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Questions tagged [distribution-tails]

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31 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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0answers
58 views

Tails for the random variable $X^2-Y^2$

I would like to calculate the following probability $$ \mathbb{P}(X^2-Y^2 \geq t) \geq ? $$ where $X= \lvert \mathbf{a}^{H}\mathbf{u} \rvert$, and $Y= \lvert \mathbf{a}^{H}\mathbf{v} \rvert$, for some ...
2
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1answer
48 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 ...
3
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1answer
80 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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25 views

Concentration/Tail Bounds for a vector of Poisson r.v.

Let $X$ be $n$-dimensional s.t. $X_j\sim Poiss(\lambda_j)$. The components are independent, but the rates are different. I am interested in bounds for $\Pr(||X-\lambda||\geq y)$, where $\lambda$ is ...
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0answers
17 views

Hypothesis testing: two tailed vs one tailed test paradox

Let $H_0$ = population mean is 50 $H_{alt1}$ = population mean is less than 50 $H_{alt2}$ = population mean is not equal to 50 for $H_{alt1}$, we do one tailed test, for $H_{alt2}$ we do two ...
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12 views

A question in random matrices

In page 78 of "An Introduction to Matrix Concentration Inequalities", it is written that if $Z$ be a random matrix, I can't understand why $$\mathbb{E}||Z||^2=\mathbb{E}\text{max}\{||ZZ^*||,||Z^*Z||\}...
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0answers
24 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
61 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
92 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
40 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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0answers
62 views

Asymptotic variance of Normal vs. Lognormal distributions truncated to a finite interval in the upper tail

How to prove analytically the following curiosities? Claim. Let $X\sim \text{Normal}(\mu,\sigma^2)$, $Y:=e^X$, and let $U\sim\text{Uniform}$ on an interval of width $w$. Then for any $w>0$, $$...
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1answer
66 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
121 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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0answers
33 views

Motivating Pearson type IV distributions

Note: this question is concerned with univariate distributions. The Pearson type IV distribution has a pdf of the form $$\dfrac{\Bigg|\frac{\Gamma(m+\frac{\nu}{2}i)}{\Gamma(m)}\Bigg|^2}{\alpha\...
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2answers
61 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
316 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 ...
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0answers
38 views

A classical result of first hitting time of simple random walk 1

We define a first hitting time of simple random walk 1D by $$\tau _z=\min\{n:S_n=z\}.$$ I read a paper which write ...a classical result for random walks in $d=1$ with zero mean adn finite variance,...
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20 views

Comparing two sum of fractal moments for heavy-tail distribution

Assume a heavy tailed distribution whose tail can be approximated as $$P(X\geq x)\sim x^{-\alpha}$$ Consider some fractal moment of iid $X_i$, we have $$\frac{1}{n}\sum_{i=1}^nX_i^{\theta}\sim O(n^{\...
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0answers
14 views

An explicit expression for tail probability using fourier transform

I am reading a paper about tail probability approximation. However, I got into trouble at the very first formula. The background setting and formula goes like this: $\bar X=\frac{1}{n}\Sigma_{i=1}^...
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0answers
78 views

Heavy tailed distributions and their sum

Let $X_{1}, X_{2}, \ldots, X_{n}$ be the sequence of i.i.d random variables with heavvy tailed distributions, i.e. $$p(x_{i}) \sim \frac{A}{x_{i}^{\alpha}}$$ as $x_{i} \rightarrow \infty$, where $p(...
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1answer
76 views

To establish an inequality using Chebyshev's probability bound

Let $X$ be a random variable with mean, $E(X)=\mu$ and variance, $E(X-\mu)^2=\sigma^2$. Then Chebyshev's inequality asserts that $$ P\{|X-\mu|\geq k\sigma\} \leq \frac{1}{k^2} $$ Using this ...
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1answer
142 views

Applying Chernoff's/Hoeffding's Tail Bounds for Bounded, Dependent Variables

Say I have $\lbrace 0, 1 \rbrace$ random variables $\lbrace Y_1,\cdots, Y_n\rbrace $, for which $P(Y_i = 1) \geq a$. Then, say I create n i.i.d. Bernoulli random variables $\lbrace X_1,\cdots,X_n \...
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1answer
115 views

Is there a way to lower bound the left tail probability of a random variable?

I am looking for a bound of the form $P(X<0) > t$ where $X$ is a general random variable with positive mean, and all of whose (or most) moments exist. $t$ is ideally a function of these moments. ...
1
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1answer
37 views

Probability that an infinite sequence of i.i.d. integers has a repetition

Let $X_i$ be a sequence of i.i.d. random variables in $\mathbb N$. According to Hewitt-Savage's 0-1 law, the probability that $X_i$ has a repetition is either $0$ or $1$. If the $X_i$ have a moment ...
2
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1answer
2k views

Tail Value at Risk of Normal Distribution

For a random variable $X$, Tail-value-at-risk is denoted as $\operatorname{TVaR}_p(X) = \operatorname E(X \mid X>\pi_p) = \dfrac{ \int_{\pi_p}^\infty xf(x) \, dx}{1-F(\pi_p)}$, where $\pi_p=\...
3
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1answer
130 views

Concentration inequality for i.i.d. negative multinomial variables

Let $\mathbf p=(p_0,p_1,\cdots,p_m)$ be a probability vector, such that $p_0+p_1+\cdots+p_m=1$. Let $X_1,X_2,\cdots$ be i.i.d. random variables distributed according to the categorical distribution ...
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0answers
33 views

Tail Bounds of Spectral Norm of unbounded Chi-Squared Vector

Let $Z=(z_0,z_2,...,z_{N-1})$ be a vector with Chi-square distributed random variables ($zi=a_ia_i^*, a_i:CN(0,1)$ :Complex scalar gaussian), and consider $f:R^N->R, (f(Z)=||Z||_2$) as function ...
2
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1answer
82 views

Stopping Time Tail Probability

Problem Let $X_1,X_2,\dots$ be independent each with $P(X_j=1)=P(X_j=-1)=1/2$. Let $S_n=X_1+X_2+\cdots +X_n$ and $N>1$ be an integer. Define the stopping time $$ T=\inf\{n: |S_n|=N\} $$ Show that ...
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1answer
276 views

maximal bound for a sub-gaussian RV

By definition a RV is $\nu$-sub-gaussian, if the log of the moment generating function is bounded such that $$ \log(\mathbb{E}[\exp(\lambda X)]) \leq \frac{\lambda^2\nu}{2} $$ Also, it can be shown ...
2
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1answer
68 views

Non Commutative Bernstein Inequality

In the proof for noncommutative Bernstein Inequality, a symbol is used whose intuition is not clear. In the picture, Pr[X symbol A] is used. What is the semantic meaning of that symbol. Can someone ...
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24 views

Large $k$ asymptotic of $\mathbb{P}[X=k]$, given the generating function $\mathbb{E}[z^X]$

Consider a non-negative integer valued random variable $X$. Suppose that we are given the generating function $\mathbb{E}[z^X]$, but we do not have it in the form $\sum_{k=0}^\infty\mathbb{P}[X=k]z^k$...
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0answers
40 views

What is the cutting point of two distributions?

I just want to know what is the cutting point between the predictive bayesian distribution and the predictive frequentist distribution (plugin) (the one that uses the MLE). I will only consider ...
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0answers
56 views

Probability of a tail event on moments of normal random variables

Suppose $a_i \sim \mathcal{N}(0,1)$ are iid. Then for any $\epsilon>0$ there is a constant $C >0$ such that $m \ge C\cdot n$ implies \begin{align} X_1 = \frac{1}{m}\sum_{i=1}^{m}\left(...
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1answer
325 views

tail probability for norm of normal random vector

If $x \sim \mathcal{N}(0, I)$ is an $n$ dimensional random vector. I am looking for a proof of $$ \mathrm{Prob}\left( \lVert x\rVert^2 \ge 6n \right) \le e^{-1.5n}$$ $\lVert x \rVert^2 \sim \chi^{2}$...
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1answer
2k views

Definition of Tail-index of a probability distribution

What is a valid definition of Tail-index of a probability distribution? I understand that it is something to do with the rate of convergence of the density function $f(x)$ $($to $0)$ as $x \to \infty$...
2
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1answer
282 views

maximum of a sequence of i.i.d. random variables

Let $X_i,i=1,2,...,$ be a sequence of i.i.d. random variables and $u_n$ be a sequence of positive real numbers. I was motivated by the fact that if $$P(\max_{1\leq i\leq n}|X_i|\geq u_n)\to 0 \quad \...
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1answer
100 views

Questions about heavy-tailed distributions

I stumbled upon a definition of a heavy tail (on the left side) as follows $\lim_{x\to\infty}( [1-F(x)]e^{\lambda x})=\infty$ for $\lambda>0$. Now on-to questions: what does it mean if I get $-\...
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1answer
120 views

Random variables with rapid variation have finite moments

This problem is from Resnick's A Probability Path (chapter 5, Integration and Expectation): Rapid variation. A distribution tail $1-F(x)$ is called rapidly varying if $$\lim_{t\to\infty} \frac{1-...
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0answers
98 views

Proving logarithm of CDF of bivariate normal is concave in $\rho$ (the correlation).

Let $X,Y$ be two random variables and let the joint distribution of $(X,Y)$ be bivariate normal with mean $[0,0]$ and covariance $\begin{bmatrix} 1 & \rho \\ \rho & 1 \end{...
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81 views

Counter example to heavy/light tailed and mean excess loss

In a course we have been given the following proposition: For a random variable $X\geq0$ with essup$(X)=\infty$ i) $F$ is heavy tailed if $\lim_{u\rightarrow\infty}e_F(u)=\infty $ ii) $F$ is light-...
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1answer
206 views

relationship between a finite moment and a polynomial tail

I see condition (C2) in the paper http://www-stat.wharton.upenn.edu/~tcai/paper/Precision-Matrix.pdf is called polynomial-type tails. I do not know why they call it as polynomial-type tails. Let $X$ ...
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0answers
96 views

Tighter McDiarmid inequality

Is there a tighter version of McDiarmid inequality $$ \Pr\left(g(X_1,\dotsc,X_n)) - \mathbb{E}[g(X_1,\dotsc,X_n)] > t\right) \le 2e^{2t^2/\sum_{i=1}^n(b_i-c_i)^2} $$ I'm referring specifically to ...
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0answers
87 views

Analyzing a randomized algorithm for constructing an independent set

Suppose that $G = (V,E)$ is a 3-regular graph on $n$ vertices and $m$ edges. Below I propose a randomized algorithm for obtaining an independent set for $G$. Step $1$: Delete each vertex (...
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1answer
127 views

Probability of a tail event

Let $X_n$ be a sequence of random variables such that $$P(X_n = 0 \text{ eventually}) = 1 $$ Does this imply that $$P(\sum_{i=1}^nX_{n,i} = 0 \text{ eventually}) = 1 $$ where $X_{n,1},X_{n,2},....
0
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1answer
101 views

estimate support (maximum) of distribution from sample

Let $X$ be a random variable. The only information we have about $X$ is that $X \leq M$ for some $M \in \mathbb R$. ($M$ is unknown.) We also have a random sample $X_1, \dots, X_n$ from $X$. I'd like ...
2
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1answer
3k views

Tail of probability distribution

I need to analyze the plot of a probability distribution for a group of random samples. The question asked: "What does the tail of probability distribution of the sample values look like?" I don't ...
4
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1answer
56 views

If $X_1, \ldots, X_n \sim t_\nu$, a t-distribution with $\nu >1$, how to show $E\left(\max_{1 \leq i \leq n}|X_i|\right) = O\left(n^{1/\nu}\right)$?

If $X_1, \ldots, X_n \sim t_\nu$, a t-distribution with $\nu >1$ degrees of free, with each of them independent, then a result from probability theory is that: $$ E\left(\max_{1 \leq i \leq n}|...
6
votes
2answers
493 views

On the variance proxy of a positive (and bounded) sub-Gaussian variable

Consider a random variable $X \ge 0$ which takes values in an interval $[0, b]$, and further $$ \text{P}(X \ge t) \le C \exp\left(\frac{-t^{2}}{B}\right), \quad \forall t \ge 0, $$ for given constants ...
1
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1answer
257 views

Bounding the eigenvalues of a matrix projected onto a random subspace

Let $\Sigma$ be a fixed $m\times m$ positive semidefinite matrix, and $X$ be an $m\times n$ matrix (with $n<m$) whose columns are random vectors drawn uniformly from the unit sphere. I need tail ...