INEQUALITIES FOR SUMS OF INDEPENDENT RANDOM VARIABLES IN LORENTZ SPACES

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1 ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 45, Number 5, 2015 INEQUALITIES FOR SUMS OF INDEPENDENT RANDOM VARIABLES IN LORENTZ SPACES GHADIR SADEGHI ABSTRACT. By using interpolation with a function parameter, we establish a moment ineuality for sums of independent random variables in Lorentz spaces Λ p (φ). These estimates generalize Rosenthal ineualities in the Lorentz- Zygmund spaces L p, (log L) γ as well as Lorentz spaces L p,. 1. Introduction. We begin our work by recalling the classical Khintchine ineualities. Let {r k k 1 be a Rademacher seuence on a probability space (Ω, F, P ). Since {r k k 1 is an orthogonal seuence in L 2 (Ω), for any finite seuence {α k C ( 2 ) 1/2 α k r k = α k 2. k The classical Khintchine ineualities assert that k α kr k 2 is uniformly euivalent to k α kr k p for any p <, namely, ( p ) 1/2 α k r k α k 2. k The euivalence A B means that c 1 A B c 2 A for some positive constants c 1 and c 2. Rosenthal [12] generalized the Khintchine ineuality by replacing {r k k 1 with an arbitrary seuence {X k k 1 of independent symmetric random variables on a probability space (Ω, F, p). More precisely, he proved that, for such a seuence {X k k 1 L p (Ω), p > 2, we have (1.1) { p n X k max k k ( 2 n X k, X k p p ) 1/p 2010 AMS Mathematics subject classification. Primary 46E30, Secondary 60G50. Keywords and phrases. Random variable, Rosenthal ineuality, Lorentz space, interpolation. Received by the editors on September 21, DOI: /RMJ Copyright c 2015 Rocky Mountain Mathematics Consortium 1631

2 1632 GHADIR SADEGHI for all n 1. Carothers and Dilworth [3] proved an analogous result for some of the Lorentz spaces, namely, for 2 < p <, 0 <, and any independent symmetric random variables X 1, X 2,..., X n, (1.2) { n X kl max X kl, XkL, p, (Ω) 2 (Ω) p, (0, ) where n Xk denotes the disjoint sum of X 1, X 2,..., X n, which is a function on (0, ) with d X (t) = n d X k (t). For example, we could take X(t) = n i=1 X i(t i+1)χ [i 1,i] for 0 t n. In the setting symmetric function spaces, Johnson and Schechtman [7] established a generalization of Rosenthal ineualities. Recently, Hu [6] generalize Rosenthal ineualities to p 0 instead of p > 2 and replaced the uantity 2 by r [1, 2] for conditionally independent mean zero random variables. In this paper, by use of interpolation with a function parameter, a moment ineuality is proved for sums of independent random variables in Lorentz spaces Λ (Ω). These estimates generalize Rosenthal ineualities in the Lorentz-Zygmund spaces L p, (log L) γ as well as Lorentz spaces L p,. 2. Lorentz spaces Λ Ω (φ). Let (Ω, Σ, µ) be a σ-finite nonatomic measure space. For a given weight ω, let L p µ(ω) denote the Lebesgue space defined by the norm f L p µ (ω) = fω Lp (µ) and L p (ω) when the measure is dt/t on R + = (0, ). Definition 2.1. We say that function f : (0, ) (0, ) belongs to the class B if f(1) = 1, f is continuous and for all 0 < t <. f(ts) f(t) = sup s>0 f(s) <, For such a function f, the Boyd upper and lower indices α f and β f ([10]) of f, which is submultiplicative and Lebesgue-measurable, are

3 ROSENTHAL INEQUALITIES 1633 then defined by with α f = lim t + log f(t) log f(t), β log t f = lim t 0 log t < β f α f < +. For example, if θ, γ R, then f(t) = t θ (1 + log t ) γ B, f(t) = t θ (1 + log t ) γ and α f = β f = θ. Let φ B and 0 < ; the Lorentz space Λ (φ) is the set of (classical of) µ-measurable functions from Ω in C such that ( f Λ := f L (φ) = (φ(t)f (t)) dt ) 1/ < 0 t (0 < < ) f Λ := f L (φ) = sup φ(t)f (t) <, t>0 where f denotes the decreasing rearrangement of f, i.e., f (t) = inf{s > 0 : d f (s) = µ({ f > s) t. It is known that Λ Ω (φ) is a rearrangement uasi Banach space. Remark 2.2. It is also well known that the inclusion relations between Lorentz spaces are determined by their fundamental functions, since Λ Ω (φ 1) Λ Ω (φ 2) if and only if ω 2 (t) Cω 1 (t) for all t > 0, and both spaces agree if and only if ω 1 ω 2, where ( t ω i (t) = 0 φ(s) ds s ) 1/ is the fundamental function for Λ Ω (φ i), i = 1, 2, [4]. Example 2.3. For φ(t) = t 1/p (1 + log t ) γ with 0 < p < and < γ < +, Λ is the Lorentz-Zygmund space Lp, (log L) γ. This is the classical Lorentz space L p, if γ = 0. We let (A 1, A 2 ) denote a compatible couple of uasi-banach spaces pair (i.e., A 1 and A 2 are uasi-banach spaces, which are continuously

4 1634 GHADIR SADEGHI embedded in some Hausdorff topological vector space) and K is the classical interpolation functional of Peetre. K(t, a) = K(t, a, A 1, A 2 ) = inf {a 1 A1 + ta 2 A2 : a = a 1 + a 2, t > 0. We can define, for each p, 0 < p and each Lebesgue-measurable function f : (0, ) (0, ), the space { (A 1, A 2 ) f,p;k = a : a A 1 + A 2, a f,p;k = K(t, a; A 1, A 2 )/f(t) L (0, ) <. The space (A 1, A 2 ) f,p;k is uasi-normed by f,p;k. To generalize to (A 1, A 2 ) f,p;k the very well known properties of this space when f(t) = t θ (i.e., (A 1, A 2 ) θ,p;k ), one takes the function f in the class B. In [10], Merucci showed that interpolation with a function parameter is perfectly suited for identifying interpolation spaces between two uasinormed Lorentz spaces Λ Ω (φ). We refer the reader to [5, 10, 11, 13] for the theory and bibliography concerning these spaces. Recall also that intersection of two Lorentz spaces Λ Ω (φ 1) and Λ Ω (φ 2) is a uasi Banach space under the uasi-norm max{ Λ Ω (φ1), Λ Ω (φ2). 3. Main results. In the seuel, we assume that (Ω, F, P ) is probability space and establish an extension of Rosenthal ineualities in Lorentz spaces Λ Ω (φ). To prove the main result, we need the following lemma. Lemma 3.1. Let 0 < r < p <, f B, and 0 <. Then (L r (0, ), L r (0, ) L p (0, )) f,;k = L r (0, ) (L r (0, ), L p (0, )) f,;k. Proof. By use of Holmsted s formula about interpolation with a function parameter [10, 11] the proof of this lemma is similar to [3, Lemma 2.1].

5 ROSENTHAL INEQUALITIES 1635 Theorem 3.2. Given 1 r 2 < p < and 0 <, let f B with 0 < β f α f < 1. Then { Λ n X k max X klr, (Ω) XkΛ (0, ) (φ) for all independent symmetric random variables X 1, X 2,..., X n in Λ, where t 1/r φ(t) = f ( t 1/r 1/p). Proof. It follows from [10, Theorem 3] that φ B. It is convenient to take Ω be [0, 1] N with the product measure and denote a typical element of Ω by the seuence t = (t 1, t 2,...). Define a linear operator T : L 0 (0, ) L 0 (Ω [0, 1]) by T (g) = g k (t k )r k (s), where g k (t k ) = g(t k + k 1) and r k (s) is the kth Rademacher function. Then, by Hu s ineuality [6], T is a bounded operator from L r (0, ) L p (0, ) into L p (Ω [0, 1]) for p > 2. So, by Lemma 3.1 and the interpolation theorem with a function parameter ([10, Theorem 3] and [5]), T is bounded from L r (0, ) Λ (0, ) (φ) into Λ Ω (φ), where φ(t) = t 1/r f ( t 1/r 1/p). Therefore, there exists a positive constant C such that (3.1) { Λ n X k C max X kl, r (Ω) It follows from Remark 2.2 that n Λ (3.2) X kl2 C 1 X k (Ω) (Ω) (φ) for a positive constant C 1., XkΛ (0, ) (φ).

6 1636 GHADIR SADEGHI Since 1 r 2 < p and α f < 1, it follows from [10, Propositions 2, 3] that ( 1 α f(t 1/r 1/p ) = r 1 ) α p f, and so α φ < 1. On the other hand, α Λ = α φ < 1, where α Λ are Boyd indices of Λ, [13]. Now, by [8, Theorem 5.8], Λ Ω (φ) has the Kalton property (that is, for φ(t) = t 1/r f ( t 1/r 1/p), Λ satisfies X CY whenever X Y (recall that X (t) = t 1 t 0 X (s) ds)). By the definition of the disjoint sum, it is easy to check that ( n ) (( n ) 1/2 ) Xk X k 2. Now, by the Kalton property, we have (3.3) ( n ) 1/2 Λ XkL C 2 X k 2 ((0, )) for some positive constant C 2. Since n X k has the same distribution as n X k(t)r k (t), by the Maurey-Khintchine ineuality [9, Theorem 1.d.6] and ineuality (3.3), we obtain (3.4) Λ XkΛ C 3 X k, (0, ) (φ) for some positive constant C 3. (3.4), we get Therefore, by ineualities (3.2) and { n C Λ max X kl, XkΛ X k, 2 (Ω) (0, ) (φ) where C = 1/max{C 1, C 3. So, the desired ineuality now follows

7 ROSENTHAL INEQUALITIES 1637 easily since 1 r 2, i.e., (3.5) { n C Λ max X kl, XkΛ X k. r (Ω) (0, ) (φ) Thus, ineualities (3.1) and (3.5) imply that { Λ n X k max X klr, XkΛ. (Ω) (0, ) (φ) Corollary 3.3. Given 1 r 2 < p < and 0 <, we then have { L n X k max X klr, Xk, p, (log L) γ (Ω) Lp,(log L)γ for all independent symmetric random variables X 1, X 2,..., X n in L p, (log L) γ. Proof. It is sufficient to consider ( f(t) = t θ 1 + pr ) γ log t p r in Theorem 3.2. Remark 3.4. In the previous corollary, if γ = 0 and r = 2 (p = ), then this corollary implies Rosenthal ineualities (1.2) in Lorentz spaces L p, (spaces L p ). REFERENCES 1. C. Bennet and R. Sharpley, Interpoloation of operators, Academic Press, Boston, C. Bergh and J. Löfström, Interpolation spaces, Springer, Berlin, N.L. Carothers and S.J. Dilworth, Ineualities for sums of independent random variables, Proc. Amer. Math. Soc. 104 (1988), M.J. Carro and J. Soria, Weighted Lorentz spaces and the Hardy operator, J. Funct. Anal. 112 (1993), J. Gustavsson, A function parameter in connection with interpolation of Banach spaces, Math. Scand. 42 (1978),

8 1638 GHADIR SADEGHI 6. X.L. Hu, An extension of Rosenthal s ineuality, Appl. Math. Comp. 218 (2011), W.B. Johnson and G. Schehtman, Sums of independent random variables in rearrangement invariant function spaces, Ann. Probab. 17 (1988), N.J. Kalton, Compact and strictly singular operators on certain function spaces, Arch. Math. 43 (1986), J. Lidenstrauss and L. Tzafriri, Classical Banach spaces II, Function spaces, Springer-Verlag, Berlin, C. Merucci, Applications of interpolation with a function parameter to Lorentz, Sobolev and Besov spaces, Lect. Notes Math (1984), L.E. Persson, Interpolation with a parameter function, Math. Scand. 59 (1986), H.P. Rosenthal, On the subspaces of L p (p > 2) spanned by seuences of independent random variables, Israel J. Math. 8 (1970), R. Sharpley, Spaces Λ α (X) and interpolation, J. Funct. Anal. 11 (1972), Department of Mathematics and Computer Sciences, Hakim Sabzevari University, P.O. Box 397, Sabzevar, Iran address: ghadir54@gmail.com, g.sadeghi@hsu.ac.ir