All Questions
Tagged with measure-concentration pr.probability
324 questions
1
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0
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98
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Small ball probabilities for functions of correlated normals
Let $f : \mathbb{R}^k \rightarrow \mathbb{R}$ and let $X$ be distributed k-dimensional normal with mean $0$ (with "arbitrary" covariance matrix). I am looking for references with bounds of the form: ...
- 111
2
votes
1
answer
274
views
Is there monotonicity of measure concentration?
Suppose $X$ and $Y$ are nonnegative random variables such that $\mathrm{Pr}(X\geq t)\leq\mathrm{Pr}(Y\geq t)$ for all $t\geq0$. Now take $X_1,\ldots,X_n$ to be independent with the same distribution ...
- 7,633
1
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0
answers
610
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Upper bound on expectations of the sum of product of a martingale difference sequence with a predictable sequence, weighted by certain random weights
Let $(\mathcal{F}_i)_{i\geq 1}$ be a filtration. Let $0\leq p_i\leq 1$, be a random variable measurable w.r.t. $\mathcal{F}_i$. Consider two sequences of random vectors $v_i\in\mathbb{R}^M,w_i\in\...
- 323
5
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1
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282
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What is the spectral norm of a random projection times a diagonal?
Take $n\ll N$. Let $P$ be an $n\times N$ matrix of iid $\mathcal{N}(0,1)$ random variables, and let $D$ be an $N\times N$ diagonal matrix.
What can be said about the distribution of the largest ...
- 7,633
5
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2
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684
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Asymptotic Expansion of Distribution in Central Limit Theorem for Non-Identically Distributed Random Variables
My question is related to the following theorem (e.g. Section XVI.4 of Feller's 1971 book): Let $Z_i$ $(i=1,\cdots,n)$ be independent and identically distributed random variables with mean zero, ...
- 325
2
votes
1
answer
439
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Upper bound on the maxima of ratio of expectation of quantities under Gaussian measure
Let $\lambda,\eta >0$ be given, and $u:\mathbb{R}\rightarrow \mathbb{R}$ be a real valued function. Define
$$\Delta(u)= \frac{\int u(h) \exp(-\eta u(h))\exp(-\frac{\lambda}{2}h^2)~\mathrm{d}h}{\...
- 323
5
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1
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765
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Measure concentration for weakly dependent random variables
For an application quite alien to probability theory, I'd like to have a kind of measure concentration estimate, in the following spirit. Suppose that to every $1\le i,j\le n$ there corresponds a zero-...
- 23k
2
votes
2
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291
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How many boxes so that there is $k$ of same of color from $n$ different colors?
Say you have $m$ boxes each of which is colored with one of $n$ colors. What should $m$ be so that the probability that there is atleast $k$ boxes with one same color is strictly greater than $\frac{1}...
- 13.9k
9
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1k
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Balls and bins -- concentration bounds pertaining to the minimal load bin
Consider the standard balls and bins process, where $m$ balls are thrown uniformly at random into $n$ bins. Previous work has been done on estimating the value of the maximum load (i.e., the number of ...
- 201
10
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1
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846
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A Johnson-Lindenstrauss lemma for finite fields?
Given $m$ points in $\mathbb{R}^N$, the Johnson-Lindenstrauss lemma guarantees the existence of a linear operator $\mathbb{R}^N\rightarrow\mathbb{R}^n$ that nearly preserves pairwise distances between ...
- 7,633
4
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0
answers
1k
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Concentration of sum of independent random variables
Let $X_1, ..., X_n$ be i.i.d. sub-Gaussian random variables with mean $0$ and variance $1$. That is, we have $\Pr[|X_i| > t] \leq \exp(1-t^2/K^2)$ for all $t>0$ and a parameter $K$.
Then we can ...
- 1,324
6
votes
0
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337
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Chernoff bound in the not-quite-sub-exponential case
In Terry Tao's notes on Concentration of measure, Exercise 7 indicates that the Chernoff bound can be generalized to sub-exponential random variables:
http://terrytao.wordpress.com/2010/01/03/254a-...
- 7,633
6
votes
2
answers
720
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Local concentration of measure on Erdos-Rényi graph
Let $G_n=(V_n,E_n)$ be an Erdos-Rényi random graph, precisely the vertex set is $V_n=(1,\dots,n)$ and the edge set is $E_n=(ij\in\mathcal{P}_2(V_n)\ |\ \epsilon_{ij}=1)$ where $(\epsilon_{ij})_{ij}$ ...
- 293
3
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1
answer
1k
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Chernoff-Hoeffding bound for complex values
Consider the Chernoff-Hoeffding bound, stated as follows: Let $X_1, \dots, X_K$ be i.i.d. real-valued random variables with expectation value
$\mu$ and satisfying $|X_i| \le b$.
Let $\epsilon > 0$. ...
- 298
3
votes
1
answer
1k
views
concentration inequality for averages of dependent random variables
Let $X \in R^n$ be a random vector such that
$$P(|X_i| > \epsilon) > e^{-\epsilon^2}$$
What is a tight bound on
$$P(\sum_{i=1}^n |X_i| > \epsilon)$$
and on
$$P(\max_{1\le i\le n} |X_i| ...
- 461
14
votes
3
answers
2k
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Concentration bounds for sums of random variables of permutations
I'm trying to find theorems regarding random variables derived from sampling permutations, specifically concentration bounds.
As an example, let $X_i$ be the $\{0,1\}$-random variable that represents ...
- 539
8
votes
2
answers
1k
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Does Multiplicative Version of Azuma's Inequality Hold?
It is known that there are multiplicative version concentration inequalities for
sums of independent random variables. For example, the following
multiplicative version Chernoff bound.
Chernoff bound:...
- 175
7
votes
2
answers
2k
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Tails of sums of Weibull random variables
Suppose that $X_1, X_2, \ldots, X_n$ are i.i.d random variables distributed according to Weibull distribution with shape $0 < \epsilon < 1$ (it means that $\mathbf{Pr}[X_i \geq t] = e^{-\Theta(t^...
- 1,791
4
votes
1
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503
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An elementary probability question
Let $X$ be a $d$-dimensional random vector distributed according to probability measure $D$. At least the second moment of the coordinates of $X$ is finite.
Consider $n+1$ samples $X_0, \ldots, X_n ...
- 461
2
votes
1
answer
635
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Azuma's Inequality when the conditions hold with high probability?
In Azuma's Inequality, is the statement true when $|X_k - X_{k-1}| < c_k$ almost surely rather than with probability 1? If not, is there another result which gives strong concentration when the ...
- 201
18
votes
1
answer
1k
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How fast can extreme eigenvalues of the average of random matrices converge to their expectation?
Suppose that $X_1,X_2,\ldots,X_m$ are independent $d\times d$ random matrices and let $\overline{X} := \frac{1}{m}\sum_{i=1}^m X_i$. One of the questions studied under the theory of random matrices is ...
- 181
5
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0
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397
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Concentration of functions of random unitary matrices
Suppose $U$ and $V$ are $n \times n$ random unitary matrices, chosen independently from the Haar measure. Is there any kind of concentration inequality which would be applicable to polynomials $p(U,V)$...
- 2,449
5
votes
3
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898
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Lower bound for Gaussian random vector with negative correlation
Let $X = (X_1,\ldots,X_n) \in \mathbb{R}^n$ be jointly Gaussian with mean $0$, covariance matrix: $Var(X_i) = 1$, $Cov(X_i, X_{i+1}) = -1/2$, and $Cov(X_i, X_j) = 0$ else.
Let $\zeta \in \mathbb{R}^...
- 1,010
9
votes
2
answers
616
views
construction of a random measure with a given mean
Let me first pose a trivial question.
Given a Borel probability measure $\mu$ on the real line, is it possible to construct a purely atomic random measure $M$ whose mean is $\mu$?
The answer is ...
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