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.

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Tail bound for gaussian on lattice

Suppose I have a Gaussian distribution $$f(x) = \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-x^2/2 \sigma^2}$$ If the gaussian is continuous, then we can retrieve a Chernoff-like bound: $$\int_t^\infty f(x) dx ...
NYG's user avatar
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Mean of samples without replacement is no less concentrated than with replacement?

Consider a population $\mathcal{C}$ of $N$ real numbers, possibly with multiplicities. For an integer $n\leq N$, let $A_n$ be the random variable denoting the mean of $n$ random samples of $\mathcal{C}...
AAA's user avatar
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Poisson Tail Estimate using the Poisson Limit Theorem

This is Exercise $2.3.3$ in Vershynin's book. Let $X\sim \operatorname{Pois}(\lambda)$. Show that for any $t > \lambda$, we have $$P(X\ge t) \le e^{-\lambda}\left(\frac{e\lambda}{t}\right)^t.$$ ...
stoic-santiago's user avatar
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Tail bound for difference of i.i.d Laplace variables

I am trying to find a tail bound for $X-Y$, where $X,Y$ are i.i.d. variables following Laplace distribution with parameter $(0,\lambda)$, i.e., the pdf is $P(X=x) = \frac{1}{2\lambda}e^{\frac{-|x|}{\...
white's user avatar
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Tail's area of a generic distribution

I have several datasets, which data can be represented by a random variable $X$. Some of the datasets follow an approximately symmetric distribution, while other ones follow a skewed distribution. For ...
Ommo's user avatar
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2 votes
1 answer
95 views

Rate of decay of the Binomial cdf

Let $X \sim \text{Bin}(n,p)$ and $t \geq 0$. I believe the following inequality holds but I don't know how to prove it: $$ \text{Pr}[X \geq 2t] \leq \left ( \text{Pr}[X \geq t] \right )^2. $$ I have ...
mstou's user avatar
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Equality case in the subadditive property of Expected Shortfall

If $Z_1$ and $Z_2$ are two $L^1$ real-values random variables, it is well known that $ES_\alpha(Z_1+Z_2) \le ES_\alpha(Z_1) + ES_\alpha(Z_2)$ for any $\alpha \in (0,1)$ However, what about the ...
Aristodog's user avatar
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From Large Deviations to Finite Time Probability Tails

Let $(B_t)$ be a standard $d$-dimensional Brownian motion. It is well-known that $$\mathbb P(\sup_{s\in[0,t]}|B_s|\ge \alpha) \le 4de^{-\alpha^2/2dt}.$$ One possibility to obtain such a result is ...
Benjamin's user avatar
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Avoiding misuse of "leading term". How to make it rigorous?

I've seen several papers on heavy-tail analysis use "leading term" to refer to something which is a factor, not a term. For instance, if we've determined the asymptotics of a tail function ...
WithinCellsInterlinked's user avatar
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Sub-gaussian norm vs. variance proxy

When studying sub-gaussian variables, I have come across several definitions. One of the most common uses the concept of a "variance proxy," i.e. a (mean-zero) random variable $X$ is sub-...
LSK21's user avatar
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Making the absolute constant in sub-gaussian characterization explicit

I have recently been reading Vershynin's "High-Dimensional Probability" (which can be found for free here), and I would very much like to find an explicit representation of the absolute ...
LSK21's user avatar
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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}\...
Small Deviation's user avatar
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Upper bound for a squared zero-mean sub-Gaussian random variable: Transforming a sub-Gaussian rv to a a Gaussian rv

I'm studying from the book "Mathematical Analysis of Machine Learning Algorithms" by Tong Zhang. Theorem 2.9 states Let $\{X_n\}_{n=1}^N$ be independent zero-mean sub-Gaussian random ...
Gerardo Duran-Martin's user avatar
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Approximating a modified Gaussian integral by considering tail-end behavior

I am considering calculating the following definite integral. The limits of integration are positive and $c > 0$. One can see that the integrand is bounded above by the Gaussian in the numerator ...
Debbie's user avatar
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Are these two versions of the Hanson-Wright Inequality Equivalent?

I have a question regarding the following two different versions of the Hanson-Wright inequality for estimating the tail behavior of (sub-)Gaussian chaos (i.e. quadratic forms). The first is Thm. 6.2....
LSK21's user avatar
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$X$ sub-Gaussian $\implies \text{Var}(X) \leq \sigma^2$

$\newcommand{\V}{\text{Var}}$ $\newcommand{\E}{\mathbb E}$ Definition: A mean zero random variable $X$ is $\sigma$ sub-Gaussian if for all $\lambda \in \mathbb R$, \begin{align} \E\left[\exp\left(\...
Novice's user avatar
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The lower bound probability of Bernoulli sum is positive [closed]

Let $X_i$ be iid random variables. It equals to $1$ with probability $(1+p)/2$ and equals to $-1$ with probability $(1-p)/2$. My question is: What is the probability that $\sum_{i=1}^nX_i>0$? I ...
happyle's user avatar
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Asymptotic expansion for the density-quantile function of a bijected truncated normal distribution

Summary How would I prove that $$ fQ(u) = \frac{1}{a} \left( 1 - \Phi^{-1}(au+b)^2 \right)\varphi\left(\Phi^{-1}(au+b)\right) \sim (1 -u), $$ as $u \to 1$, where $\Phi, \varphi$ are the CDF and PDF of ...
Tim Hargreaves's user avatar
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71 views

Supremum of tail probabilities over nonnegative random variables

Let $\xi$ be a nonnegative random variable, and $a>0$. Find $\sup P(\xi\geq a)$ over all distributions such that: (i): $E(\xi)=20$; (ii): $E(\xi)=20$, $\text{Var}(\xi)=25$; (iii): $E(\xi)=20$, $\...
aaaaaaaaaaaaaaaaaanon's user avatar
1 vote
1 answer
196 views

Bounding expectation of an expression given a tail bound

Let $X$ be a positive (i.e. nonzero) random variable which is bounded in probability as follows: there ex. constants $A, C > 0$ s.t. for every $\varepsilon > 0$ $$ \mathbb{P}(X \geq \varepsilon) ...
LSK21's user avatar
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3 votes
0 answers
119 views

Paley Zygmund inequality, why positive?

Paley Zygmund inequality states that: $$P(Z> \theta E[Z]) \ge (1-\theta)^2\frac{E[Z]^2}{E[Z^2]}.$$ Throughout the internet and text books, it is stated for a non-negative random variable Z (i.e. $Z\...
Ron's user avatar
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2 answers
210 views

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: (...
LSK21's user avatar
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0 answers
40 views

Supremum of random variables evaluated at the inverse of their densitiy

Assume $X_1$, ..., $X_n$ are random variables (might be independent or not) with the same marginal density $f$ having finite moments to any desired order. I want to reach a tight tail distribution of ...
Ibra's user avatar
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1 vote
1 answer
108 views

Sum of (non-centered) Subgaussian Random Variables

If we have a finite sum of independent subgaussian (or subexponential) random variables which are not mean zero, i.e. random variables $(X_i)_{i=1, \dots, N}$ such that $\forall i=1, \dots, N$ $$ \...
LSK21's user avatar
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1 answer
67 views

Manipulating Concentration Inequalities

I have been studying concentration inequalities for sums of random variables recently, specifically Bernstein's inequality in several forms, and have a question regarding the manipulation of such ...
LSK21's user avatar
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Distribution of $k$-matchings in a random graph

Take the Erdos-Renyi random graph $G(n,p)$, i.e. the random graph with $n$ vertices and where each possible edge has an independent probability of $p$ of being present. Recall that a $k$-matching is a ...
Harry Vinall-Smeeth's user avatar
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Deriving Concentration of Measure Given Conditional Concentration

I have a question regarding a couple of concentrated random variables. I have tried abstracting the original setting in order to ask the question here, reducing the details to what I believe is all ...
LSK21's user avatar
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3 votes
0 answers
213 views

Tightness of Bernstein's inequality and Hoeffding's inequality

Let $\{ X_t \}$ be independent zero mean random variables, and $|X_t|<R$. Bernstein's inequality yields: $$\mathbb{P} \left( \sum_{t=1}^{T}X_{i} \geq \varepsilon \right) \leq \exp \left( \frac{ -\...
RS.'s user avatar
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0 answers
254 views

Cramer Condition

I was reading the book Technical Incerto - Statistical Consequences of Fat Tails by Nassim Nicholas Taleb, in this book the autor writes the following (page 28): "Membership in the ...
CREZPO's user avatar
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Inequality of tail probabilities for to distributions of the same parametric family?

Suppose $f(X,v_i), \textrm{ for } i=1,2$ are two probability distributions of the same parametric family. Moreover, I have a function L such that: $$\sum_{x} I_{\{L(X)\leq \bar{\gamma}\}} f(X,v_1) \...
entropy's user avatar
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0 answers
34 views

Tail bound for maximum of asymmetric random walk with varying transition probability

Suppose we are considering an asymmetric random walk $X_t$ beginning at $X_0 = 1$, with varying probability on $\mathbb{Z}_{\ge 0}$ : $$P(X_{t+1}=n+1 | X_t = n) = p(n) \quad P(X_{t+1}=n-1 | X_t = n) = ...
rrr's user avatar
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Tail bound version of Cramer-Rao

The Cramer-Rao bound is well-known: for an unbiased estimator $\hat{\theta}$, $$\text{var}(\hat\theta(X_1,...,X_n)) \geq\frac{1}{n}\mathcal{I}(\theta)^{-1}),$$ where $ X_1,...,X_n\sim P_\theta$ i.i.d.,...
RS.'s user avatar
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0 answers
98 views

Concentration bounds for a sum of dependent almost rademacher variables

Let $Y_1,\ldots, Y_n \in \{\pm1\}$ be i.i.d. Rademacher variables. Define the following "Rademacher" variables: $$Y'_i=\operatorname{sign}\left(\sum_{j\neq i} Y_iY_j + C\right)$$ for some ...
bgbgtata's user avatar
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1 answer
424 views

Tail probability of summation of (possibly dependent) Bernoulli variables

We have a series of $n$ Bernoulli random variables, $X_1, X_2, \ldots, X_n$, where $X_i \sim \operatorname{Bernoulli}(p_i), \forall i \in [n]$. Let $Y = \sum_{i = 1}^n X_i$ be the random variable that ...
Vezen BU's user avatar
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1 vote
0 answers
43 views

Tail probability of first collision of 3 random walks

I am interested in finding more details about first collision of the 3 simple symmetric random walks. Let me make the problem precise: let $ \{ S_n^{(T)} : n \geq 0 \}, \{ S_n^{(M)} : n \geq 0 \} $ ...
Rana's user avatar
  • 462
2 votes
2 answers
144 views

Compare difference of probability between two Poisson distributions, evaluate at certain point.

$X,Y$ are two independent random variables such that $X\sim \mathop{\mathrm{Po}}(\lambda_1), Y\sim \mathop{\mathrm{Po}}(\lambda_2)$, where $\lambda_2>\lambda_1$. Is there a conclusion about the ...
lsstat's user avatar
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4 votes
1 answer
173 views

Tail bounds of a mean of Laplace random variables

Let's have three "true" values $\mu_1, \mu_2, \mu_3 \in \mathbb{R}$. Their "true" mean is simply $y = \frac{1}{N} \sum_{i = 1}^{N} \mu_i$ where $N = 3$. Now say I observe each &...
John Doe's user avatar
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0 answers
129 views

Recovering upper bound on MGF from tail probabilities

It is a common technique to obtain tail bounds from the MGF. But now suppose I have the other way round. Suppose it is given, $$\mathbb P [X_i \geq t] \leq e^{-t^2/2\sigma^2}$$ then what can you say ...
DuttaA's user avatar
  • 283
3 votes
1 answer
156 views

Trouble integrating tail bound

I have the following bound on the deviation of a non-negative random variable: $$\mathbb P (X \geq t) \leq ce^{-t^\alpha}$$ for some $\alpha \geq 1, c>1$. I want to show that: $$\mathbb E X \leq (\...
dmh's user avatar
  • 2,998
0 votes
2 answers
108 views

Asymptotics of tail function of product of 2 iid gamma variables

Suppose we have an integral of this form: $\overline F(x)=\frac{\beta^{2\alpha-1}}{\Gamma^2(\alpha)}\int_{0}^{\infty}x^{\alpha-1}e^{-\beta(\frac{x}{y}+y)}dy$, where $\beta>0, \alpha>0$ and $x>...
BigFun's user avatar
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0 votes
1 answer
155 views

tail estimation of normal distribution

In https://www.johndcook.com/blog/2021/11/05/normal-tail-estimate/ , why "base of our rectangle runs from x to the point t where the integrand drops by a half" ?
kevin's user avatar
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2 votes
1 answer
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A question about tail probabilities of identically distributed variables.

I have $X_1, X_2, \dots, X_n$ be identically distributed but not necessarily independent random variables with $E[X_j]=0$. I am trying to show that $$\lim_{n \to \infty}P\left[ \max_{1\leq j \leq n} ...
PSE's user avatar
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2 votes
0 answers
191 views

Chernoff Bound as an approximation for binomial distribution tightness.

I'm curious about how well the Chernoff bound approximates the value of the upper tail of a binomial distribution. It is well known that, for $X\sim B(n,p),\ \delta >0$: $$P(X \geq (1+\delta)np) \...
FH93's user avatar
  • 1,198
3 votes
2 answers
109 views

Tail Probabilities of $L^p$ bounded martingale differences

Assume that I have a probability space $(\Omega, \mathcal{A}, \mathbb{P})$ on which we define a sequence of martingale differences $X_1,X_2,\dots$ (w.r.t. to a certain filtration). Further let $p \in (...
Abel1353216381's user avatar
1 vote
1 answer
170 views

Tail behavior of a stable distribution according its moments

I'm studying about stable distributions and I would like to understand a statement that relates the moments to the behavior of their tails. More specifically, the characteristic function of a $\alpha-$...
PSE's user avatar
  • 544
1 vote
1 answer
93 views

Tail Probabilities of a martingale difference sequence

I'm currently facing the problem, that I can neither prove nor find a counterexample for the following statement. Let $q \in (1,2)$ and let $(D_n)_{n \in \mathbb{N}}$ be a martingale difference ...
Abel1353216381's user avatar
1 vote
0 answers
200 views

Gaussian tail probability

Let $z(q)$ be the quantile of a standard normal random variable $Z$, i.e., $z(q) = k$ when $\Pr(Z\geq k) = q$. Then I would like to know why the following two results hold. (a) If we hold $\alpha$ ...
Jie Wei's user avatar
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1 vote
0 answers
51 views

How to get an "average" of a distribution with no average

Modelling something where I need some sort of average time, I reached Pareto Distribution, and the PDF is $$ R(t)=\begin{cases}\sqrt{\frac{1}{2\pi t^{3}}}, & \text{for } t > \frac{2}{\pi}\\ 0 &...
ck1987pd's user avatar
  • 1,104
1 vote
1 answer
387 views

The additive Chernoff bound for the absolute value.

I am trying to derive a generic, additive Chernoff bound for $\Pr[|X-\mu|\leq a]$ with $a>0$. By generic I mean a Chernoff bound in terms of the moment generating function instead of assuming a ...
synack's user avatar
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2 votes
1 answer
372 views

Numerical simulation of SDE with Lévy noise in Python - Overflow issue

My goal is to simulate a SDE with alpha-stable noise in Python. This is my code: ...
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