All Questions
83 questions
1
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1
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335
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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 \...
CommunityBot
- 1
2
votes
1
answer
105
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Inequality for Gaussian measures
Let $\mu$ denote a centered Gaussian measure on $\mathbb{R}^k$, $K=(-\infty, a] \times \mathbb{R}^{k-1}$ ($a\ge 0$) and $L=\mathbb{R}\times C$ where $C$ is a convex set in $\mathbb{R}^{k-1}$, ...
- 109k
8
votes
2
answers
2k
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Median and mean of the sample mean of i.i.d. log-normal
Let $y:=\frac1n\sum_{i=1}^n x_i$, where $\{x_i\}_{i=1}^n$ is a set of i.i.d. random variables, and every $x_i$ has a lognormal distribution $x_i \sim\text{Lognormal}(\mu,\sigma^2)$. Let $\text{Med}[y]$...
- 4,713
2
votes
0
answers
55
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stochastic process and integral
Let $(X_n(t))_{t\in [1,+\infty], n\geqslant 1}$ be a sequence of nonnegative random variables and $(\mathcal{F}_s)$ a filtration ($\mathcal{F}_s \subset \mathcal{F}_r$ for $s\leqslant r$). We assume ...
- 29
0
votes
0
answers
87
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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 \...
- 63.9k
7
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1
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556
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A variation on the Borel–Cantelli lemma theme
Let $X,X_0,X_1,\dots$ be nonnegative independent identically distributed (i.i.d.) random variables. Let
\begin{equation*}
E:=\bigcap_{n\ge0}B_n,
\end{equation*}
where
\begin{equation*}
B_n:=\...
- 128k
1
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1
answer
186
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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 $...
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5
votes
1
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516
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Bounding the variance of a truncated Gaussian random variable
Suppose $X_1, X_2, X_3 \sim N(0, 1)$ are three independent standard normal random variables. I am interested in showing that:
$$\text{Var}[X_2\mid X_2 \geq X_1 - a, X_1 \leq X_3 + b] < 1,$$
where ...
1
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0
answers
68
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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 ...
- 5,429
3
votes
1
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346
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Simple anticoncentration bound for binomially distributed variable
The following question, which arose during my research, seems deceivingly simple to me, but I could not find any elegant and formal argument.
For a binomially distributed variable $X \sim \text{Bin} \...
- 128k
0
votes
1
answer
188
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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
$$...
- 128k
1
vote
1
answer
97
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A strict inequality for the $L^1$-norm of a symmetrized zero-mean random variable
Suppose that $Y$ is an independent copy of a random variable (r.v.) $X$ with a zero-mean nondegenerate distribution. Is it then always true that $E|X-Y|>E|X|$?
To get the non-strict version of ...
- 36
7
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2
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392
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On a von Bahr–Esseen-type inequality for pairwise independent zero-mean random variables
For $p\in(1,2)$, let $C_p$ be the smallest constant factor $C$ in the von Bahr–Esseen-type inequality
\begin{equation}\label{eq:pair}\tag{1}
E\Bigl\lvert\sum_{j=1}^n X_j\Bigr\rvert^p\le C\sum_{j=1}...
- 128k
4
votes
0
answers
131
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Log of a truncated binomial
Let $X$ follow a binomial distribution with $n$ trials and success probability $p$, and let $0\leq k\leq n$. Are there any natural approximations or bounds for the ratio $$\frac{\boldsymbol{E}\log\...
- 4,049
1
vote
1
answer
362
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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 ...
- 128k
3
votes
1
answer
3k
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Is there a tight lower bound for the expectation of the product of two positive valued random variables?
Let $X,Y$ be two (dependent) random variables with $\mathbb{P}(X\ge 0)=\mathbb{P}(Y\ge 0)=1$.
I want to find a tight lower bound of $\mathbb{E}(XY)$ when $X,Y$ are non-negative, almost surely.
...
CommunityBot
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1
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251
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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:
$...
- 93
2
votes
0
answers
69
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A distribution $\pi \propto \exp(-f)$ satisfies log-Sobolev inequality, does $\exp(-af)$ also satisfy LSI?
Assume a distribution $\pi \propto e^{-f}$ satisfies log-Sobolev inequality (LSI)
$$\forall \rho \in P(\mathbb{R}^n), \quad KL(\rho\| \pi) \le \frac{1}{2\lambda} I(\rho \| \pi)$$
with LSI constant $\...
- 21
3
votes
1
answer
142
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How does the integral of pseudo Gaussian kernel on $(0,\infty)$ depend on its variance?
Let $a, b: \mathbb R_+ \to [0,1]$ be continuous functions. Let $k: \mathbb R_+\times\mathbb R \to [1,2]$ be $1-$Lipschitz. Set, for $0<s<t$ and $y>0$,
$$A(s,t,y):=\int_s^t\frac{k(u,y)}{1+a(u)}...
- 128k
0
votes
1
answer
503
views
Asymptotics of a 1D integral, or the orthant probability of an equicorrelated random Gaussian vector
Problem: Let $\phi(x)$ be the normal probability density function (pdf), and $\Phi(x)$ the normal cumulative distribution (cdf). I'm interested in the asymptotic behavior of the following integral
$I(...
CommunityBot
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5
votes
1
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512
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Concentration inequality for Hilbert space valued random variables
I have read in a paper about the following result:
Let $V$ be a separable Hilbert space and $(\Omega,A_{\Omega},P)$ a probability space. Suppose that $Y_1,Y_2,...$ is a sequence of independent $V$-...
- 3,958
6
votes
1
answer
1k
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Variance of the norm of a random variable under finite-moment assumptions
There is the following exercise in Vershynin's book on High-Dimensional Probability.
Exercise 3.1.6:
Let $X = (X_1, \dots, X_n) \in \mathbb{R}^n$ be a random vector with independent coordinates $X_i$ ...
- 128k
2
votes
2
answers
161
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Determine the affine envelope of a random process's MGF
Suppose that a stationary random process $S(t)$ can be characterized as the figure below, which for most of the time is a straight line $S(t)=c\cdot t$, but occasionally would "stall" for a ...
- 128k
2
votes
1
answer
136
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Does higher volatility of SDE imply lower probability of staying positive?
Given two SDEs $X^1$, $X^2$ :
$$X^i_t=1+t+\int_0^t\sigma_i(s)dW_s,\quad \forall t\ge 0,$$
where $\sigma_i:\mathbb R_+\to [1/2,1]$ are non-decreasing s.t. $\sigma_1(t)\le \sigma_2(t)$ for all $t\ge 0$....
- 128k
10
votes
3
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803
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Discrete entropy of the integer part of a random variable
Let $X$ be a real valued random variable. Of course, the integer part $\lfloor X \rfloor$ of $X$ is a discrete random variable taking values in $\mathbb{Z}$. We can therefore define its discrete ...
- 128k
2
votes
1
answer
138
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Comparison between $\|X\|_2$ and $\|X\|_{2,1}$
For any real random variable $X$, define
$$\|X\|_{2,1}=\int_0^\infty \sqrt{\Pr(|X|>t)}dt.$$
This quantity (it is not a norm) appears in various problems, e.g. the multiplier central limit theorem (...
- 128k
0
votes
1
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133
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How to demonstrate a correlation inequality? [closed]
If there are 3 vectors X, Y, Z of the same length, for any $x_i \in X,y_i \in Y,z_i \in Z$, we have $0<x_i<1,0<y_i<1,0<z_i<1$.
The correlation between Z, Y is greater than between X, ...
- 63.9k
4
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0
answers
96
views
Is this conjecture about the binomial and beta distributions true?
Let $X$ follow a binomial distribution with parameters $n$ and $p$, and also fix $k$ such that $1<k<n$. Define
$$a = \mathbb{E}(X-k)^+$$
and
$$b = \mathbb{E}\log\binom{X}{(X-k)^+}$$
where the ...
3
votes
0
answers
169
views
Probabilistic behavior of greedy point selection in the plane
Let $\mathcal{X} = X_1,\dots,X_n$ be a collection of independent, uniform samples in the unit square. Let $\mathcal{S}=\{X_1\}$, and consider the following process: for $i=2,\dots,n$, let $x^*$ be ...
- 4,049
2
votes
1
answer
198
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Bounds for the beta CDF
This question is closely related to a previous question that I asked here:
An inequality involving the beta distribution
Let $a,b$ be strictly positive integers, and let $F_{a,b}(x)$ denote the CDF ...
- 128k
4
votes
0
answers
146
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An inequality for three iid random variables with a log-concave density
It was previously shown that
$$H\ge cG,\tag{1}$$
where $c:=1/14334$,
$$G:=E|X-Y|,\quad H:=E|X-Y|-\tfrac12\,E|X+Y-2Z|,$$
and $X,Y,Z$ are independent random variables with the same log-concave density.
...
- 128k
5
votes
1
answer
150
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Kullback–Leibler chains
The following question was asked and then deleted by the post author:
Let $P$ and $Q$ be two probability distributions defined over the same space, with $KL(P \parallel Q) < \infty$. For $\epsilon ...
- 128k
5
votes
1
answer
225
views
Anti-concentration of Gaussian when conditioning on event
Let $v$ be a given vector with $\|v\|_{\Sigma^{-1}} \leq 1$, where $\Sigma$ is a positive semi-definite matrix and $\|v\|_{\Sigma^{-1}} = \sqrt{v^\top\Sigma v}$. Meanwhile, let $u$ be a random vector ...
- 128k
1
vote
1
answer
207
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Expectation of the sum of the squares of the cardinal of an inverse function
I sample a random one-to-one function $\pi:\{0\,;\,1\}^n\to\{0\,;\,1\}^n$. I define $f$ as:
$$\forall x\in\left[0\,;\,2^n-1\right]\cap\mathbb{N},f(x)=x\oplus\pi(x)$$
where $\oplus$ is the bitwise XOR.
...
- 128k
1
vote
1
answer
107
views
Tail bounds on random series in Hilbert space
Tail bounds on random series in Hilbert space
Let $X_n$, $n \in \mathbb {N}$, be independent $\pm 1$ symmetric random variables, and $a_n$,
$n \in \mathbb {N}$, be a sequence in a Hilbert space $H$ ...
CommunityBot
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votes
1
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966
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Bound the norm of sum of random vector that generated from standard basis
I have a question like this:
Consider $N$ samples $X_1, X_2, ..., X_N$ that uniformly random generated from standard basis $\{e_i, i=1,2,...,d\}$, i.e. $(1,0,0,\cdots,0),(0,1,0,\cdots,0),(0,0,1,0,\...
- 128k
0
votes
1
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159
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Best bounds on the integral of an increasing function
The following question, somewhat edited here, was asked and then closed at The best bound of the integral of a nondecreasing real function in a closed interval.
Let $F\colon[0,1]\to[0,1]$ be a ...
- 128k
0
votes
1
answer
133
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Projection onto manifold of Gaussian measures by "trunction" of moments
Let $\mathcal{P}_2(\mathbb{R}^n)$ be the set of Borel probability measures on $\mathbb{R}^n$ with finite mean and variance; in the sense that
$$
\int_{x \in \mathbb{R}^n} \|x\|^p d\mathbb{P}(x) < \...
- 63.9k
0
votes
1
answer
340
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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 $v_i\in(0,1]$.
...
- 1,677
2
votes
0
answers
84
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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 ...
- 128k
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}...
- 65
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1
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199
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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 ...
- 195
3
votes
2
answers
1k
views
Expected value of a truncated binomial
Let $X\sim B(n,p)$ be a binomial random variable and fix $0<k<n$. Are there any well-known bounds for $\mathbb{E} (X-k)^+$, where $(X-k)^+ =\max\{0,X-k\}$? I am particularly interested in ...
- 128k
1
vote
1
answer
144
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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 $k$...
CommunityBot
- 1
3
votes
1
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431
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Inequality on the Hellinger distance between Poisson and mixture of Poisson
Let $H$ denote the Hellinger distance; i.e., for two discrete distributions $p,q$ (identified with their pmf) over $\mathbb{N}$,
$$
H(p,q)^2 = \frac{1}{2}\sum_{n=0}^\infty \left(\sqrt{p(n)}-\sqrt{q(n)}...
- 1,372
4
votes
0
answers
205
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Dimension reduction for low-order moments of Rademacher-weighted sums of vectors
Let $x_1,\dots,x_n$ be vectors in a Euclidean space $H$. Let $\varepsilon_1,\dots,\varepsilon_n$ be independent Rademacher random variables (r.v.'s), so that $P(\varepsilon_i=\pm1)=1/2$ for all $i$.
...
- 2,288
4
votes
3
answers
161
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Find distribution that minimises a function of its moments
Imagine a probability density function $f(x)$, defined for positive $x$, and let's note its $n$th non-centred moment $x_{n}$. The mean $x_{1}$ is fixed (and positive).
How can I find $f(x)$ that ...
- 17.2k
4
votes
3
answers
300
views
Reconstructing probability distribution with high probability
Sample $m$ times from unknown probability distribution $p=(p_1,p_2,\cdots,p_n)$, we can construct a probability distribution $q=(q_1.q_2,\cdots,q_n)$.
How large $m$ should be to achieve that the ...
- 128k
2
votes
1
answer
90
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A probability inequality: $p+(1-p)E[v|v\geq a] \geq E[v|v \geq p+(1-p)a]$
There is a random variable $v \sim F(\cdot)$ with support $[0,1]$. For a parameter $p \in (0,1)$ and $a \in (0,1)$. Define $A$ and $B$ as the following:
$$A=p+(1-p)E[v|v\geq a], B=E[v|v \geq p+(1-p)a]$...
- 128k
7
votes
3
answers
3k
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expected value of squared infinity norm of vector of iid gaussians
Given a random vector
\begin{equation}
x=(x_1, \ldots, x_n)
\end{equation}
with independent and identically distributed entries $x_i \sim \mathcal{N}(0,\sigma^2)$, I would like to find a lower ...
- 128k