Sub-Gaussian distribution explained

In probability theory, a subgaussian distribution, the distribution of a subgaussian random variable, is a probability distribution with strong tail decay. More specifically, the tails of a subgaussian distribution are dominated by (i.e. decay at least as fast as) the tails of a Gaussian. This property gives subgaussian distributions their name.

Often in analysis, we divide an object (such as a random variable) into two parts, a central bulk and a distant tail, then analyze each separately. In probability, this division usually goes like "Everything interesting happens near the center. The tail event is so rare, we may safely ignore that." Subgaussian distributions are worthy of study, because the gaussian distribution is well-understood, and so we can give sharp bounds on the rarity of the tail event. Similarly, the subexponential distributions are also worthy of study.

Formally, the probability distribution of a random variable

X is called subgaussian if there is a positive constant C such that for every t \geq 0 ,

$\backslash operatorname(|X|\; \backslash geq\; t)\; \backslash leq\; 2\; \backslash exp$.There are many equivalent definitions. For example, a random variable

X is sub-Gaussian iff its distribution function is bounded from above (up to a constant) by the distribution function of a Gaussian:P(|X| \geq t) \leq cP(|Z| \geq t) \forall t > 0

where

c\ge 0 is constant and Z is a mean zero Gaussian random variable.

Definitions

Subgaussian norm

The subgaussian norm of

X , denoted as \Vert X \Vert _{\psi _{2}} , is$$\backslash Vert\; X\; \backslash Vert\_\; =\; \backslash inf\backslash left\backslash .$$In other words, it is the Orlicz norm of X generated by the Orlicz function \Phi(u) = e ^{u ^{2}} -1 . By condition ( 2 ) below, subgaussian random variables can be characterized as those random variables with finite subgaussian norm.

Variance proxy

If there exists some

s ^{2} such that \operatorname{E} [e ^{(X-\operatorname{E} [X])t}] \leq e ^{(s ^{2t} ^{2})/(2)} for all t , then s ^{2} is called a variance proxy, and the smallest such s ^{2} is called the optimal variance proxy and denoted by \Vert X\Vert ^{2} _{vp} .

Since

\operatorname{E} [e ^{(X-\operatorname{E} [X])t}] = e ^{(\sigma ^{2} t ^{2})/(2)} when X \sim l{N}(\mu, \sigma ^{2)} is Gaussian, we then have \|X\| ^{2} _{vp} = \sigma ^{2} , as it should.

Equivalent definitions

Let

X be a random variable. The following conditions are equivalent: (Proposition 2.5.2 ^[1])

1. Tail probability bound:

\operatorname{P}(|X| \geq t) \leq 2 \exp{( - t ^{2 )}} _{1} for all t \geq 0 , where K _{1} is a positive constant;

1. Finite subgaussian norm:

\Vert X \Vert _{\psi _{2}} = K _{2} < inf ty .

1. Moment:

\operatorname{E} |X| ^{p} \leq 2 K ^{p} _{3} \Gamma\left( (p)/(2) +1 \right) for all p \geq 1 , where K _{3} is a positive constant and \Gamma is the Gamma function.

1. Moment:

\operatorname{E}|X| ^{p\leq} K ^{p} p ^{p/2} for all p \geq 1 ,

1. Moment-generating function (of

X ), or variance proxy : \operatorname{E} [e ^{(X-\operatorname{E} [X])t}] \leq e ^{(K ^{2t} ^{2})/(2)} for all t , where K is a positive constant.

1. Moment-generating function (of

X ^{2} ): for some K > 0 , \operatorname{E}[e ^{X ^{2t} ^{2}} ] \leq e ^{K ^{2t} ^{2}} for all t \in [ -1 /K, +1 /K] .

1. Union bound: for some c > 0,

\operatorname{E}[ max \{|X _{1} - \operatorname{E}[X]|,\ldots,|X _{n} - \operatorname{E}[X]|\}] \leq c \sqrt{ log n} for all n > c, where X _{1,} \ldots, X _{n} are i.i.d copies of X.

1. Subexponential:

X ^{2} has a subexponential distribution.Furthermore, the constant K is the same in the definitions (1) to (5), up to an absolute constant. So for example, given a random variable satisfying (1) and (2), the minimal constants K _{1,} K _{2} in the two definitions satisfy K _{1} \leq cK _{2,} K _{2} \leq c' K _{1} , where c, c' are constants independent of the random variable.

Proof of equivalence

As an example, the first four definitions are equivalent by the proof below.

Proof.

( 1 )\implies( 3 ) By the layer cake representation,

After a change of variables

u = t ^{2} _{1} , we find that( 3 )\implies( 2 ) By the Taylor series $e^x\; =\; 1\; +\; \backslash sum\_^\backslash infty\; \backslash frac,$which is less than or equal to 2 for λ \leq (1)/(3 K ^{2} _{3}) . Let K _{2} \geq 3 ^{(1)/(2)} K _{3} , then $\backslash operatorname[\backslash exp\{(X^2/K\_2^2)\}]\; \backslash leq\; 2.$( 2 )\implies( 1 ) By Markov's inequality,$$\backslash operatorname(|X|\backslash geq\; t)\; =\; \backslash operatorname\backslash left(\backslash exp\backslash left(\backslash frac\backslash right)\; \backslash geq\; \backslash exp\backslash left(\backslash frac\backslash right)\; \backslash right)\; \backslash leq\; \backslash frac\; \backslash leq\; 2\; \backslash exp\backslash left(-\backslash frac\backslash right).$$( 3 ) \iff ( 4 ) by asymptotic formula for gamma function: \Gamma(p/ 2 + 1 ) \sim \sqrt{\pi p} \left( (p)/(2 e) \right) ^{p/2} .

From the proof, we can extract a cycle of three inequalities:

• If

\operatorname{P}(|X| \geq t) \leq 2 \exp{( - t ^{2/K} ^{2)}} , then \operatorname{E} |X| ^{p} \leq 2 K ^{p} \Gamma\left( (p)/(2) +1 \right) for all p \geq 1 .

• If

\operatorname{E} |X| ^{p} \leq 2 K ^{p} \Gamma\left( (p)/(2) +1 \right) for all p \geq 1 , then \|X \| _{\psi _{2}} \leq 3 ^{(1)/(2)} K .

• If

\|X \| _{\psi _{2}} \leq K , then \operatorname{P}(|X| \geq t) \leq 2 \exp{( - t ^{2/K} ^{2)}} .

In particular, the constant

K provided by the definitions are the same up to a constant factor, so we can say that the definitions are equivalent up to a constant independent of X .

Similarly, because up to a positive multiplicative constant,

\Gamma(p/ 2 + 1 ) = p ^{p/2} x (( 2 e) ^{-1/2} p ^{1/2p} ) ^{p} for all p \geq 1 , the definitions (3) and (4) are also equivalent up to a constant.

Basic properties

Proposition.

• If

X is subgaussian, and k > 0 , then \|kX\| _{\psi _{2}} = k \|X\| _{\psi _{2}} and \|kX\| _{vp} = k \|X\| _{vp} .

• If

X, Y are subgaussian, then \|X + Y\| ^{2} _{vp} \leq (\|X\| _{vp} + \|Y\| _{vp} ) ^{2} .Proposition. (Chernoff bound) If X is subgaussian, then Pr(X \geq t) \leq e ^{- (t ^{2})/(2 \|X\| ^{2} _{vp})} for all t \geq 0 .

Definition.

X \lesssim X' means that X \leq CX' , where the positive constant C is independent of X and X' .

Proposition. If

X is subgaussian, then \|X - E[X]\| _{\psi _{2}} \lesssim \|X\| _{\psi _{2}} .

Proof. By triangle inequality,

\|X - E[X]\| _{\psi _{2}} \leq \|X\| _{\psi _{2}} + \|E[X]\| _{\psi _{2}} . Now we have \|E[X]\| _{\psi _{2}} = \sqrt{ ln 2 } |E[X]| \leq \sqrt{ ln 2 } E[|X|] \sim E[|X|] . By the equivalence of definitions (2) and (4) of subgaussianity, given above, we have E[|X|] \lesssim \|X\| _{\psi _{2}} .

Proposition. If

X, Y are subgaussian and independent, then \|X + Y\| ^{2} _{vp} \leq \|X\| ^{2} _{vp} + \|Y\| ^{2} _{vp} .

Proof. If independent, then use that the cumulant of independent random variables is additive. That is,

ln \operatorname{E}[e ^{t(X+Y)} ] = ln \operatorname{E}[e ^{tX} ] + ln \operatorname{E}[e ^{tY} ] .

If not independent, then by Hölder's inequality, for any

1 /p + 1 /q = 1 we have$$E[e^\{t(X+Y)\}]\; =\; \backslash |e^\backslash |\_1\; \backslash leq\; e^$$Solving the optimization problem \begin{cases} min p \|X\| ^{2} _{vp} + q \|Y\| ^{2} _{vp} \\ 1 /p + 1 /q = 1 \end{cases} , we obtain the result.

Corollary. Linear sums of subgaussian random variables are subgaussian.

Strictly subgaussian

Expanding the cumulant generating function:$$\backslash frac\; 12\; s^2\; t^2\; \backslash geq\; \backslash ln\; \backslash operatorname[e^\{tX\}]\; =\; \backslash frac\; 12\; \backslash mathrm[X]\; t^2\; +\; \backslash kappa\_3\; t^3\; +\; \backslash cdots$$we find that

Var [X] \leq \|X\| ^{2} _{vp} . At the edge of possibility, we define that a random variable X satisfying Var [X] = \|X\| ^{2} _{vp} is called strictly subgaussian.

Properties

Theorem.^[2] Let

X be a subgaussian random variable with mean zero. If all zeros of its characteristic function are real, then X is strictly subgaussian.

Corollary. If

X _{1,} ... , X _{n} are independent and strictly subgaussian, then any linear sum of them is strictly subgaussian.

Examples

By calculating the characteristic functions, we can show that some distributions are strictly subgaussian: symmetric uniform distribution, symmetric Bernoulli distribution.

Since a symmetric uniform distribution is strictly subgaussian, its convolution with itself is strictly subgaussian. That is, the symmetric triangular distribution is strictly subgaussian.

Since the symmetric Bernoulli distribution is strictly subgaussian, any symmetric Binomial distribution is strictly subgaussian.

Examples

X\

_	\	X\	_^2	strictly subgaussian?
gaussian distribution l N ( 0 , 1 )	\sqrt{ 8 / 3 }	1	Yes
mean-zero Bernoulli distribution p\delta _{q} + q \delta _{-p}	solution to pe ^{(q/t) ^{2}} + qe ^{(p/t) ^{2}} = 2	(p - q)/(2 ( log p -log q))	Iff p =0 , 1 / 2 , 1
symmetric Bernoulli distribution (12)/(\delta) _{1/2} + (12)/(\delta) _{-1/2}	(1)/(2 \sqrt{ ln2) }	1 / 4	Yes
uniform distribution U( 0 , 1 )	solution to \int ^{1} _{0} e ^{x ^{2/t} ^{2}} dx = 2 , approximately 0.7727	1 / 3	Yes
arbitrary distribution on interval [a, b]		\leq \left( (b - a)/(2) \right) ^{2}

The optimal variance proxy \Vert X\Vert ^{2} _{vp} is known for many standard probability distributions, including the beta, Bernoulli, Dirichlet, Kumaraswamy, triangular, truncated Gaussian, and truncated exponential.

Bernoulli distribution

Let

p + q = 1 be two positive numbers. Let X be a centered Bernoulli distribution p\delta _{q} + q \delta _{-p} , so that it has mean zero, then \Vert X\Vert ^{2} _{vp} = (p - q)/(2 ( log p -log q)) .^[3] Its subgaussian norm is t where t is the unique positive solution to pe ^{(q/t) ^{2}} + qe ^{(p/t) ^{2}} = 2 .

Let

X be a random variable with symmetric Bernoulli distribution (or Rademacher distribution). That is, X takes values -1 and 1 with probabilities 1 / 2 each. Since X ^{2=1} , it follows that$$\backslash Vert\; X\; \backslash Vert\_\; =\; \backslash inf\backslash left\backslash \; =\; \backslash inf\backslash left\backslash =\backslash frac,$$and hence X is a subgaussian random variable.

Bounded distributions

Bounded distributions have no tail at all, so clearly they are subgaussian.

If

X is bounded within the interval [a, b] , Hoeffding's lemma states that \Vert X\Vert ^{2} _{vp} \leq \left( (b - a)/(2) \right) ^{2} . Hoeffding's inequality is the Chernoff bound obtained using this fact.

Convolutions

Since the sum of subgaussian random variables is still subgaussian, the convolution of subgaussian distributions is still subgaussian. In particular, any convolution of the normal distribution with any bounded distribution is subgaussian.

Mixtures

Given subgaussian distributions

X _{1,} X _{2,} ... , X _{n} , we can construct an additive mixture X as follows: first randomly pick a number i \in \{ 1 , 2 , ... , n\} , then pick X _{i} .

Since

\operatorname{E}\left[\exp{\left( (X ^{2})/(c ^{2}) \right)}\right] = \sum _{i} p _{i} \operatorname{E}\left[\exp{\left( (X ^{2} _{i})/(c ^{2}) \right)}\right] we have \|X\| _{\psi _{2}} \leq max _{i} \|X _{i\|} ^{\psi _{2}} , and so the mixture is subgaussian.

In particular, any gaussian mixture is subgaussian.

More generally, the mixture of infinitely many subgaussian distributions is also subgaussian, if the subgaussian norm has a finite supremum:

\|X\| _{\psi _{2}} \leq \sup _{i} \|X _{i\|} ^{\psi _{2}} .

Subgaussian random vectors

So far, we have discussed subgaussianity for real-valued random variables. We can also define subgaussianity for random vectors. The purpose of subgaussianity is to make the tails decay fast, so we generalize accordingly: a subgaussian random vector is a random vector where the tail decays fast.

Let

X be a random vector taking values in \R ^{n} .

Define.

\|X\| _{\psi _{2}} : = \sup _{v \in S ^{n-1}} \|v ^{T} X\| _{\psi _{2}} , where S ^{n-1} is the unit sphere in \R ^{n} .X is subgaussian iff \|X\| _{\psi _{2}} < inf ty .

Theorem. (Theorem 3.4.6) For any positive integer

n , the uniformly distributed random vector X \sim U(\sqrt{n} S ^{n-1} ) is subgaussian, with \|X\| _{\psi _{2}} \lesssim{} 1 .

This is not so surprising, because as

n \to inf ty , the projection of U(\sqrt{n} S ^{n-1} ) to the first coordinate converges in distribution to the standard normal distribution.

Maximum inequalities

Proposition. If

X _{1,} ... , X _{n} are mean-zero subgaussians, with \|X _{i} \| ^{2} _{vp} \leq \sigma ^{2} , then for any \delta > 0 , we have max (X _{1,} ... , X _{n)} \leq \sigma\sqrt{ 2 ln (n)/(\delta) } with probability \geq 1- \delta .

Proof. By the Chernoff bound,

Pr(X _{i} \geq \sigma \sqrt{ 2 ln (n/\delta)}) \leq \delta/n . Now apply the union bound.

Proposition. (Exercise 2.5.10) If

X _{1,} X _{2,} ... are subgaussians, with \|X _{i} \| _{\psi _{2}} \leq K , then $$E\backslash left[\backslash sup\_n\; \backslash frac\{|X\_n|\}\{\backslash sqrt\{1+\backslash ln\; n\}\}\backslash right]\; \backslash lesssim\; K,\; \backslash quad\; E\backslash left[\backslash max\_\{1\; \backslash leq\; n\; \backslash leq\; N\}\; |X\_n|\backslash right]\; \backslash lesssim\; K\; \backslash sqrt$$Further, the bound is sharp, since when X _{1,} X _{2,} ... are IID samples of l N( 0 , 1 ) we have E\left[ max _{1} |X _{n|\right]} \gtrsim \sqrt{ ln N} .^[4]

^[5]

Theorem. (over a finite set) If

X _{1,} ... , X _{n} are subgaussian, with \|X _{i} \| ^{2} _{vp} \leq \sigma ^{2} , thenTheorem. (over a convex polytope) Fix a finite set of vectors v _{1,} ... , v _{n} . If X is a random vector, such that each \| v ^{T} _{i} X \| ^{2} _{vp} \leq \sigma ^{2} , then the above 4 inequalities hold, with max _{v \in conv (v _{1,} ... , v _{n)}} v ^{T} X replacing max _{i} X _{i} .

Here,

conv (v _{1,} ... , v _{n)} is the convex polytope spanned by the vectors v _{1,} ... , v _{n} .

Theorem. (over a ball) If

X is a random vector in \R ^{d} , such that \|v ^{T} X\| ^{2} _{vp} \leq \sigma ^{2} for all v on the unit sphere S , then $$E[\backslash max\_\{v\; \backslash in\; S\}\; v^T\; X]\; =\; E[\backslash max\_\{v\; \backslash in\; S\}\; |v^T\; X|]\; \backslash leq\; 4\backslash sigma\; \backslash sqrt$$For any \delta > 0 , with probability at least 1- \delta ,$$\backslash max\_\; v^T\; X\; =\; \backslash max\_\; |\; v^T\; X\; |\; \backslash leq\; 4\; \backslash sigma\; \backslash sqrt+2\; \backslash sigma\; \backslash sqrt$$

Inequalities

Theorem. (Theorem 2.6.1) There exists a positive constant

C such that given any number of independent mean-zero subgaussian random variables X _{1,} ... ,X _{n} , $$\backslash left\backslash |\backslash sum\_^n\; X\_i\backslash right\backslash |\_^2\; \backslash leq\; C\; \backslash sum\_^n\backslash left\backslash |X\_i\backslash right\backslash |\_^2$$Theorem. (Hoeffding's inequality) (Theorem 2.6.3) There exists a positive constant c such that given any number of independent mean-zero subgaussian random variables X _{1,} ... ,X _{N} ,$$\backslash mathbb\backslash left(\backslash left|\backslash sum\_^N\; X\_i\backslash right|\; \backslash geq\; t\backslash right)\; \backslash leq\; 2\; \backslash exp\; \backslash left(-\backslash frac\backslash right)\backslash quad\; \backslash forall\; t\; >\; 0$$Theorem. (Bernstein's inequality) (Theorem 2.8.1) There exists a positive constant c such that given any number of independent mean-zero subexponential random variables X _{1,} ... ,X _{N} ,$$\backslash mathbb\backslash left(\backslash left|\backslash sum\_^N\; X\_i\backslash right|\; \backslash geq\; t\backslash right)\; \backslash leq\; 2\; \backslash exp\; \backslash left(-c\; \backslash min\; \backslash left(\backslash frac,\; \backslash frac\backslash right)\backslash right)$$Theorem. (Khinchine inequality) (Exercise 2.6.5) There exists a positive constant C such that given any number of independent mean-zero variance-one subgaussian random variables X _{1,} ... ,X _{N} , any p \geq 2 , and any a _{1,} ... , a _{N} \in \R ,$$\backslash left(\backslash sum\_^N\; a\_i^2\backslash right)^\; \backslash leq\backslash left\backslash |\backslash sum\_^N\; a\_i\; X\_i\backslash right\backslash |\_\; \backslash leq\; C\; K\; \backslash sqrt\backslash left(\backslash sum\_^N\; a\_i^2\backslash right)^$$

Hanson-Wright inequality

The Hanson-Wright inequality states that if a random vector

X is subgaussian in a certain sense, then any quadratic form A of this vector, X ^{TAX} , is also subgaussian/subexponential. Further, the upper bound on the tail of X ^{TAX} , is uniform.

A weak version of the following theorem was proved in (Hanson, Wright, 1971).^[6] There are many extensions and variants. Much like the central limit theorem, the Hanson-Wright inequality is more a cluster of theorems with the same purpose, than a single theorem. The purpose is to take a subgaussian vector and uniformly bound its quadratic forms.

Theorem.^{[7] [8]} There exists a constant

c , such that:

Let

n be a positive integer. Let X _{1,} ..., X _{n} be independent random variables, such that each satisfies E[X _{i]} = 0 . Combine them into a random vector X = (X _{1,} ... , X _{n)} . For any n x n matrix A , we have$$P(|X^T\; AX\; -\; E[X^TAX]|\; >\; t)\; \backslash leq\; \backslash max\backslash left(2\; e^,\; 2\; e^\; \backslash right)\; =\; 2\; \backslash exp\; \backslash left[-c\; \backslash min\; \backslash left(\backslash frac\{t^2\}\{K^4\backslash |A\backslash |\_F^2\},\; \backslash frac\{t\}\{K^2\backslash |A\backslash |\}\backslash right)\backslash right]$$where K = max _{i} \|X _{i\|} ^{\psi _{2}} , and \|A\| _{F} = \sqrt{\sum _{ij} A ^{2 }} _{ij} is the Frobenius norm of the matrix, and \|A\| = max _{\|x\| _{2=1}} \|Ax\| _{2} is the operator norm of the matrix.

In words, the quadratic form

X ^{TAX} has its tail uniformly bounded by an exponential, or a gaussian, whichever is larger.

In the statement of the theorem, the constant

c is an "absolute constant", meaning that it has no dependence on n, X _{1,} ... , X _{n,} A . It is a mathematical constant much like pi and e.

Consequences

Theorem (subgaussian concentration). There exists a constant

c , such that:

Let

n, m be positive integers. Let X _{1,} ..., X _{n} be independent random variables, such that each satisfies E[X _{i]} = 0 , E[X ^{2 ]} _{i} = 1 . Combine them into a random vector X = (X _{1,} ... , X _{n)} . For any m x n matrix A , we have$$P(|\; \backslash |\; AX\backslash |\_2\; -\; \backslash |A\backslash |\_F\; |\; >\; t)\; \backslash leq\; 2\; e^$$In words, the random vector A X is concentrated on a spherical shell of radius \|A \| _{F} , such that \| AX\| _{2} - \|A \| _{F} is subgaussian, with subgaussian norm \leq \sqrt{ 3 /c} \|A\| K ^{2} .

See also

• Platykurtic distribution

References

• J.P. . Kahane . Propriétés locales des fonctions à séries de Fourier aléatoires . . 19 . 1–25 . 1960 . 10.4064/sm-19-1-1-25 . free .
• V.V. . Buldygin . Yu.V. . Kozachenko . Sub-Gaussian random variables . Ukrainian Mathematical Journal . 32 . 483–489 . 1980 . 6 . 10.1007/BF01087176 .
• Book: Ledoux . Michel . Talagrand . Michel . Probability in Banach Spaces . 1991 . Springer-Verlag .
• Book: Stromberg . K.R. . Probability for Analysts . 1994 . Chapman & Hall/CRC .
• A.E. . Litvak . A. . Pajor . M. . Rudelson . N. . Tomczak-Jaegermann . Smallest singular value of random matrices and geometry of random polytopes . . 195 . 491–523 . 2005 . 2 . 10.1016/j.aim.2004.08.004 . free .
• Mark . Rudelson . Roman . Vershynin . Non-asymptotic theory of random matrices: extreme singular values . Proceedings of the International Congress of Mathematicians 2010 . 1576–1602 . 1003.2990 . 10.1142/9789814324359_0111 . 2010 .
• News: O. . Rivasplata . Subgaussian random variables: An expository note . Unpublished . 2012 .
• Vershynin, R. (2018). "High-dimensional probability: An introduction with applications in data science" (PDF). Volume 47 of Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge.
• Zajkowskim, K. (2020). "On norms in some class of exponential type Orlicz spaces of random variables". Positivity. An International Mathematics Journal Devoted to Theory and Applications of Positivity. 24(5): 1231--1240. . .

Notes and References

1. Book: Vershynin, R. . High-dimensional probability: An introduction with applications in data science . Cambridge University Press . 2018 . Cambridge.
2. Bobkov . S. G. . Strictly subgaussian probability distributions . 2023-08-03 . 2308.01749 . Chistyakov . G. P. . Götze . F.. math.PR .
3. Bobkov . S. G. . Strictly subgaussian probability distributions . 2023-08-03 . 2308.01749 . Chistyakov . G. P. . Götze . F.. math.PR .
4. Kamath, Gautam. "Bounds on the expectation of the maximum of samples from a gaussian." (2015)
5. Web site: MIT 18.S997 Spring 2015 High-Dimensional Statistics, Chapter 1. Sub-Gaussian Random Variables . 2024-04-03 . MIT OpenCourseWare . en.
6. Hanson . D. L. . Wright . F. T. . 1971 . A Bound on Tail Probabilities for Quadratic Forms in Independent Random Variables . The Annals of Mathematical Statistics . 42 . 3 . 1079–1083 . 10.1214/aoms/1177693335 . 2240253 . 0003-4851. free .
7. Rudelson . Mark . Vershynin . Roman . January 2013 . Hanson-Wright inequality and sub-gaussian concentration . Electronic Communications in Probability . 18 . none . 1–9 . 10.1214/ECP.v18-2865 . 1083-589X. 1306.2872 .
8. Book: Vershynin, Roman . High-Dimensional Probability: An Introduction with Applications in Data Science . 2018 . Cambridge University Press . 978-1-108-41519-4 . Cambridge Series in Statistical and Probabilistic Mathematics . Cambridge . 6. Quadratic Forms, Symmetrization, and Contraction . 127–146 . 10.1017/9781108231596.009 . https://doi.org/10.1017/9781108231596.009.