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2 votes
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approximate the square of 2-norm distance between binary distributions with high probability

Suppsose we take $m$ samples from a Bernoulli distribution with probability $p$, and $m$ samples from another probability distribution with probability $q$. We want to calculate a statistic $x$ from ...
gondolf's user avatar
  • 1,503
1 vote
2 answers
4k views

Variance of truncated normal distribution

Let $ X \sim \mathcal{N} ( \mu, \sigma^2 ) $, $ - \infty \leqslant a < b \leqslant +\infty $ ($ a, b \ne \infty $ simultaneously) and $ Y $ has a truncated normal distribution on $ (a, b )$, i.e. $...
user47855's user avatar
1 vote
1 answer
125 views

A differential inequality and a special value

Let $G \colon [0,1] \to [0,1]$ be a monotonically decreasing function with $G(0) = 1$ and $G(1) = 0$. Suppose that $G$ is differentiable infinitely many times, and that: $$G(x)G''(X) \leq 2{G'(x)}^2.$$...
Pablo's user avatar
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1 vote
1 answer
362 views

An inequality involving the Wasserstein distance and chi-squared distance

$\newcommand{\N}{\mathbb N}$Let $P$ be the set of all probability mass functions on $\N_0:=\{0\}\cup\N$, where $\N:=\{1,2,\dots\}$. Let $P_{>0}$ denote the set of all $q=(q_0,q_1,\dots)\in P$ such ...
Iosif Pinelis's user avatar
1 vote
1 answer
144 views

A uniform mixture of order statistics

Let $0<k<n$ be integers, and let $X$ be a random variable obtained as follows: sample $n$ points independently and uniformly at random in the unit interval, and select (uniformly) one of the $k$...
Tom Solberg's user avatar
  • 4,049
1 vote
2 answers
462 views

lower bound the probability of at least L collisions

Lets say we get a list $M$ containing $|M|=\sqrt{L\cdot N}$ randomly and independtly drawn elements from a set of size $N$. And lets denote the $i$-th element of the list $M$ by $M[i]$. If we now ask ...
Memphisd's user avatar
  • 123
1 vote
1 answer
499 views

property of iid random variable

Let $ (\xi_i)_{i \ge 1} $ be independent identically distributed random variables, taking values in $ (1,3]$. Can we show: $P( \exists N \in \mathbb{N}, \text{ s.t. } \forall k \ge 0, \prod_{i=1}^{...
jason's user avatar
  • 553
1 vote
1 answer
97 views

A strict inequality for the $L^1$-norm of a symmetrized zero-mean random variable

Suppose that $Y$ is an independent copy of a random variable (r.v.) $X$ with a zero-mean nondegenerate distribution. Is it then always true that $E|X-Y|>E|X|$? To get the non-strict version of ...
Iosif Pinelis's user avatar
1 vote
1 answer
251 views

Using Hoeffding inequality for risk / loss function

I've got a question to the Hoeffding Inequality which states, that for data points $X_1, \dots, X_n \in X$, which are i.i.d. according to a probability measure $P$ on $X$, we find an upper bound for: $...
Mathematiger's user avatar
1 vote
1 answer
207 views

Expectation of the sum of the squares of the cardinal of an inverse function

I sample a random one-to-one function $\pi:\{0\,;\,1\}^n\to\{0\,;\,1\}^n$. I define $f$ as: $$\forall x\in\left[0\,;\,2^n-1\right]\cap\mathbb{N},f(x)=x\oplus\pi(x)$$ where $\oplus$ is the bitwise XOR. ...
Tristan Nemoz's user avatar
1 vote
1 answer
107 views

Tail bounds on random series in Hilbert space

Tail bounds on random series in Hilbert space Let $X_n$, $n \in \mathbb {N}$, be independent $\pm 1$ symmetric random variables, and $a_n$, $n \in \mathbb {N}$, be a sequence in a Hilbert space $H$ ...
Yilmis's user avatar
  • 11
1 vote
1 answer
249 views

On concentration of a sum random variable

Take a random variable defined as $$r=u_{11}v_{1}v_{1}+u_{12}v_{1}v_{2}+\dots+u_{n,n-1}v_{n}v_{n-1}+u_{nn}v_{n}v_{n}$$ where $v_{i}$ are independent uniform random variables from $\{0,\dots,b\}$, $u_{...
Turbo's user avatar
  • 13.9k
1 vote
1 answer
186 views

Kolmogorov inequality for Bernoulli random variables

This question is also asked on math stackexchange. The question is about one inequality which shows in Kolmogorov's paper (inequality (3.1)) but is not proved. The inequality says that, if we assume $...
Greenhand's user avatar
1 vote
1 answer
285 views

Exponential upper bounds for sums of martingale differences

Let $(X_{i})_{i\geq 1}$ be a sequence of centered real-valued martingale-differences with respect to some filtration $(\mathcal{F}_{i})_{i \geq 1}$. Define $S_{n} = \sum_{i=1}^{n}X_{i}$ and $\Sigma^{2}...
Oleksandr Z.'s user avatar
1 vote
1 answer
478 views

Distance between the product of marginal distributions and the joint distribution

Given a joint distribution $P(A,B,C)$, we can compute various marginal distributions. Now suppose: \begin{align} P1(A,B,C) &= P(A) P(B) P(C) \\ P2(A,B,C) &= P(A,B) P(C) \\ P3(A,B,C) &= P(...
took's user avatar
  • 49
1 vote
0 answers
68 views

A one-sided/monotone version of min/max-stable distributions -- does this have a name?

In a couple of papers I am working on (in random graph theory) I have encountered the following property of certain probability distributions, which I will describe shortly, and I am wondering if this ...
Joel Ottar's user avatar
1 vote
1 answer
335 views

Finding a connection between two types of convergence

Please, help me find connections between two types of convergence: Let $\{X_n\}_{n\ge1}: (\Omega,F,P) \rightarrow (\mathbb{R},Bor)$ be a sequence of r.v., there are two convergences: 1) $X_n \...
Ivan Petrov's user avatar
1 vote
0 answers
98 views

Joint distribution of two weighted sums of IID random variables

Let $X_1, X_2, \dots$ be independently uniformly distributed random variables in $\{-1, +1\}$ and let $a_1, b_1,a_2,b_2, \ldots \in \mathbb{R}$ be fixed, bounded and of non-zero average. Let $Y_n=...
Penchez's user avatar
  • 341
1 vote
0 answers
376 views

Anti-concentration bounds for folded normal and inverse of gaussian variables

Are there any easy to use bounds on sums of the following kind : $$ \sum_{i = 1}^{i = N} |a_i| \geq P \\ a_i \sim \mathcal{N}(0, 1) \\ $$ and also for sums of the form : $$ \sum_{i = 1}^{i = M} \...
Govind Gopakumar's user avatar
1 vote
0 answers
171 views

An inequality for moments of a random variable

I'm interested in a class C of $R^1$-valued random variables $\xi$ which satisfy an inequality of the type $$ (1) \qquad E|\xi|^p \leq F(E|\xi|^2), $$ where $p>2$, $F$ is a certain non-...
Ievgen's user avatar
  • 195
0 votes
1 answer
213 views

Show that interval of maximum probability grows no faster than $\sqrt{n}$ for binomial distribution

Let $X \sim \text{Binom}(n, p)$ a binomial random variable. I want to show that : $$\forall 0 < t < 0.9, \quad \exists C, \quad \forall n >1, \quad \mathbb P\bigg(|X-np| \leq C\sqrt{n}\bigg) \...
Julien__'s user avatar
  • 119
0 votes
1 answer
199 views

Is this probability inequality true?

This question may be simple, though I'm not managing to find an answer. Let $X$ and $Y$ be two dependent random vectors in in $\mathbb{R}^d$, with joint probability density $\mu(x,y)$ (with respect to ...
Jack London's user avatar
0 votes
1 answer
133 views

How to demonstrate a correlation inequality? [closed]

If there are 3 vectors X, Y, Z of the same length, for any $x_i \in X,y_i \in Y,z_i \in Z$, we have $0<x_i<1,0<y_i<1,0<z_i<1$. The correlation between Z, Y is greater than between X, ...
Mac Zhang's user avatar
0 votes
1 answer
188 views

Equality cases in a certain case of Jensen's inequality

Suppose that $Y$ is an independent copy of a random variable (r.v.) $X$ with a zero-mean nondegenerate distribution. Is there a non-tautological, preferably simple characterization of the cases when $$...
Iosif Pinelis's user avatar
0 votes
1 answer
967 views

Bound the norm of sum of random vector that generated from standard basis

I have a question like this: Consider $N$ samples $X_1, X_2, ..., X_N$ that uniformly random generated from standard basis $\{e_i, i=1,2,...,d\}$, i.e. $(1,0,0,\cdots,0),(0,1,0,\cdots,0),(0,0,1,0,\...
Betty's user avatar
  • 25
0 votes
1 answer
159 views

Best bounds on the integral of an increasing function

The following question, somewhat edited here, was asked and then closed at The best bound of the integral of a nondecreasing real function in a closed interval. Let $F\colon[0,1]\to[0,1]$ be a ...
Iosif Pinelis's user avatar
0 votes
1 answer
133 views

Projection onto manifold of Gaussian measures by "trunction" of moments

Let $\mathcal{P}_2(\mathbb{R}^n)$ be the set of Borel probability measures on $\mathbb{R}^n$ with finite mean and variance; in the sense that $$ \int_{x \in \mathbb{R}^n} \|x\|^p d\mathbb{P}(x) < \...
ABIM's user avatar
  • 5,405
0 votes
1 answer
340 views

Expectation of the ratio of two discrete random variables with combinatorial constraints

We are given a set $S=\{1, 2, \ldots, n\}$ where $n\gg 1$, and for all indices $1\le i \le n$, $i$ is associated with a real value $\alpha_i\!\cdot\! v_i$, where $\alpha_i\in[0,1]$ and $v_i\in(0,1]$. ...
Penelope Benenati's user avatar
0 votes
1 answer
339 views

Expectations, double integrals and Jensen's inequality

$\def\anonfunc#1{#1(\cdot)}$Consider two random variables distributed $v\sim \anonfunc G$ and $c \sim \anonfunc F$ with pdfs $\anonfunc g$ and $\anonfunc f$. Let the supports of $c$ and $v$ be $[x,y]$....
carlogambino's user avatar
0 votes
1 answer
503 views

Asymptotics of a 1D integral, or the orthant probability of an equicorrelated random Gaussian vector

Problem: Let $\phi(x)$ be the normal probability density function (pdf), and $\Phi(x)$ the normal cumulative distribution (cdf). I'm interested in the asymptotic behavior of the following integral $I(...
Daniel Soudry's user avatar
0 votes
0 answers
87 views

Comparison between the expected values of the inverse of the CDF of binomial-distributed random variables

Let us denote with $F(x;j,\mu)$ the cdf of a Binomial distributed random variable with $j$ trial with success probability $\mu$ considered in $x$, and let $f(x;j,\mu)$ be the pmf. Defining $0\leq \...
Marco Max Fiandri's user avatar
0 votes
0 answers
141 views

Effect of partitioning the realizations of random variables on the total variation distance?

Let $X$ and $Y$ be two random variables with joint pmf $p(x,y)=p(x)\cdot p(y|x)$ and $X$ has uniform distribution. Also assume that the following relation is satisfied: \begin{align} \lVert p(y|x)-p(y)...
Math_Y's user avatar
  • 287
-1 votes
1 answer
76 views

Transforming random variables for having good property?

For arbitrary functions $A$ and $B$ and independent random variables $X$ and $Y$, assume that \begin{align} \Omega&\triangleq \{(x,y): A(x,y)=1\},\\ \Lambda&\triangleq \{x: B(x)=1\}. \end{...
Math_Y's user avatar
  • 287

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