- Email: [email protected]
On the concentration phenomenon for ϕ -subgaussian random elements
ARTICLE IN PRESS
Statistics & Probability Letters 76 (2006) 465–469 www.elsevier.com/locate/stapro
On the concentration phenomenon for j-subgaussian random elements Rita Giuliano Antoninia,1, Tien-Chung Hub,2, Andrei Volodinc,,3 a
Dipartimento di Matematica, Universita` di Pisa, Largo B. Pontecorvo 5, 56100 Pisa, Italy Department of Mathematics, National Tsing Hua University, Hsinchu 30043, Taiwan, Republic of China c Department of Mathematics and Statistics, University of Regina, Regina, Saskatchewan, Canada S4S 0A2 b
Received 28 March 2005 Available online 15 September 2005
Abstract We study the deviation probability PfjkXk EkXkj4tg where X is a j-subgaussian random element taking values in the Hilbert space l 2 and jðxÞ is an N-function. It is shown that the order of this deviation is expfj ðCtÞg, where C depends on the sum of j-subgaussian standard of the coordinates of the random element X and j ðxÞ is the Young–Fenchel transform of jðxÞ. An application to the classically subgaussian random variables (jðxÞ ¼ x2 =2) is given. r 2005 Elsevier B.V. All rights reserved. MSC: primary 60F15 Keywords: Concentration of measure phenomenon; j-Subgaussian random variables; N-function; Young–Fenchel transform; Exponential inequalities
1. Introduction The concentration of measure phenomenon roughly describes how a well-behaved function is almost a constant on almost all the space. We refer to the monographs Ledoux (2001), and Ledoux and Talagrand (1991, Chapter 1), where this phenomenon is discussed in detail and important examples are provided. The purpose of this note is to further complete our understanding of the concentration phenomenon by obtaining the exponential estimate of the behaviour of the deviation probability with respect to the mean PfjkXk EkXkj4tg where X is a j-subgaussian random element taking values in the Hilbert space l 2 and jðxÞ is an N-function. It is shown that the order of this deviation is expfj ðCtÞg, where C depends on the jsubgaussian standard of the coordinates of the random element X and j ðxÞ is the Young–Fenchel transform of jðxÞ. An application to the classically subgaussian random variables (jðxÞ ¼ x2 =2) is given. Corresponding author. Tel.: +1 306 585 4771; fax: +1 306 585 4020.
E-mail address: The research of 2 The research of 3 The research of 1
[email protected] (A. Volodin). R. Giuliano has been partially supported by M.I.U.R., Italy. T.-C. Hu has been partially supported by the National Science Council of Taiwan. A. Volodin has been partially supported by the National Science and Engineering Research Council of Canada.
0167-7152/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2005.08.035
ARTICLE IN PRESS R.G. Antonini et al. / Statistics & Probability Letters 76 (2006) 465–469
466
2. Definitions and a few technical lemmas In this section we present definitions and a few technical results that we will use in the proof of the main result of the paper. A continuous even convex function jðxÞ; x 2 R, is called an N-function, if (a) jð0Þ ¼ 0 and jðxÞ is monotone increasing for x40; jðxÞ (b) limx!0 jðxÞ x ¼ 0 and limx!1 x ¼ 1. In the following the notation jðxÞ always stands for an N-function. It is obvious that the function jðxÞ ¼ jxjp =p; p41 is an example of N-function. The function j ðxÞ; x 2 R, defined by j ðxÞ ¼ supy2R ðxy jðyÞÞ is called the Young– Fenchel transform of jðxÞ. It is well known that j ðxÞ is an N-function, too, and if jðxÞ ¼ jxjp =p; p41 for all x, then j ðxÞ ¼ jxjq =q for all x, where 1=p þ 1=q ¼ 1. A random variable X is said to be j-subgaussian if there exists a constant a40 such that, for every t 2 R, we have E expftX gp expfjðatÞg. The j-subgaussian standard tj ðX Þ is defined as tj ðX Þ ¼ inffa40 : E expftX gp expfjðatÞg; t 2 Rg. We refer to the monograph Buldygin and Kozachenko (2000) and the paper Giuliano Antonini et al. (2003) where this notion is discussed in detail and important examples are provided. In the case jðxÞ ¼ x2 =2 the notion of j-subgaussian gives us the notion of classical subgaussian random variable, cf. for example in Hoffmann-Jørgensen (1994, Section 4.29). Let X ¼ ðX 1 ; X 2 ; . . .Þ be an l 2 -valued random element defined on a probability space ðO; F; PÞ, that is, for almost all o 2 O the norm !1=2 1 X 2 kXðoÞk ¼ X k ðoÞ k¼1
is finite. A random element X taking values in l 2 is said to be scalarly j-subgaussian if each coordinate X k ; kX1; is a j-subgaussian random variable and tj ðXÞ ¼
1 X
tj ðX k Þo1.
k¼1
Now we will present a few technical lemmas which are important for the proof of the main result. Lemma 1. Let X and Y be two independent identically distributed j-subgaussian random variables, and a and b be constants. If for some pX1 the function jðjxj1=p Þ is convex, then tj ðaX þ bY Þpðjajp þ jbjp Þ1=p tj ðX Þ. Proof. According to Buldygin and Kozachenko (2000, Chapter 2, Theorem 5.2), we can give the following estimate tpj ðaX þ bY Þptpj ðaX Þ þ tpj ðbY Þ since X and Y are independent ¼ jajp tpj ðX Þ þ jbjp tpj ðY Þ ¼ ðjajp þ jbjp Þtpj ðX Þ since X and Y are identically distributed.
&
The second lemma is well known. We present its proof for the sake of completeness. Lemma 2. Let X and Y be two independent identically distributed random variables and c : R ! R be a convex function. Then E½cðX EX ÞpEcðX Y Þ.
ARTICLE IN PRESS R.G. Antonini et al. / Statistics & Probability Letters 76 (2006) 465–469
467
Proof. Let F X ðxÞ and F Y ðyÞ be the distribution functions of X and Y, respectively. Then their joint distribution function is F X ;Y ðx; yÞ ¼ F X ðxÞF Y ðyÞ. For each x we have that x EX ¼ Eðx X Þ, hence by Jensen’s inequality cðx EX ÞpEcðx X Þ. Integrating this inequality, using the change of variables formula and the Fubini theorem, we obtain that Z 1 E½cðX EX Þ ¼ cðx EX Þ dF X ðxÞ Z1 1 Z 1 p cðx yÞ dF Y ðyÞ dF X ðxÞ ¼ EcðX Y Þ: & 1
1
Lemma 3. Let X ¼ ðX 1 ; X 2 ; . . .Þ be an l 2 -valued scalarly j-subgaussian random element. Then for all t 2 R E expftkXkgp2 expfjðttj ðXÞÞg. Proof. First of all, note that for any t 2 R and kX1 E expftjX k jgpE expfjtX k jg ¼ E expfjtjX k gIfX k X0g þ E expfjtjX k gIfX k o0g pE expfjtjX k g þ E expfjtjX k gp2 expfjðttj ðX k ÞÞg.
ð1Þ
Next, since kXk ¼
1 X
!1=2 X 2k
p
k¼1
1 X
jX k j
k¼1
and P1 from the generalized Holder inequality, for any sequence of positive numbers fpk ; kX1g such that k¼1 1=pk ¼ 1, we get ( ) 1 X E expftkXkgpE exp t jX k j k¼1
p p
1 Y k¼1 1 Y k¼1
½EðexpftjX k jgÞpk 1=pk ¼
1 Y
½E expftpk jX k jg1=pk
k¼1
½2 expfjðtpk tj ðX k ÞÞg1=pk
by ð1Þ
) 1 X jðtpk tj ðX k ÞÞ ¼ 2 exp . pk k¼1 Taking pk ¼ ð
P1
(
i¼1 tj ðX i ÞÞ=tj ðX k Þ
we obtain the result.
&
The next lemma is only a slight modification of Buldygin and Kozachenko (2000, Chapter 2, Lemma 4.3). We again give a proof for the sake of completeness. Lemma 4. Let X be a random variable such that for each t 2 R we have that E expftjX jgpA expfjðtBÞg, where A and B are positive constants. Then for any t n t o PfjX j4tgp2A exp j . B Proof. By Markov’s inequality, for any l40 and t PfX 4tgp expfltgE expflX gpA expfjðlBÞ ltg.
ARTICLE IN PRESS R.G. Antonini et al. / Statistics & Probability Letters 76 (2006) 465–469
468
Next, inf ½jðlBÞ lt ¼ sup ½lt jðlBÞ l40 ht i ¼ sup m jðmÞ where m ¼ lB m40 B t ¼ j . B The same argument shows that n t o : & PfX o tgpA exp j B l40
3. The main result With the preliminaries accounted for, we can now state and prove the main results of the paper. Theorem 1. Assume that for some pX1 the function jðjxj1=p Þ is convex. Let X ¼ ðX 1 ; X 2 ; . . .Þ be an l 2 -valued scalarly j-subgaussian random element. Then for any t 2t PfjkXk EkXkj4tgp4 exp j , pC p tj ðXÞ where C p ¼ maxð1; 21=p
1=2
Þ.
Proof. Let Y ¼ ðY 1 ; Y 2 ; . . .Þ be an independent copy of X and denote XðsÞ ¼ X sinðsÞ þ Y cosðsÞ;
0pspp=2.
Then Xð0Þ ¼ Y and Xðp=2Þ ¼ X. Moreover, X0 ðsÞ ¼ X cosðsÞ Y sinðsÞ. Note that by Lemma 1 tj ðX0 ðsÞÞpðj cosðsÞjp þ j sinðsÞjp Þ1=p tj ðXÞpC p tj ðXÞ. Next, if we denote by hx; yi the inner product of two elements x; y 2 l 2 , then Z p=2 Z p=2 d XðsÞ kXðsÞk ds ¼ ; X0 ðsÞ ds kXk kYk ¼ ds kXðsÞk 0 0 Z p=2 p ds kX0 ðsÞk . p 2 p=2 0 Fix any t 2 R and let cðuÞ ¼ expftug. Then c is convex and by Jensen inequality Z p=2 ds p cðkXk kYkÞp c kX0 ðsÞk 2 p=2 0 Z p=2 2 p ¼ c kX0 ðsÞk ds. p 0 2 According to Lemma 2 we can write E expftðkXk EkXkÞg ¼ EcðkXk EkXkÞ pEcðkXk kYkÞ Z p 2 p=2 p Ec kX0 ðsÞk ds. p 0 2
(2)
ARTICLE IN PRESS R.G. Antonini et al. / Statistics & Probability Letters 76 (2006) 465–469
469
Hence by Lemma 3 and (2) Z Z n p o n p o 2 p=2 2 p=2 ¼ E exp t kX0 ðsÞk dsp 2 exp j t C p tj ðXÞ ds p 0 2 p 0 2 n p o ¼ 2 exp j t C p tj ðXÞ . 2 The conclusion of the theorem now follows from Lemma 4. & Remark 1. Note that in the proof of the theorem we established the following inequality that may be of an independent interest: n p o E expftðkXk EkXkÞgp2 exp j t C p tj ðXÞ . 2 As a simple corollary of the theorem we can present the following application to the classically subgaussian random elements. In the case jðxÞ ¼ x2 =2, x 2 R, we will say that X ¼ ðX 1 ; X 2 ; . . .Þ is an l 2 -valued scalarly classically subgaussian random element if tx2 =2 ðXÞ ¼
1 X
tx2 =2 ðX k Þo1.
k¼1
Corollary 1. Let X ¼ ðX 1 ; X 2 ; . . .Þ be an l 2 -valued scalarly classically subgaussian random element. Then ( ) 2t2 PfjkXk EkXkj4tgp4 exp 2 2 . p tx2 =2 ðXÞ Proof. Note that in this case j ðxÞ ¼ x2 =2 and we can apply theorem with p ¼ 2, hence C p ¼ 1.
&
References Buldygin, V.V., Kozachenko, Yu.V., 2000. Metric Characterization of Random Variables and Random Processes. American Mathematical Society, Providence, RI. Giuliano Antonini, R., Kozachenko, Yu., Nikitina, T., 2003. Spaces of j-sub-Gaussian random variables. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. 27 (5), 95–124. Hoffmann-Jørgensen, J., 1994. Probability with a view toward statistics. Chapman & Hall Probability Series, vol. 1. Chapman & Hall, New York. Ledoux, M. (2001). The concentration of measure phenomenon. Mathematical Surveys and Monographs, vol. 89. American Mathematical Society, Providence, RI. Ledoux, M., Talagrand, M., 1991. Probability in Banach spaces. Isoperimetry and processes. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) (Results in Mathematics and Related Areas (3)), vol. 23. Springer, Berlin.
Statistics & Probability Letters 76 (2006) 465–469 www.elsevier.com/locate/stapro
On the concentration phenomenon for j-subgaussian random elements Rita Giuliano Antoninia,1, Tien-Chung Hub,2, Andrei Volodinc,,3 a
Dipartimento di Matematica, Universita` di Pisa, Largo B. Pontecorvo 5, 56100 Pisa, Italy Department of Mathematics, National Tsing Hua University, Hsinchu 30043, Taiwan, Republic of China c Department of Mathematics and Statistics, University of Regina, Regina, Saskatchewan, Canada S4S 0A2 b
Received 28 March 2005 Available online 15 September 2005
Abstract We study the deviation probability PfjkXk EkXkj4tg where X is a j-subgaussian random element taking values in the Hilbert space l 2 and jðxÞ is an N-function. It is shown that the order of this deviation is expfj ðCtÞg, where C depends on the sum of j-subgaussian standard of the coordinates of the random element X and j ðxÞ is the Young–Fenchel transform of jðxÞ. An application to the classically subgaussian random variables (jðxÞ ¼ x2 =2) is given. r 2005 Elsevier B.V. All rights reserved. MSC: primary 60F15 Keywords: Concentration of measure phenomenon; j-Subgaussian random variables; N-function; Young–Fenchel transform; Exponential inequalities
1. Introduction The concentration of measure phenomenon roughly describes how a well-behaved function is almost a constant on almost all the space. We refer to the monographs Ledoux (2001), and Ledoux and Talagrand (1991, Chapter 1), where this phenomenon is discussed in detail and important examples are provided. The purpose of this note is to further complete our understanding of the concentration phenomenon by obtaining the exponential estimate of the behaviour of the deviation probability with respect to the mean PfjkXk EkXkj4tg where X is a j-subgaussian random element taking values in the Hilbert space l 2 and jðxÞ is an N-function. It is shown that the order of this deviation is expfj ðCtÞg, where C depends on the jsubgaussian standard of the coordinates of the random element X and j ðxÞ is the Young–Fenchel transform of jðxÞ. An application to the classically subgaussian random variables (jðxÞ ¼ x2 =2) is given. Corresponding author. Tel.: +1 306 585 4771; fax: +1 306 585 4020.
E-mail address: The research of 2 The research of 3 The research of 1
[email protected] (A. Volodin). R. Giuliano has been partially supported by M.I.U.R., Italy. T.-C. Hu has been partially supported by the National Science Council of Taiwan. A. Volodin has been partially supported by the National Science and Engineering Research Council of Canada.
0167-7152/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2005.08.035
ARTICLE IN PRESS R.G. Antonini et al. / Statistics & Probability Letters 76 (2006) 465–469
466
2. Definitions and a few technical lemmas In this section we present definitions and a few technical results that we will use in the proof of the main result of the paper. A continuous even convex function jðxÞ; x 2 R, is called an N-function, if (a) jð0Þ ¼ 0 and jðxÞ is monotone increasing for x40; jðxÞ (b) limx!0 jðxÞ x ¼ 0 and limx!1 x ¼ 1. In the following the notation jðxÞ always stands for an N-function. It is obvious that the function jðxÞ ¼ jxjp =p; p41 is an example of N-function. The function j ðxÞ; x 2 R, defined by j ðxÞ ¼ supy2R ðxy jðyÞÞ is called the Young– Fenchel transform of jðxÞ. It is well known that j ðxÞ is an N-function, too, and if jðxÞ ¼ jxjp =p; p41 for all x, then j ðxÞ ¼ jxjq =q for all x, where 1=p þ 1=q ¼ 1. A random variable X is said to be j-subgaussian if there exists a constant a40 such that, for every t 2 R, we have E expftX gp expfjðatÞg. The j-subgaussian standard tj ðX Þ is defined as tj ðX Þ ¼ inffa40 : E expftX gp expfjðatÞg; t 2 Rg. We refer to the monograph Buldygin and Kozachenko (2000) and the paper Giuliano Antonini et al. (2003) where this notion is discussed in detail and important examples are provided. In the case jðxÞ ¼ x2 =2 the notion of j-subgaussian gives us the notion of classical subgaussian random variable, cf. for example in Hoffmann-Jørgensen (1994, Section 4.29). Let X ¼ ðX 1 ; X 2 ; . . .Þ be an l 2 -valued random element defined on a probability space ðO; F; PÞ, that is, for almost all o 2 O the norm !1=2 1 X 2 kXðoÞk ¼ X k ðoÞ k¼1
is finite. A random element X taking values in l 2 is said to be scalarly j-subgaussian if each coordinate X k ; kX1; is a j-subgaussian random variable and tj ðXÞ ¼
1 X
tj ðX k Þo1.
k¼1
Now we will present a few technical lemmas which are important for the proof of the main result. Lemma 1. Let X and Y be two independent identically distributed j-subgaussian random variables, and a and b be constants. If for some pX1 the function jðjxj1=p Þ is convex, then tj ðaX þ bY Þpðjajp þ jbjp Þ1=p tj ðX Þ. Proof. According to Buldygin and Kozachenko (2000, Chapter 2, Theorem 5.2), we can give the following estimate tpj ðaX þ bY Þptpj ðaX Þ þ tpj ðbY Þ since X and Y are independent ¼ jajp tpj ðX Þ þ jbjp tpj ðY Þ ¼ ðjajp þ jbjp Þtpj ðX Þ since X and Y are identically distributed.
&
The second lemma is well known. We present its proof for the sake of completeness. Lemma 2. Let X and Y be two independent identically distributed random variables and c : R ! R be a convex function. Then E½cðX EX ÞpEcðX Y Þ.
ARTICLE IN PRESS R.G. Antonini et al. / Statistics & Probability Letters 76 (2006) 465–469
467
Proof. Let F X ðxÞ and F Y ðyÞ be the distribution functions of X and Y, respectively. Then their joint distribution function is F X ;Y ðx; yÞ ¼ F X ðxÞF Y ðyÞ. For each x we have that x EX ¼ Eðx X Þ, hence by Jensen’s inequality cðx EX ÞpEcðx X Þ. Integrating this inequality, using the change of variables formula and the Fubini theorem, we obtain that Z 1 E½cðX EX Þ ¼ cðx EX Þ dF X ðxÞ Z1 1 Z 1 p cðx yÞ dF Y ðyÞ dF X ðxÞ ¼ EcðX Y Þ: & 1
1
Lemma 3. Let X ¼ ðX 1 ; X 2 ; . . .Þ be an l 2 -valued scalarly j-subgaussian random element. Then for all t 2 R E expftkXkgp2 expfjðttj ðXÞÞg. Proof. First of all, note that for any t 2 R and kX1 E expftjX k jgpE expfjtX k jg ¼ E expfjtjX k gIfX k X0g þ E expfjtjX k gIfX k o0g pE expfjtjX k g þ E expfjtjX k gp2 expfjðttj ðX k ÞÞg.
ð1Þ
Next, since kXk ¼
1 X
!1=2 X 2k
p
k¼1
1 X
jX k j
k¼1
and P1 from the generalized Holder inequality, for any sequence of positive numbers fpk ; kX1g such that k¼1 1=pk ¼ 1, we get ( ) 1 X E expftkXkgpE exp t jX k j k¼1
p p
1 Y k¼1 1 Y k¼1
½EðexpftjX k jgÞpk 1=pk ¼
1 Y
½E expftpk jX k jg1=pk
k¼1
½2 expfjðtpk tj ðX k ÞÞg1=pk
by ð1Þ
) 1 X jðtpk tj ðX k ÞÞ ¼ 2 exp . pk k¼1 Taking pk ¼ ð
P1
(
i¼1 tj ðX i ÞÞ=tj ðX k Þ
we obtain the result.
&
The next lemma is only a slight modification of Buldygin and Kozachenko (2000, Chapter 2, Lemma 4.3). We again give a proof for the sake of completeness. Lemma 4. Let X be a random variable such that for each t 2 R we have that E expftjX jgpA expfjðtBÞg, where A and B are positive constants. Then for any t n t o PfjX j4tgp2A exp j . B Proof. By Markov’s inequality, for any l40 and t PfX 4tgp expfltgE expflX gpA expfjðlBÞ ltg.
ARTICLE IN PRESS R.G. Antonini et al. / Statistics & Probability Letters 76 (2006) 465–469
468
Next, inf ½jðlBÞ lt ¼ sup ½lt jðlBÞ l40 ht i ¼ sup m jðmÞ where m ¼ lB m40 B t ¼ j . B The same argument shows that n t o : & PfX o tgpA exp j B l40
3. The main result With the preliminaries accounted for, we can now state and prove the main results of the paper. Theorem 1. Assume that for some pX1 the function jðjxj1=p Þ is convex. Let X ¼ ðX 1 ; X 2 ; . . .Þ be an l 2 -valued scalarly j-subgaussian random element. Then for any t 2t PfjkXk EkXkj4tgp4 exp j , pC p tj ðXÞ where C p ¼ maxð1; 21=p
1=2
Þ.
Proof. Let Y ¼ ðY 1 ; Y 2 ; . . .Þ be an independent copy of X and denote XðsÞ ¼ X sinðsÞ þ Y cosðsÞ;
0pspp=2.
Then Xð0Þ ¼ Y and Xðp=2Þ ¼ X. Moreover, X0 ðsÞ ¼ X cosðsÞ Y sinðsÞ. Note that by Lemma 1 tj ðX0 ðsÞÞpðj cosðsÞjp þ j sinðsÞjp Þ1=p tj ðXÞpC p tj ðXÞ. Next, if we denote by hx; yi the inner product of two elements x; y 2 l 2 , then Z p=2 Z p=2 d XðsÞ kXðsÞk ds ¼ ; X0 ðsÞ ds kXk kYk ¼ ds kXðsÞk 0 0 Z p=2 p ds kX0 ðsÞk . p 2 p=2 0 Fix any t 2 R and let cðuÞ ¼ expftug. Then c is convex and by Jensen inequality Z p=2 ds p cðkXk kYkÞp c kX0 ðsÞk 2 p=2 0 Z p=2 2 p ¼ c kX0 ðsÞk ds. p 0 2 According to Lemma 2 we can write E expftðkXk EkXkÞg ¼ EcðkXk EkXkÞ pEcðkXk kYkÞ Z p 2 p=2 p Ec kX0 ðsÞk ds. p 0 2
(2)
ARTICLE IN PRESS R.G. Antonini et al. / Statistics & Probability Letters 76 (2006) 465–469
469
Hence by Lemma 3 and (2) Z Z n p o n p o 2 p=2 2 p=2 ¼ E exp t kX0 ðsÞk dsp 2 exp j t C p tj ðXÞ ds p 0 2 p 0 2 n p o ¼ 2 exp j t C p tj ðXÞ . 2 The conclusion of the theorem now follows from Lemma 4. & Remark 1. Note that in the proof of the theorem we established the following inequality that may be of an independent interest: n p o E expftðkXk EkXkÞgp2 exp j t C p tj ðXÞ . 2 As a simple corollary of the theorem we can present the following application to the classically subgaussian random elements. In the case jðxÞ ¼ x2 =2, x 2 R, we will say that X ¼ ðX 1 ; X 2 ; . . .Þ is an l 2 -valued scalarly classically subgaussian random element if tx2 =2 ðXÞ ¼
1 X
tx2 =2 ðX k Þo1.
k¼1
Corollary 1. Let X ¼ ðX 1 ; X 2 ; . . .Þ be an l 2 -valued scalarly classically subgaussian random element. Then ( ) 2t2 PfjkXk EkXkj4tgp4 exp 2 2 . p tx2 =2 ðXÞ Proof. Note that in this case j ðxÞ ¼ x2 =2 and we can apply theorem with p ¼ 2, hence C p ¼ 1.
&
References Buldygin, V.V., Kozachenko, Yu.V., 2000. Metric Characterization of Random Variables and Random Processes. American Mathematical Society, Providence, RI. Giuliano Antonini, R., Kozachenko, Yu., Nikitina, T., 2003. Spaces of j-sub-Gaussian random variables. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. 27 (5), 95–124. Hoffmann-Jørgensen, J., 1994. Probability with a view toward statistics. Chapman & Hall Probability Series, vol. 1. Chapman & Hall, New York. Ledoux, M. (2001). The concentration of measure phenomenon. Mathematical Surveys and Monographs, vol. 89. American Mathematical Society, Providence, RI. Ledoux, M., Talagrand, M., 1991. Probability in Banach spaces. Isoperimetry and processes. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) (Results in Mathematics and Related Areas (3)), vol. 23. Springer, Berlin.