(\alpha/t) - 1)$, where $\alpha > 0$ and $t \in (0, 1)$

Let $\alpha > 0$ and $ t \in (0, 1)$. For simplicity, take $\alpha=1$. Let $\Phi$ be the normal cumulative distribution function. Of course, the core of the problem is the term $t\Phi(\alpha/t)$. ...
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7 views

Tail Bound for Squared Noncentered Gaussian

I am trying to upper bound the event $$P((x-\lambda)^2 < c)$$ where $x \sim N(\mu, 1)$, $c > 0$, and $\lambda \in [0, 1]$. While I am aware of chi-squared tail bounds for standard squared ...
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0answers
11 views

Is stochastic integral subgaussian

Consider the solution $X$ of the stochastic differential equation $$ \mathrm{d}X_t = \sigma(X_t) \mathrm{d}W_t, $$ where $W$ is a Wiener process and assume the standard growth conditions ($X$ is a ...
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1answer
26 views

Equivalent definitions of a heavy-tailed distribution?

$\int_{-\infty}^{\infty}e^{tx}dF(x)=\infty \text{ for all } t>0 \Leftrightarrow \lim_{x \rightarrow \infty}e^{tx}\mathbb{P}[X>x]= \infty \text{ for all }t >0$. I proved the $\Leftarrow $ ...
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1answer
29 views

Finding dominating terms

1) Let $f(t) = \frac{exp(-\sqrt{t})}{\sqrt{t}}, t>0$. Show that $\underset{t \rightarrow \infty}{lim} f(t) \sim O(exp(-\sqrt{t}))$. 2) Let $f(t) = exp(-1/t)\frac{1}{t^2}, t>0$. Show that $\...
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1answer
21 views

concentration of function of Chi squared random variables

Let $X, Y$ be iid Chi-squared random variables with parameter $k$ and consider, \begin{align*} Z = \frac{X-Y}{X+Y}. \end{align*} I am after bounds for the tail: $\mathbb{P}[ |Z| > t ]$. I know the ...
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0answers
15 views

Tail bounds for sum of rectified gaussian variables

Is there a concentration inequality for the rectified gaussian distribution? For example, let $u_i$, $i=1,...,n$ be a sequence of i.i.d. standard normal random variables. Is it possible to bound $P\...
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8 views

tail distribution equals to its n-scaled tail distribution raised to power n

The question requires to find all probability measures P on $[0,\infty), B_{[0,\infty)}$ satisfying the property: for every $n\in \mathbb{N}$ and every sequence X1,...,Xn of independent RVs with ...
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13 views

Tail distribution subject to a condition

Consider a sequence of iid scalar Gaussian random variables $\{x_{i}\}_{i=1}^{i=m}$ with mean $\mu$ and standard deviation $\sigma$, i.e. $x\sim\mathcal{N}(0,\sigma^{2})$. Denote by $\operatorname{Lap}...
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12 views

Parameters of product of scalar sub-gaussian independent random variables

I know that the product of two sub-gaussian random variables is a sub-exponential random variable and it's easy to show that. What I want is to find parameters of resulting sub-exponential R.V. namely ...
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28 views

Comparing Tail Probabilities of Sum of Dominated Random Variables

I have been thinking about the following problem for quite a while, but did not get much progress of how to approach it: Assume that $X$, $Y$ are independent random variables with N(0, 1) ...
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22 views

Tail bound for integral product of two functions

(I am reformulating this question which did not get much attention) Suppose I have a well defined probability density function $f$ over $(0,\infty)$. I would like to find an upper bound for the tail ...
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0answers
53 views

Regular tails of a probability distribution

In a text I read, the distribution of a random variable $X$ has so-called regular tails, if the following property holds: The distribution of $X$ belongs without centering to the domain of attraction ...
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24 views

Tail bound on difference of shifted binomials (generalization)

I have a post Tail bound on difference of shifted binomials answered before and now I want to consider I slightly generalization of it which can't be solved using the methods in the previous thread. ...
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1answer
52 views

Tail bound on difference of shifted binomials

I would like to derive an upper bound on $\mathsf{Pr}\{(\frac{X}{n}-\frac{1}{2})^2 \leq (\frac{Y}{n}-\frac{1}{2})^2\}$ where $X,Y$ are independent and $X\sim$ Bin($n,p$) where $p\neq \frac{1}{2}$, and ...
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0answers
26 views

Hoeffding-Type Bounds for Noncentered Variables

Hoeffding's Tail Bound is well-known for subgaussian variables. It can be written in the following way: Assume $X_i$ for $1\leq i\leq n$ satisfies: $$ \mathbb{P}(|X_i-\mu|>t)\leq 2\exp\left(\frac{-...
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1answer
55 views

One-sided heavy tailed distribution

I seek a univariate distribution with analytically expressible density function that approximates (a vertically scaled version of) the standard normal distribution around the origin, but with a heavy ...
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1answer
35 views

Error in calculation of second moment

What is wrong with the following method of calculating second moments using integration by parts $$\int_0^\infty x^2f(x)dx=[x^2F(x)]^\infty_0-2\int^\infty_0xF(x)dx=\infty-\infty$$ On the other hand ...
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1answer
33 views

Finding second moment given $P(X>t)$

I know that for nonnegative continuous R.V, $E[X]=\int_0^\infty P(X>t)dt$. Is there a formula for $E[X^2]$ when we only have $P(X>t)$?
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90 views

Bounds on the Chi-Square Distribution Tail, need upper bound for probability

I have the following probability bound I want to prove but don't know what bound the author is using. say $N_k \sim \mathcal{N}(0,\sigma^2)$ Then, we are interested in upper bounding the probability:...
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26 views

Prove that for Bernoulli distribution, the normal-based CI is shorter than the one given by Hoeffding's inequality

This is a result claimed in Example 6.17 of All of Statistics. Basically, we need to prove that: $$ z_{\alpha/2} \sqrt{\frac{\hat p_n (1- \hat p_n)}{n}} < \sqrt{\frac 1 {2n} \log(\frac 2 \alpha)} ...
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23 views

a probability bound related to conditional 1-subgaussian variables

I am taking an intermediate probability course and this problem is assigned as a challenge (no ready solution). It goes like this: $\{z_i\}_{i=1}^{N}$ is a conditional 1-subGaussian sequence. $d$ is ...
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42 views

whats is the Chernoff bound for an uniform distribution variable

Let $X$ be an uniform random variable between $[0,K]$. I want to find an upper bound for a state that $X> \alpha$. So I have used blow Chernoff bound: \begin{align*} \mathbb{P}(X\ge \alpha)&\...
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1answer
59 views

Tail bound for the maximum of a reduced-rank Gaussian random vector

Suppose that $X_i, i=1,...,n$ are Gaussian random variables, each with mean equal to $0$ and variance equal to $1$. However, suppose that their covariance matrix is reduced rank (assume that such a ...
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0answers
36 views

Concentration of centered variable, based on un-centered moments and tails

For a random variable $X$ and positive integers $m,k$ the tail bound is: $$P(|X|>t) \le \exp(- m t^{2/m} )$$ And for any $k\in\mathbb{N}^{+}$ the moments are given by: $$E X^{2k+1} = 0$$ $$E X^{2k} ...
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0answers
35 views

Why is the gaussian part of this Bernstein-type inequality not trivial?

The following Bernstein-type inequality can be found in Introduction to the non-asymptotic analysis of random matrices. Theorem Let $X_1,\ldots X_n$ be mean-zero sub-exponential random variables with $...
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0answers
64 views

Probability density function with exponential tail

While analyzing data regarding the pore size distribution of a disordered material (see for example S. Bhattacharya and K.E. Gubbins, Langmuir 2006 22 18 7726-7731), I found a probability density (...
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1answer
22 views

Limit distribution of the second maximum for standard normal random variates

Suppose $X_1, \dots, X_n$ are independent, standard normal random variables, and let $X_{(1)} \leq \dots \leq X_{(n)}$ denote their order statistics. I am interested in the joint distribution of $X_{(...
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1answer
43 views

p-norm inequality for two random variables

I read the following result in a book, however I believe that there is a mistake in the proof. Do you know of any book that proves this result, or do you have an idea on how to prove it? Let $X$ and $...
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1answer
145 views

Concentration bound for square of sum of Rademacher random varialbes

For $i\in[n]$ let us define iid random variables $u_i\sim \text{Unif}(\{-1,1\})$, i.e. $Pr(u_i=1)=Pr(u_i=-1)=1/2$. Also let $X=\sum_{i=1}^n u_i$. My question is, how can we compute the concentration ...
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1answer
73 views

Lower bound for concentration probability of Rademacher sum

Suppose that $\epsilon_1, \dots, \epsilon_n$ are $n$, iid Rademacher random variables (equally likely to be $+1, -1$). I would like to know what the tightest result is for the following ...
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68 views

How big are the exponential moments of a truncated normal distribution?

Given a random variable $X$ valued on $[-1,1]$ with mean zero. We can use say Hoeffding's Lemma to get $$ \mathbb E[e^{\lambda X}] \le e^{\lambda^2/2}$$ I believe this bound cannot be improved much ...
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15 views

Intuition of TailVaR

As per the actuarial guide I have called the CMP - from Acted - tailVaR is the expected loss in excess of the benchmark value L. I don't really get that, so I tried splitting the equation into: $...
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1answer
108 views

Tail bound for sum of random variables satisfying subgaussian upper tail bound

So suppose you have a collection of random variables $X_1, \cdots X_n$ that are iid and they all satisfy the tail bound $$ P(X_i-L>u)\leq \exp(-\frac{u^2}{2\sigma^2})$$ for all $u>0$. Is it ...
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0answers
35 views

Upper tail bound on product of 2 independent chi^2 or Gamma variables

We have two independent Gaussian vectors: $X \sim N(0,I_k)$, $Y \sim N(0,I_r)$ independent of $X$. I'm looking for a tail bound on the product of their square norms. $Pr_{X,Y}[\|X\|^2\cdot \|Y\|^2 &...
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1answer
122 views

Negatively correlated random variables Chebyshev bound?

It is quite well known that the Chernoff bound applies to negatively correlated binary random variables (see e.g. Theorem 1.16 here). Does there exist a reference for Chebyshev-type bound for ...
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1answer
172 views

$\Pr[\sum_i X_i^2 Y_i^2\ge t]$, Chernoff bound for sum of pairs of squared Normal random variables

I'm interested in finding tail bound for $\sum_{i=1}^k X_i^2 Y_i^2$, where $X_i$ and $Y_i$ are independent standard normal random variables. It should be roughly as tight as the standard Chernoff ...
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0answers
30 views

Tail Bound for normal random variable.

I want to show that if $g \ \sim N(0,1)$, for all $t>0$ we have $P(g\geq t)\leq e^{-t^2/2}$. My solution: Let $\lambda>0$. $P(g\geq t)=P(e^{\lambda g}\geq e^{\lambda t})\leq \frac{E[e^{\lambda ...
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0answers
66 views

Tail probability of sum of order statistics of distance from point to a set

Let $P$ be a distribution on a metric space $(\mathcal X, d)$. For a point $x \in \mathcal X$ and a Borel $B \subseteq \mathcal X$, let $d(x,B) := \inf_{y \in B}d(x,y)$ be the distance of $x$ from $B$....
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3answers
115 views

Inequality for the maximum of the absolute value of two normal distributed random-variables

I would like to show following statement: For $M\geq 2,\ X_1,\dots,X_M\sim^{iid}\mathcal{N}(0,1)$ independent, it holds $P(\max_{i=1,\dots,M}\lvert X_i\rvert\geq y)\leq Me^{-y^2/2}$. I think it ...
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1answer
68 views

Deduce upper bound of variance from Chernoff-type tail bound

For a random variable $X$, I have a large deviation inequality of the form \begin{equation} P(|X-\mathbb EX|\geq r)\leq ce^{-\alpha r}\,. \end{equation} Consider a sample mean $S_n=\frac{1}{n}(X_1+...
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1answer
219 views

Tight upper tail bound for Normal distribution

The following is a well-known chain of inequalities for the tail of the normal distribution when $a = 1:$ $$ \Big(\frac{1}{x} - \frac{a}{x^3}\Big) \phi(x) \leq \Big(\frac{x}{a + x^2}\Big) \phi(x) \...
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0answers
235 views

Large deviation upper bound for Chi-squared random variable

Let $X \sim \chi^2_n$ random variable. I am looking for a large deviation upper bound for $X$. The answer here, says that Since you said that you're looking for an upper bound, it should also be ...
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2answers
131 views

Probability of Overlap of Sample Subjects from Two Groups 4 SDs Apart

This question came up a little while ago but unfortunately was put on hold. However, I found it intriguing as I had never come across a question like this before. There are $2$ groups of $30$ people ...
4
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1answer
678 views

Value at Risk: Coherent risk measure for normal distribution

I know that there are cases where VaR does not satisfy the subadditivity property (coherent risk measure properties) for coherent risk measures. But I would like to show that in the case of normal ...