Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [distribution-tails]

This tag is for questions relating to "tail-distribution" which essentially means how much probability is distributed over the largest values(usually) of the random variable.

The tail behavior of a probability distribution is known to be closely related to the behavior of the characteristic function of the distribution in the neighborhood of the origin.

A distribution may be viewed as A fat-tailed distribution is a probability distribution that exhibits a large skewness or kurtosis, relative to that of either a normal distribution or an exponential distribution. In common usage, the terms fat-tailed and heavy-tailed are sometimes synonymous; fat-tailed is sometimes also defined as a subset of heavy-tailed. - if it assigns smaller probabilities for larger values of the variable, or heavy tailed - if it assigns larger probabilities for larger values of the variable.
Again a fat-tailed distribution is a probability distribution that exhibits a large skewness or kurtosis, relative to that of either a normal distribution or an exponential distribution. In common usage, the terms fat-tailed and heavy-tailed are sometimes synonymous; fat-tailed is sometimes also defined as a subset of heavy-tailed.

For more about this you can see the following (also the references therein):

https://en.wikipedia.org/wiki/Heavy-tailed_distribution

https://en.wikipedia.org/wiki/Long_tail

https://en.wikipedia.org/wiki/Fat-tailed_distribution

235 questions
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Looking for a limit

Looking for a limiting value: $$\lim_{K\to \infty } \, -\frac{x \sum _{j=0}^K x (a+1)^{-3 j} \left(-(1-a)^{3 j-3 K}\right) \binom{K}{j} \exp \left(-\frac{1}{2} x^2 (a+1)^{-2 j} (1-a)^{2 j-2 K}\right)}{\sum _{j=0}^K (a+1)^{-j} (1-a)^{j-K}…
Nero
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1 answer

Tail Lower Bounds using Moment Generating Functions

Given a random variable $X>0$ with Moment Generating Function $m(s)=E[e^{sX}]$ I'm interested in finding a lower bound $$\Pr[X \ge t] \ge 1-\varepsilon(t),$$ where $t>E[X]$. A classic technique for finding upper bounds for $\Pr[X \ge t]$ is using…
Thomas Ahle
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10
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2 answers

Tighter tail bounds for subgaussian random variables

Let $X$ be a random variable on $\mathbb{R}$ satisfying $\mathbb{E}\left[e^{tX}\right] \leq e^{t^2/2}$ for all $t \in \mathbb{R}$. What is the best explicit upper bound we can give on $\mathbb{P}[X \geq x]$ for $x > 0$? A well-known upper bound…
Thomas Steinke
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Sharper Lower Bounds for Binomial/Chernoff Tails

The Wikipedia page for the Binomial Distribution states the following lower bound, which I suppose can also be generalized as a general Chernoff lower bound. $$\Pr(X \le k) \geq \frac{1}{(n+1)^2}…
Thomas Ahle
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3 answers

Proof that $E(X)<\infty$ entails $\lim_{n\to\infty}n\Pr(X\ge n) = 0$?

As the title says. I think this should follow straightforwardly but I can't find a proof. My random variable of interest $X$ takes values in the non-negative integers. The only other assumption on its distribution is that $E(X)<\infty$. I want to…
Remco
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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 $\pi_p=\operatorname{VaR}_p=$ the value-at-risk $=$ the value such…
EllipticalInitial
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8
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2 answers

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 $x \to \infty$. I tried searching google and I do find a lot…
user405743
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3 answers

Probability of more than ${3\over 4}N$ heads in $N$ flips of a coin?

What is the probability of getting more than $ \frac { 3 N } 4 $ heads in $ N $ flips of coins? I know we need to use binomial distribution formula for this and sum it from $ N = \frac { 3 N } 4 $ to $ N $. I can solve this when numbers are given…
Hitesh Khandelwal
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Tail Probabilities of Multi-Variate Normal

For a standard normal random variable $X \sim \mathcal{N}(0,1)$, we have the simple upper-tail bound of $$\mathbb{P} (X > x) \leq \frac{1}{x \sqrt{2\pi}} e^{-x^2 / 2}$$ and thus from this we can deduce the general upper-tail bound for $X' \sim…
paulinho
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3 answers

Mnemonic for platykurtic and leptokurtic

I keep confusing terms leptokurtic and platykurtic. Is there a good mnemonic to help remember which is which? "Lepto" means "little", "platy" means "flat", and both are equally unrelated to thickness of a tail. It also does not help that "lepto"…
nsg
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Regular Variation and Maximal Moments

Let $X$ be a non-negative random variable. We call $X$ regularly varying with tail index $\alpha>0$ if $$\lim_{u\to\infty}\frac{\mathbb P[X>ut]}{\mathbb P[X>u]}=t^{-\alpha}, \hspace{1cm}\forall t>0.$$ It is well-known, see e.g. Heavy-Tailed Time…
Small Deviation
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Question Regarding Vershynin's Proof of Bernstein's Inequality

I have been studying Vershynin's "High-dimensional Probability," and I have some confusion regarding the proof of Bernstein's inequality (Thm 2.8.2). It concerns the following step: (Perhaps note that $(X_i)_i$ is a finite sequence of independent,…
LSK21
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2 answers

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 $C \gg 1$ and $B >0$. Since $X$ is bounded, it…
megas
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6
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2 answers

Repeatedly rolling a die and the tails of the multinomial distribution.

For $1\leq i\leq n$ let $X_i$ be independent random variables, and let each $X_i$ be the uniform distribution on the set ${0,1,2,\dots,m}$ so that $X_i$ is like an $m+1$ sided die. Let $$Y=\frac{1}{n}\sum_{i=1}^n \frac{1}{m} X_i,$$ so that…
Eric Naslund
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5
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My data is not normally distributed: what can I do to estimate a tail probability?

Continuing on from my earlier question, I'm attempting to analyse the data qualitatively. In the following plot, I make $10000$ samples where I count "the number of clashes". I plot $n$ vs. the number of times $n$ clashes occurred. (The number of…
Rebecca J. Stones
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