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On the Concentration of Measure in Orlicz Spaces of Exponential Type

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On the Concentration of Measure in Orlicz Spaces of Exponential Type ON THE CONCENTRATION OF MEASURE IN ORLICZ SPACES OF EXPONENTIAL TYPE VIET NGUYEN Introduction The study of Orlicz spaces, first described in 1931 by Wladyslaw Orlicz and Zygmunt Wilhelm Birnbaum [1], became popular in the empirical processes and functional analysis literature, due to the rising interest in chaining arguments to derive probabilistic bounds for stochastic processes, and generalizations of Lp spaces as well as Sobolev space embeddings, respectively. Orlicz spaces exhibit strong concentration phenomena, inherited from their construction. In particular, they are associated to the sub-Exponential and sub-Gaussian classes of random variables. In this article, we aim to provide a brief introduction to concentration of measure in Orlicz spaces, in particular, Orlicz spaces of exponential type. We begin by the construction of these spaces, and delve into certain concentration guarantees and applications. 1. Kσ-spaces, N-functions, Orlicz spaces, norm We will closely follow the construction in [4]. Let (Ω; F; P) be our underlying probability space. 0 Definition 1.1. (Kσ-space, or K-lineal) A lattice (K(Ω); ≤) ⊂ L (Ω) is a Kσ-space with norm k·k such that: a) 8ξ; η 2 K(Ω); jξj ≤ jηj =) kξk ≤ kηk (monotonicity of norms) b) fξng ⊂ K(Ω); 9η 2 K(Ω) such that supn≥1 jξnj < η a.s. =) supn≥1 jξnj 2 K(Ω) One can see that a Kσ-space is a Banach lattice with an additional condition b). As an example, p the function spaces (L (Ω); p ≥ 1) is a Kσ-space. For our future purposes, we write K(Ω) not to denote a vector lattice, but to denote a Kσ-space. We state here without proof that in a Kσ-space, convergence in norm implies convergence in probability, namely: Lemma 1.2. Let fξng ⊂ K(Ω); ξ 2 K(Ω). Then, lim kξn − ξk = 0 =) 8 > 0; lim (jξn − ξj > ) = 0 n−!1 n−!1 P We introduce here the class of N-functions, sometimes called Young's functions and Young-Orlicz modulus beyond Soviet literature, that we will use to construct interesting Kσ-spaces, namely Orlicz spaces. Definition 1.3. (N-functions) A continuous, even, convex function = (x); x 2 R is an N- function if (x) is monotonically increasing for x > 0; (0) = 0, and satisfies: (x) (n0) limx−!0 x = 0 (x) (n1) limx−!1 x = 1 1 2 VIET NGUYEN is an N-function if and only if there exists a c`adl`agdensity p(t); t ≥ 0 such that 8t > 0; p(t) > 0; p(0) = 0; p(t) −!1 as t −!1: Z jxj (x) = p(t) dt 0 A really important example of N-functions are functions of the type (x) = exp f'(x)g − 1, such that '(x) is an N-function that is not xp for any p, essential in constructing Orlicz spaces of exponential type. The following are important properties of N-functions that we shall use in later proofs. Lemma 1.4. Let be an N-function. Then, the following hold: a) (jxj + jyj) ≥ (jxj) + jyj. b) If a > 1, (ax) ≥ a (x). We are now ready to define Orlicz spaces, whose elements are random variables defined on our underlying probability space: Definition 1.5. (Orlicz space) Let be an arbitrary N-function. The Orlicz space of ran- dom variables L (Ω) is the family of equivalence classes of random variables where for each ξ 2 L (Ω); 9rξ > 0 such that: ξ Z ξ E = ◦ dP < 1 rξ Ω rξ Two random variables are equivalent if they are equal almost surely. The norm ξ kξk = inf r > 0 : ≤ 1 E r is the Orlicz norm (sometimes Luxemburg norm) with the understanding that inf ; = 1. This makes L (Ω) a Banach space. Furthermore, one can show that L (Ω) is a lattice, and hence a 0 n 0 o Kσ-space. We can characterize L (Ω) ⊂ L (Ω) by L (Ω) = ξ 2 L (Ω) : kξk < 1 . p p Example 1.6. Suppose (x) = jxj . Then, L (Ω) = L (Ω) and kξk = kξkp. Finiteness of the norm immediately implies a tail bound: Lemma 1.7. Let ξ 2 L (Ω) where is a N-function. Then, ! −1 ξ P(jξj > x) ≤ kξk Proof. By Markov's generalized inequality, since ◦ ξ 7! 1 is nonnegative and monotonically kξk increasing, we have: h i E (ξ= kξk ) 1 P(jξj > x) ≤ ≤ (ξ= kξk ) (ξ= kξk ) ON THE CONCENTRATION OF MEASURE IN ORLICZ SPACES OF EXPONENTIAL TYPE 3 2. Orlicz spaces of exponential type, concentration Definition 2.1. (Orlicz space of exponential type) Suppose that ' is a N-function that does not necessarily satisfy (n0) or (n1), and let (x) = exp f'(x)g − 1. We call the Orlicz space generated by of exponential type, denoting it by Exp'(Ω), and the corresponding norm k·kE'. Elements of Exp'(Ω) enjoy an exponential tail bound: Theorem 2.2. Let ξ 2 Exp'(Ω). Then, ( !) x P(jξj ≥ x) ≤ 2 exp −' kξkE' The converse is also true; if n x o (jξj ≥ x) ≤ C exp −' P D for some C; D > 0, then ξ 2 Exp'(Ω) with kξkE' ≤ (1 + C)D. Proof. Assume kξkE' > 0. Generalized Markov yields: " ( !)# ( !) ξ x P(jξj ≥ x) ≤ E exp ' exp −' kξkE' k'kE' " ! # ( !) ' x = E + 1 exp −' k'kE' k'kE' ( !) x ≤ 2 exp −' k'kE' n x o For the converse statement, assume (jξj ≥ x) ≤ C exp −' holds for C; D > 0. Assume jξj P D has a density w.r.t. the Lebesgue measure, let F be the CDF of jξj and G = 1 − F , and let a > D (in particular, '(1=a) < '(1=D)). Remark that by the concentration assumption, we have: n xo n x xo exp ' G(x) ≤ C exp − ' − ' a D a 1 1 ≤ C exp −' x − D a x x x x since for N-function ', ' − ≤ ' − ' holds, from Lemma 1.4. As x −!1; we D a D a x have that exp ' a G(x) −! 0. Now, consider: ξ Z 1 n xo E exp ' = exp ' dF (x)(1) a 0 a n xo i1 Z 1 n xo x (2) = − exp ' G(x) + exp ' G(x) d' a 0 0 a a Z 1 n x x o x (3) ≤ −(−1) + C exp ' − ' d' 0 a D a Z 1 a − D x x CD (4) ≤ 1 + C exp − ' d' = 1 + 0 D a a a − D x where (2) follows from change of variables, (3) by substituting our control on exp ' a G(x), x x a a x and (4) follows from the substitution ' D = ' a D ≥ D ' a , from Lemma 1.4. We see that 4 VIET NGUYEN h ξ i h n ξ oi CD E a = E exp ' a − 1 ≤ a−D < 1, and thus ξ 2 Exp'(Ω). Choosing a = (1 + C)D, h ξ i since for this choice of a, E a < 1, and the norm being the infimum over all such a, it follows that kξkE' ≤ (1 + C)D, giving us the desired conclusion. From Theorem 3.2., in particular, we can see that choosing '(x) = jxj2, hence (x) = exp jxj2 −1, the finiteness of the Orlicz exponential type norm guarantees the sub-Gaussian property of ξ. Controlling the Orlicz norm directly enables us to control tail bounds in an Orlicz space. 3. Application: norm control for the maxima of finitely many random variables We can obtain maximal inequalities using Cram´ertransforms and applying Chernoff bounds to control the maximum of a finite number of random variables. Using Orlicz norm, one is able to achieve the same feat. Theorem 3.1. Let be a N-function for which there exists c > 0 such that (x) (y) ≤ (cxy), for all x; y ≥ 1. Then, for m random variables ξ1; : : : ; ξm, −1 max jξij ≤ K (m) max kξik i≤m i≤m Given some control on the individual Orlicz norms of each random variable, one is able to bound the Orlicz norm of the maximum. x (cx) Proof. We have from assumption ≤ . For any C > 0, y (y) jξ j (cjξ j=C) jξ j jξ j jξ j (1) max i ≤ max i 1 i ≥ 1 + i 1 i < 1 i≤m Cy i≤m (y) Cy Cy Cy (cjξ j=C) ≤ max i + (1)(2) i≤m (y) m X (cjξij=C) ≤ + (1)(3) (y) i=1 where (2) follows from convexity of . We now set C := maxi≤m kξik , and take expectations: maxi≤m jξij jξij (4) E ≤ E max Cy i≤m Cy 2 3 m jξ j= max kξ k X i i≤m i (5) ≤ E 4 5 + (1) (y) i=1 m ≤ + (1)(6) (y) where (6) follows from properties of N-functions w.r.t. their corresponding Orlicz norms. We note 1 m that when (1) ≤ 2 , we have that (y) + (1) ≤ 1 m m + (1) ≤ 1 () (y) ≥ (y) 1 − (1) m () y ≥ −1 ≥ −1(2m) 1 − (1) −1 () Cy ≥ (2m)c max kξik i≤m ON THE CONCENTRATION OF MEASURE IN ORLICZ SPACES OF EXPONENTIAL TYPE 5 h maxi≤m jξij i −1 Therefore, E Cy ≤ 1 when Cy ≥ (2m)c maxi≤m kξik . Hence, the norm being the infimum of the previous expectation, we have that: −1 max jξij ≤ (2m)c max kξik i≤m i≤m −1 ≤ 2c (m) max kξik i≤m which follows again from convexity of . Recall that the property of being sub-Gaussian is related to how one is able to control the even mo- ments of a random variable. With the Orlicz norm, we are able to make a similar statement.
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