Questions tagged [distribution-tails]
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132
questions
0
votes
1answer
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 ...
0
votes
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 \...
0
votes
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+...
2
votes
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 ...
0
votes
0answers
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 $...
0
votes
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 $t\Phi(\alpha/t)$. ...
0
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0answers
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 ...
1
vote
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 ...
0
votes
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 $ ...
0
votes
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 $\...
0
votes
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 ...
0
votes
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\...
0
votes
0answers
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 ...
0
votes
0answers
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}...
0
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0answers
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 ...
1
vote
0answers
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) ...
0
votes
0answers
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 ...
2
votes
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 ...
1
vote
0answers
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.
...
3
votes
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 ...
2
votes
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:
$$
\mathbb{P}(|X_i-\mu|>t)\leq 2\exp\left(\frac{-...
1
vote
1answer
55 views
One-sided heavy tailed distribution
I seek a univariate distribution with analytically expressible density function that approximates (a vertically scaled version of) the standard normal distribution around the origin, but with a heavy ...
1
vote
1answer
35 views
Error in calculation of second moment
What is wrong with the following method of calculating second moments using integration by parts
$$\int_0^\infty x^2f(x)dx=[x^2F(x)]^\infty_0-2\int^\infty_0xF(x)dx=\infty-\infty$$
On the other hand ...
0
votes
1answer
33 views
Finding second moment given $P(X>t)$
I know that for nonnegative continuous R.V, $E[X]=\int_0^\infty P(X>t)dt$. Is there a formula for $E[X^2]$ when we only have $P(X>t)$?
0
votes
0answers
90 views
Bounds on the Chi-Square Distribution Tail, need upper bound for probability
I have the following probability bound I want to prove but don't know what bound the author is using.
say $N_k \sim \mathcal{N}(0,\sigma^2)$
Then, we are interested in upper bounding the probability:...
3
votes
0answers
26 views
Prove that for Bernoulli distribution, the normal-based CI is shorter than the one given by Hoeffding's inequality
This is a result claimed in Example 6.17 of All of Statistics.
Basically, we need to prove that:
$$
z_{\alpha/2} \sqrt{\frac{\hat p_n (1- \hat p_n)}{n}} < \sqrt{\frac 1 {2n} \log(\frac 2 \alpha)}
...
0
votes
0answers
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 ...
0
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0answers
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)&\...
0
votes
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 ...
1
vote
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:
$$E X^{2k+1} = 0$$
$$E X^{2k} ...
2
votes
0answers
35 views
Why is the gaussian part of this Bernstein-type inequality not trivial?
The following Bernstein-type inequality can be found in Introduction to the non-asymptotic analysis of random
matrices.
Theorem Let $X_1,\ldots X_n$ be mean-zero sub-exponential random variables with $...
2
votes
0answers
64 views
Probability density function with exponential tail
While analyzing data regarding the pore size distribution of a disordered material (see for example S. Bhattacharya and K.E. Gubbins, Langmuir 2006 22 18 7726-7731), I found a probability density (...
0
votes
1answer
22 views
Limit distribution of the second maximum for standard normal random variates
Suppose $X_1, \dots, X_n$ are independent, standard normal random variables, and let $X_{(1)} \leq \dots \leq X_{(n)}$ denote their order statistics. I am interested in the joint distribution of $X_{(...
1
vote
1answer
43 views
p-norm inequality for two random variables
I read the following result in a book, however I believe that there is a mistake in the proof. Do you know of any book that proves this result, or do you have an idea on how to prove it?
Let $X$ and $...
1
vote
1answer
145 views
Concentration bound for square of sum of Rademacher random varialbes
For $i\in[n]$ let us define iid random variables $u_i\sim \text{Unif}(\{-1,1\})$, i.e. $Pr(u_i=1)=Pr(u_i=-1)=1/2$. Also let $X=\sum_{i=1}^n u_i$. My question is, how can we compute the concentration ...
1
vote
1answer
73 views
Lower bound for concentration probability of Rademacher sum
Suppose that $\epsilon_1, \dots, \epsilon_n$ are $n$, iid Rademacher random variables (equally likely to be $+1, -1$).
I would like to know what the tightest result is for the following ...
2
votes
0answers
68 views
How big are the exponential moments of a truncated normal distribution?
Given a random variable $X$ valued on $[-1,1]$ with mean zero. We can use say Hoeffding's Lemma to get $$ \mathbb E[e^{\lambda X}] \le e^{\lambda^2/2}$$
I believe this bound cannot be improved much ...
0
votes
0answers
15 views
Intuition of TailVaR
As per the actuarial guide I have called the CMP - from Acted - tailVaR is the expected loss in excess of the benchmark value L.
I don't really get that, so I tried splitting the equation into:
$...
1
vote
1answer
108 views
Tail bound for sum of random variables satisfying subgaussian upper tail bound
So suppose you have a collection of random variables $X_1, \cdots X_n$ that are iid and they all satisfy the tail bound
$$ P(X_i-L>u)\leq \exp(-\frac{u^2}{2\sigma^2})$$
for all $u>0$. Is it ...
0
votes
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 &...
2
votes
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 ...
5
votes
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 ...
1
vote
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 ...
3
votes
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$....
1
vote
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 ...
1
vote
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+...
2
votes
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) \...
2
votes
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 ...
0
votes
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
votes
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 ...
