All Questions
Tagged with pr.probability measure-concentration
324 questions
2
votes
1
answer
302
views
Concentration on discrete probability estimator
Let $t>1$ and $X_1,..., X_t$ a set of real random variables from a discrete distribution, whose pmf is $p(x)$, supported on the points $1,...,k$.
Let $N_t(x) = \sum_{i = 1}^t \mathbb{1}_{X_i =\, x}....
- 245
4
votes
2
answers
175
views
Almost independence of $x^\top a$ and $x^\top b$ for $x$ uniform on the sphere in $\mathbb R^d$ and $a,b \in \mathbb R^d$ with $a^\top b = 0$
Let $d$ be a large positive integer. Let $x$ be uniformly distributed on the unit-sphere in $\mathbb R^d$ and let $a$ and $b$ be perpendicular vectors in $\mathbb R^d$, i.e such that $a^\top b=0$. Let ...
- 6,853
1
vote
1
answer
475
views
Convergence of quadratic form $y^T Q y$ where $y$ is a random iid sequence of length $n$ and $Q$ is an $n \times n$ random matrix independent of $y$
For each positive integer, let $Q_n=(q_{i,j})_{i,j \in [n]}$ be a random $n \times n$ psd matrix. In the limit $n \to \infty$, suppose the eigenvalues of this sequence of matrices are uniformly ...
- 6,853
2
votes
0
answers
172
views
Asymptotic lower and upper bounds for the eigenvalues of hadamard product $W \circ W$, where $W$ is a large Wishart matrix
Let $n$ and $d$ be large positive integers with $n,d \to \infty$ such that $n/d \to \gamma \in (0,\infty)$. Let $X$ be a random $n \times d$ random matrix with iid copies of log-concave isotropic ...
- 6,853
2
votes
1
answer
415
views
High-probability lower bound for norm of least squares solution when both design matrix $X$ and response vector $y$ are random (and independent)
Let $n,d \to \infty$ with $n/d \to \gamma \in (0,\infty)$. Let $X$ be a random $n \times d$ matrix independent rows uniformly distributed on the the unit-sphere in $\mathbb R^d$ and let $y$ be a ...
- 6,853
3
votes
1
answer
553
views
How did the story of Kim-Vu type inequalities continue?
I am interested in the concentration of polynomials of random variables. I have been reading Boucheron, Lugosi, and Massart's "Concentration inequalities" and they give some references. ...
- 233
1
vote
1
answer
144
views
Bounds for the extreme singular-values of random matrix with thresholded entries
Let $n,d,k$ be large positive integers such that $\max(n/d,k/d) =: \lambda < 1$. Let $X$ be a random $n \times d$ matrix with entries drawn iid from $N(0,1/d)$ and let $W$ be a $k \times d$ random ...
- 6,853
1
vote
0
answers
155
views
Relation between the class $\mathcal{M}(m,\sigma)$ and subgaussianity
In this paper, Adamczak defines, for $m>0$ and $\sigma\geq 0$, the class of probability distributions $\mathcal{M}(m,\sigma)$ over $\mathbb{R}$ as those $\mu$ satisfying the tail conditions
$$\nu^+(...
- 1,372
2
votes
0
answers
83
views
Concentration inequalities for sets
Assume that we have a random set $B$ which is constructed by selecting elements from $U = \{ X_1, \dots, X_n \}$ where $X_i$ are independent samples from Gaussians with means $\mu_i$ and variances $\...
- 143
2
votes
1
answer
668
views
Lower-bound for smallest eigenvalue of random $k \times $k matrix $C(W)$ defined by $C(W)_{i,j} := 2(w_i^\top w_j)^2 + \|w_i\|^2\|w_j\|^2$
Let $k$ and $d$ be positive integers such that $d/k:=\lambda > 1$. Let $W$ be $k \times d$ random matrix with rows $w_1,\ldots,w_k \in \mathbb R^d$ drawn iid from $N(0,(1/d)I_d)$, and define the $k ...
- 6,853
1
vote
0
answers
96
views
Concentration for $\sum_{i=1}^n y_i \psi(x_i^\top u)$, for $y_1,\ldots,y_n \sim \{\pm 1\}$ and $x_1,\ldots,x_n$ uniform iid on hypersphere
Let $y_1,\ldots,y_n$ be drawn iid uniformly from $\{\pm 1\}$ and let $x_1,\ldots,x_n$ be drawn iid uniformly from the unit-sphere $(d-1)$-dimensional sphere $\mathbb S_{d-1}$, and independently from ...
- 6,853
3
votes
1
answer
379
views
Concentration inequality for norm of solution to nonlinear least-squares problem
Define the piecewise-linear function $\psi(t):=\max(t,0)$ for all $t \in \mathbb R$.
Let $d,n,k \to \infty$ at the same rate (i.e $n \asymp k \asymp d$).
Let $y_1,\ldots,y_n \in \{-1,1\}$ uniformly ...
- 6,853
1
vote
0
answers
68
views
(Anti-)concentration of gap between largest and second largest component of multivariate random gaussian vector
Let $n$ be a large positive integer and let $Y=(Y_1,\ldots,Y_n)$ be a zero-centered random $n$-dmensional real vector with covariance matrix $\Sigma$, an $n$-by-$n$ positive definite matrix with ...
- 6,853
0
votes
1
answer
966
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,\...
- 25
3
votes
1
answer
176
views
Gaussian concentration/isoperimetric inequality with correlated Gaussian measure
Famous Gaussian concentration inequality states that:
If $\mathrm{F}$ is 1 -Lip, and $\mathbb{E} F(X)=0,$ and $X=(X_1,...,X_n) \sim N\left(0, I_{n}\right),$ then we have for some absolute constant $C&...
- 519
2
votes
2
answers
690
views
Concentration and anti-concentration of gap between largest and second largest value in Gaussian iid sample
Let $n \ge 3$ be an integer and let $X=(X_1,\ldots,X_n)$ be random vector with iid coordinates from $N(0,1)$. For $1 \le k \le n$, let $X_{(k)}$ be the value of the $k$th largest coordinate of $X$.
...
- 6,853
1
vote
1
answer
261
views
Concentration inequality for a function whose parameter depends on input samples
Concentration inequalities can be used to establish results such as sample mean cannot be too far from the actual population mean, and so on. For example, let $X_1 \ldots X_n$ be i.i.d instances of a ...
- 615
3
votes
1
answer
114
views
Lower-bound for $\underset{p \le \gamma_d(A) \le q}{\inf} \gamma(A^\epsilon)$, where $\gamma_d$ is the standard gaussian distribution on $\mathbb R^d$
Let $\gamma_d = \gamma_1 \otimes \ldots \otimes \gamma_1$ be the standard Gaussian distribution on $\mathbb R^d$, where $d$ is a large positive integer. Given $\epsilon \ge 0$ and a measurable $A \...
- 6,853
9
votes
1
answer
350
views
Concentration inequalities for very rare events on a multiplicative scale
Let $E_1, \dots, E_N$ be independent events, each of probability $p$, where $p$ is very close to $0$. Let $A_N = \frac{1}{N} ( 1_{E_1} + \dots + 1_{E_N} )$ be the proportion of the events $E_i$ that ...
- 323
1
vote
0
answers
57
views
Good lower-bound for $\inf_{x \in \Delta_n} \|Gx\|$ where $G$ is an $N \times n$ random matrix with iid entries from $\mathcal N(0,1/\sqrt{N})$
Let $G$ be an $N \times n$ random matrix with independent entries distributed according to a centered Gaussian with variance $1/\sqrt{N}$ and let $n/N = \lambda \in (0, 1)$. Let $\Delta_n$ be the $(n-...
- 6,853
3
votes
2
answers
636
views
Exponential inequality for the sum of martingale differences $X_1, \dots, X_n$ when $\sum_{i=1}^{n} \operatorname{Var}(X_i) \leq B^2$
Let $X_1, X_2, \dots, X_n$ be a martingale difference sequence such that
$$
X_i \leq y \quad \text{and} \quad \sum_{i=1}^{n} \operatorname{Var}(X_i) \leq B^2.
$$
Question 1: Does the following hold?
$$...
- 33
1
vote
1
answer
141
views
Central limit theorem for chi-squared random field on $\mathbb R^p$
Let $X:x \mapsto X(x)$ be a centered stationary Gaussian process on the $\Omega:=\mathbb R^p$, such that $X(x) \overset{d}{=}X(x')$ for all $x,x' \in \Omega$. Set $\sigma^2 := \mbox{Var}(X(0)) = \...
- 6,853
1
vote
1
answer
343
views
Concentration inequality for the supremum of $L_2$ norm of a vector-valued Gaussian process with iid components
Let $\Omega$ be a compact subset of $\mathbb R^p$ and let $f_1,\ldots,f_k$ be zero mean identically distrubuted Gaussian processes on $\Omega$ such that $f_1(x),\ldots,f_k(x)$ are independent $x \in \...
- 6,853
1
vote
1
answer
59
views
Characterization of random variables whose tensor powers have subexponential "small-ball" probabilities
Is there a succinct characterization of all random variables $\zeta$ on $\mathbb R$ with the following properties
1. Symmetry: $\zeta \overset{d}{=} - \zeta$.
2. Small-ball probability: there exists ...
- 6,853
2
votes
1
answer
212
views
Prove / disprove: If $1 \le n < N$ and $A$ is an $N \times n$ matrix with iid from $\mathcal N(0,1)$, then $s_\min(A) \ge c\sqrt{N}$ w.p $1-2e^{-N}$
Let $1 \le n < N$ be integers and $A$ be a random $N\times n$ matrix with iid entries from $\mathcal N(0,1)$. This paper (Rudelson and Vershynin) claims in the paragraph just before formula (3.4) ...
- 6,853
0
votes
1
answer
280
views
Lower-bound on smallest singular-value of rectangular random matrix
Let $X$ be a random $N \times n$ matrix with iid entries from $\mathcal N(0, 1)$ and with $n/N =: \lambda(N,n) \le \lambda_0$, for some $\lambda_0 \in (0, 1)$. That is, $X$ is genuinely rectangular (...
- 6,853
3
votes
0
answers
103
views
Concentration inequalities for gradient flows induced by random fields
Let $G=(G(x))_{x \in \mathbb R^m}$ be a conservative random field with values in $\mathbb R^m$, for large positive integer $m$. That is, there exists a scalar random field $g=(g(x))_{x \in \mathbb R^m}...
- 6,853
3
votes
1
answer
1k
views
Extension of Bernstein’s Inequality when the random variable is bounded with large probability
Bernstein’s Inequality can be stated as follows : Let $x_1, x_2, \dots, x_n$ be independent bounded random variables such that $\mathbb{E}[x_i] = 0$ and $|x_i| \leq \zeta$ with probability $1$ and let ...
- 33
3
votes
1
answer
88
views
If $X \sim N(0,I_m)$, what is a necessary and sufficient condition on $u_m > 0$ such that $\lim\sup_{m\to \infty} P(\|X\|^2 \ge u_m|X_1|) = 1$
Let $m$ be a large positive integer and $X=(X_1,\ldots,X_m) \sim N(0,I_m)$. I wish to show that the squared norm of $X$ is much much bigger than the absolute value of any of the $X_j$'s. For example, ...
- 6,853
5
votes
1
answer
273
views
Chernoff-style concentration inequality for k-tuples
I'm looking for a seemingly natural generalization of a Chernoff bound.
In many scenarios, we have a distribution $D$ with support $\mathsf{Supp}(D)$, and some event $E \subset \mathsf{Supp}(D)$ ...
- 161
0
votes
1
answer
806
views
Concentration of $\ell_2$ norm of a vector sampled from a distribution
Let $X=(X_1,\ldots,X_n)$, where $X_i \sim P_{p_i}(0,\frac{1}{\lambda})$ are iid, $P_{p_i}$ is sub gaussian distribution for $i^\text{th}$ element, and 0 and $1/\lambda$ are mean and variance.
I'm ...
- 61
3
votes
1
answer
182
views
How tight is the bound $P(\|X\|^2 \ge t |\langle a,X\rangle|) \ge 1 - t\sqrt{\frac{2}{m-1}}$, where $X \sim N(0, I_m)$ and $\|a\| = 1$?
Let $X$ be a random vector in $\mathbb R^m$ with iid $N(0,1)$ coordinates and let $a$ be a fixed unit vector in $\mathbb R^m$. In another post (SE link here https://math.stackexchange.com/a/3792730/...
- 6,853
24
votes
1
answer
1k
views
A Rademacher ‘root 7’ anti-concentration inequality
Let $r_1,r_2,r_3,\dotsc$ be an IID sequence of Rademacher random variables, so that $\mathbb P(r_n=\pm1)=1/2$, and $a_1,a_2,\dotsc$ be a real sequence with $\sum_na_n^2=1$. For $S=\sum_na_nr_n$, does ...
- 17.1k
2
votes
0
answers
68
views
Approximate any point of the interval $[-1/2,1/2]$ by the sum of $n$ iid uniform random variables from $[-1,1]$
Let $x \in [-1/2,1/2]$ and $X_1,\ldots,X_n$ be drawn iid from the uniform distribution on $[-1,1]$.
Question. Given $\varepsilon \ge 0$ an integer $k \in [1,n]$, what is a good lower-bound on the ...
- 6,853
1
vote
1
answer
201
views
Upper bound for $\mathbb P(|f(A+XX^T)-f(A)| > \epsilon)$, where $A$ is a fixed pd matrix and $X$ has random iid entries
Let $A$ be a fixed $n$ by $n$ real symmetric positive definite matrix with eigenvalues $\lambda_1 \ge \lambda_2 \ge \ldots \ge \lambda_n > 0$, and let $f(A):=\sum_{i=1}^n\log\lambda_i$, and let $X$ ...
- 6,853
3
votes
2
answers
402
views
Something between the Chernoff and Hoeffding bounds
Suppose I have $n$ independent 0-1 random variables $X_1, \cdots, X_n$ and I want to show a concentration of $X = \sum_i X_i$.
I can use either the Chernoff bound or the Hoeffding bound.
Suppose $E[...
- 233
-1
votes
1
answer
138
views
On the concentration of Lipschitz functions near its expectation, where the vector has identical but not independent, components
Consider the random vector $X:=(X_1\dots X_1) \in \mathbb{R}^n, X_1 \sim \mathcal{N}(0,1).$ Notice the identical components, they're identically distributed but not independent.
Now, I was wondering ...
- 1,467
5
votes
1
answer
1k
views
Explicit constant for Carbery–Wright inequality
The Carbery–Wright inequality is a seminal result about the anti-concentration of polynomials of Gaussian random variables.
See e.g. Meka, Nguyen, and Vu - Anti-concentration for polynomials of ...
- 111
0
votes
1
answer
378
views
Concentration of norm of linearly transformed normal random vector as dimension go to infinity
Earlier asked on MSE, but didn't get an answer, so posting here:
Let $X=(X_1 \dots X_n) \in \mathbb{R}^n, X_i\sim N(0,1), iid.$ Let $B: \mathbb{R}^n \to \mathbb{R}^n $ be the diagonal linear map: $...
- 1,467
0
votes
1
answer
58
views
Good upper-bound for $\mathbb E_A[e^{-t\|A\|_2}]$, for $t\ge0$ and random m by n matrix with iid entries with law $N(0,1)$
Let $A$ be a random $m$-by-$n$ matrix with iid $N(0,1)$ entries, $m$ and $n$ large with $n/m \longrightarrow \alpha \in (0, 1)$ . Let $\|A\|_2$ be the largest singular value of $A$ (i.e the spectral ...
- 6,853
3
votes
0
answers
307
views
Upper-bound for eigenvalues of $E [UU^T]$, where $U$ is uniformly distributed on the unit $n$-sphere
Let $X$ be a $\sigma$-subGaussian random vector on $\mathbb R^n$ (for large $n \ge 3$), meaning that the random variable $X^Tv$ is $\sigma$-subGaussian for every unit vector $v \in \mathbb R^n$. ...
- 6,853
4
votes
0
answers
638
views
Comparison of concentrations of different $L^p$-norms of (sub) Gaussian distributions
It's well-known that the Euclidean $2$-norm of subgaussian random vectors concentrates in high dimensions, e.g. when $X \sim \mathcal{N}(0,I_n),$ (or in general $X$ is subgaussian with independent co-...
- 1,467
5
votes
1
answer
165
views
Is there an i.i.d sequence in the unit cube $[-1,1]^d$ with $\mathbb E \left[ \Big \| \sum_{i=1}^N X_N \Big \|_\infty\right] = \sqrt {dN}$?
There are loads of concentration results for sums of scalar-valued independent sums $X_1,X_2,\ldots, X_N$ with $\mathbb E[X_n]=0$. For example Hoeffding's Inequality says if all $|X_1|\le 1$ then $\...
- 1,955
0
votes
1
answer
140
views
Tail bounds for the absolute difference of a coupled pair of sub-Gaussian random variables
Let $P$ and $Q$ are sub-Gaussian distributions on $\mathbb R$, and $(X,X')$ be a coupling of $P$ and $Q$, i.e $(X,X') \sim \pi$ for some distribution on $\mathbb R^2$ with marginals $P$ and $Q$.
...
- 6,853
1
vote
1
answer
89
views
Use $\mathbb{P}(\vert \hat{s}_n-s\vert > x)\leq a(n,x)$ and $\mathbb{P}(\vert \hat{s}_n-s_n\vert > x)\leq b(n,x)$ to bound $\vert s_n - s\vert$
Let $s, s_n\in\mathbb{R}$ and $\hat{s}_n$ be a random variable.
I have two concentration inequalities:
$$\mathbb{P}(\vert \hat{s}_n-s\vert > x)\leq a(n,x)$$ for all $n\geq1$ and $x>0$;
and
$$\...
- 11
2
votes
0
answers
58
views
An upper bound on $\mathbb{E}\bigg[\bigg(\sum_{i=1}^{k}(X^{\top}A_{i}X)^{2}\bigg)^{q}\bigg]$
Let $X\in\mathbb{R}^{d}$ have independent, mean zero subgaussian entries, and $A_{1},\ldots,A_{k}$ be fixed $d\times d$ matrices that have zeros on the diagonal. I would like to upper bound the ...
- 129
3
votes
1
answer
1k
views
Chernoff-type bound for sum of Bernoulli random variables, with outcome-dependent success probabilities
Let $X = (X_1, X_2, \ldots, X_n)$ be a sequence of (not necessarily independent) Bernoulli random variables where for each $i$, the success probability $\Pr[X_i = 1]$ itself is a random variable ...
- 153
0
votes
1
answer
273
views
Sum of sequences of random variables, with variable success probabilities
Consider two sequences of (not necessarily independent) Bernoulli random variables $X_1, X_2, \ldots, X_n$ and $Y_1, Y_2, \ldots, Y_n$. Suppose that for any $i$, we have $\Pr[X_i = 1] = \Pr[Y_i = 1] = ...
- 153
7
votes
3
answers
496
views
Chernoff-type bounds for a stopped sum of independent random variables
Let $Y_1, \ldots, Y_n$ and $X_1, \ldots, X_n$ be i.i.d. $p$-Bernoulli random variables and let $T \in \{0, \ldots, n\}$ be a stopping time for the process. From Wald's equation, we know
$$
E\left[\...
- 153
3
votes
1
answer
2k
views
Gaussian concentration inequality
Recently I found a concentration inequality for infinite dimensional Gaussian r.v.s in this paper. Specifically, Lemma 4 on page 307 states (without a proof) that
There exists a universal constant $...
- 185