# 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 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 ...
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### uniformly bounded tail probabilities

While reading a book from the 1980's I stumbled across the term of "uniformly bounded tail probabilities". Consider a sequence of random variables $(X_j)_{j \in \mathbb{N}}$. We shall say ...
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### Gaussian tail probability

Let $z(q)$ be the quaitle 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 $\alpha$ fixed,...
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### Estimating $n^2$ after observing $X\sim Bin(n,p)$ and knowing $p$

Let $n\in\mathbb N$ be unknown and let $p\in (0,1]$ be known. Suppose that we observe $X\sim Bin(n,p)$. This allows us to estimate $\widehat n= X/p$ and we can use the Chernoff inequality to bound the ...
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### Coupling for tail bounds

Let $X_1, X_2$ be two random variables such that $\mathbf{E}X_1=\mathbf{E}X_2=0$ and $$\forall x\geq 0, \quad \mathbf{P}[X_1\geq x]\leq \mathbf{P}[X_2\geq x].$$ Can we construct a coupling $(Y_1,Y_2)$ ...
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### Concentration bound for the distribution of the difference of two random variables

If we use $\Rightarrow$ to represent convergence in distribution and suppose that $X_n \Rightarrow N(0,\sigma_1)$ and $Y_n \Rightarrow N(0,\sigma_2)$, and $X_n$ and $Y_n$ are independent, then we all ...
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### Expectation of inverse of sum of iid random variables

Description: There is an indicator function called $I_{i}^{k}$ as follows: \begin{align*} \begin{split} I_{i}^{k}= \left\{ \begin{array}{lr} 1, \; {\rm if \; the \; event \; happened\; at\; time ...
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### Tails of function and Fourier transform

I have a questions about Fourier transforms of functions in general. Let $f$ be a function and $\phi$ its Fourier transform. If $f$ has fat tails, must $\phi$ then necessarily have thin tails? ...
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### How to show that for any random variable $X$ and $t > 0$, $\Pr(X - E[X] \geq t\sigma[X]) \leq \frac{1}{1 + t^2}$? [duplicate]

The question in the title is from exercise 3.18 in the first edition of Mitzenmacher and Upfal's Probability and Computing. It's essentially asking to prove a bound slightly tighter than Chebyshev's ...
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### Anti concentration of negatively correlated random variables

Let $X_1,\cdots,X_n$ are binary random variables. We say that they are negatively correlated if for any $I \subset \{1,\cdots,n\}$ and $b\in \{0,1\}$ we have \begin{align*} Pr\big[\forall i\in I: X_i =...
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### An interesting Bound for Binomial Distribution

I have seen the following bound somewhere, but could not find it. Let $n$ be an non negative integer and $X \sim Bin(n,\frac{1}{2})$. Show that there exists an abosolute constant $c > 0$, such ...
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### existence of sequence $\{a_n\}$ that make the tail probability decrease at a certain rate

This appeared in example (b), Section 8.8 of An Introduction to Probability Theory and Its Applications, Vol 2, by Feller. Let $F(x)$ be the cumulative distribution function of some random variable. ...
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### Sample size required to hit at least a constant fraction of a set.

I have been trying to figure out the following problem for quite some time but have not succedded. May be I am missing something very obvious. Let $M$ be and integer and define $[M] = \{1,\cdots,M\}$. ...
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### Occuppation time bound on supermartingale

I want to ask about the bound on the behavior of such a supermartingale $\{X_t\}$, where for each $t$, $X_t\in[0,1]$. Moreover, there exists $0<a_1<1$ such that for each $t$, it is satisfied ...
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### How this limit prove the Frechet Distribution has Heavy Tails?

First, we start from the GEV-distribution function: Theorem 1.1.3 (Fisher and Tippett (1928), Gnedenko (1943)) The class of extreme value fistributions is $G_\gamma(ax+b)$ with $a>0$, $b$ real, ...
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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 ...
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### Is the following tail bound on the Binomial distribution accurate?

The Poisson distribution has the following tail-bound: Given $X \leftarrow Po(\lambda)$, for any $t \geq 1$ $Pr(X \geq t) \leq (\frac{e \lambda}{t})^t$ (For $t > \lambda$, this follows directly ...
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