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Questions tagged [distribution-tails]

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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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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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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
17 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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56 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
79 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
25 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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39 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
46 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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47 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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17 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
38 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 ...
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1answer
130 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
19 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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79 views

Conditional distribution and Tail sum formula problem

Each time you spin the wheel, it comes up on a number that is uniformly distributed over {1, . . . , n}, and independently of all other spins. You spin the wheel once, and that comes up on a ...
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0answers
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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
54 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
44 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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35 views

Weighted chi-square large deviations bound?

For a fixed $n$, let $x_1,\dots,x_n$ be $n$ i.i.d. draws from a chi-square distribution with $n$ degrees of freedom. Let $\Delta^{n-1}$ be the $(n-1)$-simplex and $g_n$ a function mapping a point in $\...
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1answer
50 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
61 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. ...
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0answers
18 views

Why moment generating function for heavy tailed function is unbounded

How infiniteness for moment generating function for every $x\ge 0$ follows from this?
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1answer
34 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 ...
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1answer
1k 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=\...
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0answers
57 views

Exponentially decreasing bound for hypergeometric random variables

I have a set $S$ of random variables $X_i$ that follows the hypergeometric distribution and I would like to find exponentially decreasing bounds of the type: $$Pr \biggl[\sum_{i=1}^{|S|}X_i > (1+\...
3
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1answer
87 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
27 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 ...
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1answer
66 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
127 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 ...
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1answer
48 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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54 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
212 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 $x \to \infty$...
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1answer
238 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
85 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
98 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
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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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71 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
138 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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84 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
77 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
118 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},....
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1answer
74 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
2k 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 ...
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1answer
52 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}|...
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2answers
374 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 ...
0
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1answer
210 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 ...
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1answer
54 views

Is $\lim_{x\to\infty} x\overline{F}(x)=0$?

With partial integration I wanted to prove that for non-negative random variable with CDF F(x) holds $$ \int_0^{\infty}\overline{F}(x)dx=E[X]. $$ Here is $\overline{F}(x)= 1-F(x)$. I got this far $$ \...