Questions tagged [distribution-tails]
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2
votes
1answer
38 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
votes
1answer
47 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 ...
0
votes
0answers
22 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 ...
0
votes
0answers
15 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 ...
0
votes
0answers
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
$$\mathbb{E}||Z||^2=\mathbb{E}\text{max}\{||ZZ^*||,||Z^*Z||\}...
1
vote
0answers
19 views
Tail Bound for normal random variable.
I want to show that if $g \ \sim N(0,1)$, for all $t>0$ we have $P(g\geq t)\leq e^{-t^2/2}$.
My solution: Let $\lambda>0$. $P(g\geq t)=P(e^{\lambda g}\geq e^{\lambda t})\leq \frac{E[e^{\lambda ...
3
votes
0answers
58 views
Tail probability of sum of order statistics of distance from point to a set
Let $P$ be a distribution on a metric space $(\mathcal X, d)$. For a point $x \in \mathcal X$ and a Borel $B \subseteq \mathcal X$, let $d(x,B) := \inf_{y \in B}d(x,y)$ be the distance of $x$ from $B$....
1
vote
3answers
84 views
Inequality for the maximum of the absolute value of two normal distributed random-variables
I would like to show following statement:
For $M\geq 2,\ X_1,\dots,X_M\sim^{iid}\mathcal{N}(0,1)$ independent, it holds $P(\max_{i=1,\dots,M}\lvert X_i\rvert\geq y)\leq Me^{-y^2/2}$.
I think it ...
1
vote
1answer
32 views
Deduce upper bound of variance from Chernoff-type tail bound
For a random variable $X$, I have a large deviation inequality of the form
\begin{equation}
P(|X-\mathbb EX|\geq r)\leq ce^{-\alpha r}\,.
\end{equation}
Consider a sample mean $S_n=\frac{1}{n}(X_1+...
0
votes
0answers
46 views
Asymptotic variance of Normal vs. Lognormal distributions truncated to a finite interval in the upper tail
How to prove analytically the following curiosities?
Claim. Let $X\sim \text{Normal}(\mu,\sigma^2)$, $Y:=e^X$, and let $U\sim\text{Uniform}$ on an interval of width $w$. Then for any $w>0$,
$$...
1
vote
1answer
55 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 $a = 1:$
$$
\Big(\frac{1}{x} - \frac{a}{x^3}\Big) \phi(x) \leq
\Big(\frac{x}{a + x^2}\Big) \phi(x) \...
1
vote
0answers
60 views
Large deviation upper bound for Chi-squared random variable
Let $X \sim \chi^2_n$ random variable. I am looking for a large deviation upper bound for $X$. The answer here, says that
Since you said that you're looking for an upper bound, it should also be ...
0
votes
0answers
30 views
Motivating Pearson type IV distributions
Note: this question is concerned with univariate distributions.
The Pearson type IV distribution has a pdf of the form $$\dfrac{\Bigg|\frac{\Gamma(m+\frac{\nu}{2}i)}{\Gamma(m)}\Bigg|^2}{\alpha\...
0
votes
2answers
47 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
212 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 ...
1
vote
0answers
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
$$\frac{1}{n}\sum_{i=1}^nX_i^{\theta}\sim O(n^{\...
0
votes
0answers
113 views
Conditional distribution and Tail sum formula problem
Each time you spin the wheel, it comes up on a number that is uniformly distributed over {1, . . . , n}, and
independently of all other spins.
You spin the wheel once, and that comes up on a ...
1
vote
0answers
13 views
An explicit expression for tail probability using fourier transform
I am reading a paper about tail probability approximation. However, I got into trouble at the very first formula.
The background setting and formula goes like this:
$\bar X=\frac{1}{n}\Sigma_{i=1}^...
3
votes
0answers
64 views
Heavy tailed distributions and their sum
Let $X_{1}, X_{2}, \ldots, X_{n}$ be the sequence of i.i.d random variables with heavvy tailed distributions, i.e.
$$p(x_{i}) \sim \frac{A}{x_{i}^{\alpha}}$$
as $x_{i} \rightarrow \infty$, where $p(...
0
votes
1answer
58 views
To establish an inequality using Chebyshev's probability bound
Let $X$ be a random variable with mean, $E(X)=\mu$ and variance, $E(X-\mu)^2=\sigma^2$. Then Chebyshev's inequality asserts that
$$ P\{|X-\mu|\geq k\sigma\} \leq \frac{1}{k^2} $$
Using this ...
0
votes
0answers
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 $\...
0
votes
1answer
73 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 \...
0
votes
1answer
76 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. ...
0
votes
0answers
19 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?
1
vote
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 ...
1
vote
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=\...
3
votes
1answer
100 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 ...
1
vote
0answers
31 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
votes
1answer
77 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\}
$$
Show that ...
0
votes
1answer
171 views
maximal bound for a sub-gaussian RV
By definition a RV is $\nu$-sub-gaussian, if the log of the moment generating function is bounded such that
$$ \log(\mathbb{E}[\exp(\lambda X)]) \leq \frac{\lambda^2\nu}{2} $$ Also, it can be shown ...
2
votes
1answer
56 views
Non Commutative Bernstein Inequality
In the proof for noncommutative Bernstein Inequality, a symbol is used whose intuition is not clear.
In the picture, Pr[X symbol A] is used. What is the semantic meaning of that symbol.
Can someone ...
0
votes
0answers
24 views
Large $k$ asymptotic of $\mathbb{P}[X=k]$, given the generating function $\mathbb{E}[z^X]$
Consider a non-negative integer valued random variable $X$.
Suppose that we are given the generating function $\mathbb{E}[z^X]$,
but we do not have it in the form $\sum_{k=0}^\infty\mathbb{P}[X=k]z^k$...
2
votes
0answers
40 views
What is the cutting point of two distributions?
I just want to know what is the cutting point between the predictive bayesian distribution and the predictive frequentist distribution (plugin) (the one that uses the MLE). I will only consider ...
0
votes
0answers
54 views
Probability of a tail event on moments of normal random variables
Suppose $a_i \sim \mathcal{N}(0,1)$ are iid. Then for any $\epsilon>0$
there is a constant $C >0$ such that $m \ge C\cdot n$
implies
\begin{align}
X_1 = \frac{1}{m}\sum_{i=1}^{m}\left(...
0
votes
1answer
242 views
tail probability for norm of normal random vector
If $x \sim \mathcal{N}(0, I)$ is an $n$ dimensional random vector. I am looking for a proof of
$$ \mathrm{Prob}\left( \lVert x\rVert^2 \ge 6n \right) \le e^{-1.5n}$$
$\lVert x \rVert^2 \sim \chi^{2}$...
5
votes
1answer
2k views
Definition of Tail-index of a probability distribution
What is a valid definition of Tail-index of a probability distribution?
I understand that it is something to do with the rate of convergence of the density function $f(x)$ $($to $0)$ as $x \to \infty$...
2
votes
1answer
256 views
maximum of a sequence of i.i.d. random variables
Let $X_i,i=1,2,...,$ be a sequence of i.i.d. random variables and $u_n$ be a sequence of positive real numbers. I was motivated by the fact that if
$$P(\max_{1\leq i\leq n}|X_i|\geq u_n)\to 0 \quad \...
0
votes
1answer
90 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 $-\...
1
vote
1answer
109 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
$$\lim_{t\to\infty} \frac{1-...
1
vote
0answers
83 views
Proving logarithm of CDF of bivariate normal is concave in $\rho$ (the correlation).
Let $X,Y$ be two random variables and let the joint distribution of $(X,Y)$ be bivariate normal with mean $[0,0]$ and covariance $\begin{bmatrix}
1 & \rho \\
\rho & 1
\end{...
1
vote
0answers
75 views
Counter example to heavy/light tailed and mean excess loss
In a course we have been given the following proposition:
For a random variable $X\geq0$ with essup$(X)=\infty$
i) $F$ is heavy tailed if $\lim_{u\rightarrow\infty}e_F(u)=\infty $
ii) $F$ is light-...
0
votes
1answer
158 views
relationship between a finite moment and a polynomial tail
I see condition (C2) in the paper http://www-stat.wharton.upenn.edu/~tcai/paper/Precision-Matrix.pdf
is called polynomial-type tails. I do not know why they call it as polynomial-type tails.
Let $X$ ...
2
votes
0answers
90 views
Tighter McDiarmid inequality
Is there a tighter version of McDiarmid inequality
$$
\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}
$$
I'm referring specifically to ...
1
vote
0answers
82 views
Analyzing a randomized algorithm for constructing an independent set
Suppose that $G = (V,E)$ is a 3-regular graph on $n$ vertices and $m$ edges. Below I propose a randomized algorithm for obtaining an independent set for $G$.
Step $1$: Delete each vertex (...
0
votes
1answer
121 views
Probability of a tail event
Let $X_n$ be a sequence of random variables such that
$$P(X_n = 0 \text{ eventually}) = 1 $$
Does this imply that
$$P(\sum_{i=1}^nX_{n,i} = 0 \text{ eventually}) = 1 $$
where $X_{n,1},X_{n,2},....
0
votes
1answer
82 views
estimate support (maximum) of distribution from sample
Let $X$ be a random variable. The only information we have about $X$ is that $X \leq M$ for some $M \in \mathbb R$. ($M$ is unknown.) We also have a random sample $X_1, \dots, X_n$ from $X$. I'd like ...
2
votes
1answer
2k views
Tail of probability distribution
I need to analyze the plot of a probability distribution for a group of random samples. The question asked:
"What does the tail of probability distribution of the sample values look like?"
I don't ...
4
votes
1answer
53 views
If $X_1, \ldots, X_n \sim t_\nu$, a t-distribution with $\nu >1$, how to show $E\left(\max_{1 \leq i \leq n}|X_i|\right) = O\left(n^{1/\nu}\right)$?
If $X_1, \ldots, X_n \sim t_\nu$, a t-distribution with $\nu >1$ degrees of free, with each of them independent, then a result from probability theory is that:
$$
E\left(\max_{1 \leq i \leq n}|...
6
votes
2answers
411 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
votes
1answer
230 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 ...
