BOUNDS FOR TAIL PROBABILITIES OF MARTINGALES USING SKEWNESS AND KURTOSIS. January 2008

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1 BOUNDS FOR TAIL PROBABILITIES OF MARTINGALES USING SKEWNESS AND KURTOSIS V. Betkus 1,2 ad T. Juškevičius 1 Jauary 2008 Abstract. Let M = X X be a sum of idepedet radom variables such that X k 1, E X k = 0 ad E Xk 2 = σ2 k for all k. Hoeffdig 1963, Theorem 3, proved that P {M t} H (t, p), H(t, p) = ( 1 + qt/p ) p qt ( 1 t ) qt q with q = σ 2, p = 1 q, σ2 = σ σ2, 0 < t < 1. Betkus 2004 improved Hoeffdig s iequalities usig biomial tails as upper bouds. Let γ k = E Xk 3/σ3 k ad κ k = E Xk 4/σ4 k stad for the skewess ad kurtosis of X k. I this paper we prove (improved) couterparts of the Hoeffdig iequality replacig σ 2 by certai fuctios of γ 1,..., γ respectively κ 1,..., κ. Our bouds exted to a geeral settig where X k are martigale differeces, ad they ca combie the kowledge of skewess ad/or kurtosis ad/or variaces of X k. Up to factors bouded by e 2 /2 the bouds are fial. All our results are ew sice o iequalities icorporatig skewess or kurtosis cotrol so far are kow. 1. Itroductio ad results I a celebrated paper of Hoeffdig 1963 several iequalities for sums of bouded radom variables were established. For improvemets of the Hoeffdig iequalities ad related results see, for example, Talagrad 1995, McDiarmid 1989, Godbole ad Hitczeko 1998, Pielis , Laib 1999, B , va de Geer 2002, Perro 2003, BGZ , BGPZ 2006, BKZ 2006, 2007, BZ 2003, etc. Up to certai costat factors, some of these improvemets are close to the fial optimal iequalities, see B 2004, BKZ However so far o bouds takig ito accout iformatio related to skewess ad/or kurtosis are kow, ot to metio certai results related to symmetric radom variables, see BGZ 2006, BGPZ I this paper we prove geeral ad optimal couterparts of Hoeffdig s 1963 Theorem 3, usig assumptios related to skewess ad/or kurtosis. 1 Vilius Istitute of Mathematics ad Iformatics. 2 The research was partially supported by the Lithuaia State Sciece ad Studies Foudatio, grat No T-15/ Mathematics Subject Classificatio. 60E15. Key words ad phrases. skewess, kurtosis, Hoeffdig s iequalities, sums of idepedet radom variables, martigales, bouds for tail probabilities, Beroulli radom variables, biomial tails, probabilities of large deviatios, method of bouded differeces. 1 Typeset by AMS-TEX

2 2 V. BENTKUS Let us recall Hoeffdig s 1963 Theorem 3. Let M = X X be a sum of idepedet radom variables such that X k 1, E X k = 0, ad E Xk 2 = σ2 k for all k. Write σ 2 = σ σ2, p = σ2 1 + σ 2, q = 1 p. Hoeffdig 1963, Theorem 3, established the iequality P {M t} H (t, p), H(t, p) = ( 1 + qt/p ) p qt( 1 t ) qt q (1.1) assumig that 0 < t < 1. Oe ca rewrite H (t, p) as H (t, p) = if h>0 exp{ ht} E exp{ht }, where T = ε ε is a sum of idepedet copies of a Beroulli radom variable, say ε = ε(σ 2 ), such that E ε 2 = σ 2, ad P {ε = σ 2 } = q, P {ε = 1} = p, p = σ2 1 + σ 2, q = σ 2. (1.2) Usig the shorthad x = t, we ca rewrite the Hoeffdig result as I B 2004 the iequality (1.3) is improved to P {M x} if h>0 e hx E e ht, (1.3) P {M x} if h<x (x h) 2 E (T h) 2 +, for x R, (1.4) where z + = max{0, z}, ad z β + = (z + ) β. Actually, iequalities (1.1), (1.3) ad (1.4) exted to cases where M is a martigale or eve super-martigale, see B 2004 for a proof. I the case of (1.1) ad (1.3) this was oted already by Hoeffdig The right had side of (1.4) satisfies if (t h<x h) 2 E (T h) 2 + e2 2 P {T x}, e = (1.5) for all iteger x Z. For o-iteger x oe has to iterpolate the probability log-liearly, see B 2004 for details. The right-had side of (1.4) ca be give explicitly as a fuctio of x, p ad, see BKZ 2006, as well as Sectio 2 of the preset paper. To have bouds as tight as possible is essetial for statistical applicatios, like those i audit, see BZ Our itetio i this paper is to develop methods leadig to couterparts of (1.1), (1.3) ad (1.4) such that iformatio related to the skewess ad kurtosis γ k = E (X k E X k ) 3 σ 3 k, κ k = E (X k E X k ) 4 σ 4 k (1.6) of X k is take ito accout (i this paper we defie γ k = ad κ k = 1 if σ k = 0). All our results hold i geeral martigale settig.

3 SKEWNESS AND CURTOSIS 3 All kow proofs of iequalities of type (1.3) ad (1.4) start with a applicatio of Chebyshev s iequality. For example, i the case of (1.4) we ca estimate P {M x} if h<x (x h) 2 E (M h) 2 + (1.7) sice the idicator fuctio t I{t x} obviously satisfies I{t x} (x h) 2 (t h) 2 + for all t R, h < x. The further proof of (1.4) cosists i showig that E (M h) 2 + E (T h) 2 + for all h R. We would like to emphasize that all our proofs are optimal i the sese that o further improvemets are possible i estimatio of E (M h) 2 +. Ideed, i the special case M = T the iequality E (M h) 2 + E (T h) 2 + turs ito the equality E (T h) 2 + = E (T h) 2 +. I view of (1.7) it is atural to itroduce ad to study trasforms G G β of survival fuctios G(x) = P {X x} defied by G β (x) = if h<x (x h) β E (X h) β +, β > 0, (1.8) settig G 0 = G i the case β = 0. See Pielis 1988, 1989, B 2004, BKZ 2006 for related kow results. The paper is orgaized as follows. I the Itroductio we provide ecessary defiitios ad formulatios of our results, icludig their versios for sums of martigale differeces. I Sectio 2 we recall a descriptio of the trasform G G 2 of biomial survival fuctios our bouds are give usig G 2. Sectio 3 cotais proofs of the results. Heceforth M = X X stads for a martigale such that the differeces X k are uiformly bouded (we set M 0 = X 0 = 0). Without loss of geerality we ca assume that the boudig costat is 1, that is, that X k 1. Let F 0 F 1 F be a related sequece of σ-algebras such that M k are F k -measurable. Itroduce the coditioal variace s 2 k, skewess g k ad kurtosis c k of X k by s 2 k = E ( X 2 k F k 1 ), gk = E ( X 3 k F k 1 ) /s 3 k, Note that s 2 k, g k, c k are F k 1 -measurable radom variables. c k = E ( X 4 k F k 1 ) /s 4 k. (1.9) Remark 1.1. We prove our results usig (1.4) for martigales. It is proved i B 2004 that all three iequalities (1.1), (1.3) ad (1.4) hold with σ 2 = (σ σ)/ 2 if M is a martigale with differeces X k 1 such that the coditioal variaces s 2 k satisfy s2 k σ2 k for all k. It is easy to check that Beroulli radom variables ε = ε(σ 2 ) with distributio (1.2) have variace σ 2 ad skewess γ related as γ = 1 σ σ, σ2 = u 2 (γ), where u(x) = 1 + x2 4 x 2. (1.10)

4 4 V. BENTKUS Theorem 1.2. Assume that the differeces X k of a martigale M satisfy X k 1, ad that the coditioal skewess g k of X k are bouded from below by some o-radom γ k, that is, that g k γ k, k = 1, 2,...,. (1.11) The (1.3) ad (1.4) hold if T is a sum of idepedet copies of a Beroulli radom variable ε = ε(σ 2 ) with distributio (1.2) such that the skewess γ ad the variace σ 2 of ε satisfy γ = 1 γ 1 u(γ 1 ) + + γ u(γ ), u 2 (γ 1 ) + + u 2 (γ ) σ2 = u2 (γ 1 ) + + u 2 (γ ). (1.12) I the special case where all γ k are equal, γ 1 = = γ = γ, the Beroulli radom variable has skewess γ ad variace σ 2 = u 2 (γ). It is easy to see that Beroulli radom variables ε = ε(σ 2 ) with distributio (1.2) have variace σ 2 ad kurtosis κ related as κ = 1 σ σ2, 2σ 2 = κ + 1 ± (κ + 1) 2 4. (1.13) I particular σ 2 v(κ), where 2v(t) = t (t + 1) 2 4. (1.14) Theorem 1.3. Assume that the differeces X k of a martigale M satisfy X k 1, ad that the coditioal kurtosis c k of X k are bouded from above by some o-radom κ k, that is, that c k κ k, k = 1, 2,...,. (1.15) The (1.3) ad (1.4) hold if T is a sum of idepedet copies of a Beroulli radom variable ε = ε(σ 2 ) with distributio (1.2) such that the kurtosis κ ad variace σ 2 of ε satisfy κ = 1 σ σ2, σ 2 = v(κ 1) + + v(κ ), (1.16) where the fuctio v is give i (1.14). I the special case where κ 1 = = κ = κ, the Beroulli radom variable has kurtosis κ ad variace σ 2 = v(κ). The ext Theorem 1.4 allows to combie our kowledge about variaces, skewess ad kurtosis. Theorems 1.2, 1.3 ad (1.3), (1.4) for martigales (see Remark 1.1) are special cases of Theorem 1.4 settig i various combiatios σ 2 k =, γ k =, κ k =. Theorem 1.4. Assume that the differeces X k of a martigale M satisfy X k 1, ad that their coditioal variaces s 2 k, skewess g k ad kurtosis c k satisfy s 2 k σ 2 k, g k γ k, c k κ k, k = 1, 2,..., (1.17) with some o-radom σ 2 k 0, γ k ad 1 κ k. Assume that umbers α 2 k satisfy α 2 k mi{σ 2 k, u 2 (γ k ), v(κ k )}, k = 1, 2,...,.

5 SKEWNESS AND CURTOSIS 5 The (1.3) ad (1.4) hold if T is a sum of idepedet copies of a Beroulli radom variable ε = ε(σ 2 ) with distributio (1.2) ad σ 2 = α α2, where fuctios u ad v are defied i (1.10) ad (1.14) respectively. Remark 1.5. All our iequalities ca be exteded to the case where M is a supermartigale. Furthermore, their maximal versios { hold, that is, } i the left had sides of these iequalities we ca replace P {M x} by P max M k x. 1 k Remark 1.6. Oe ca estimate the right had sides of our iequalities usig Poisso distributios. I the case of Hoeffdig s fuctios (1.1) this is doe by Hoeffdig I otatio of (1.1) his boud is H (t, p) if h>0 e hx E e h(η λ) = exp { x (x + λ) l x + λ λ }, (1.18) where x = t, λ = σ 2, ad η is a Poisso radom variable with the parameter λ. It is show i the proof of Theorem 1.1 i B 2004, that if T is a sum of idepedet copies of a Beroulli radom variable ε = ε(σ 2 ), the if (t h<x h) 2 E (T h) 2 + if (t h<x h) 2 E (η λ h) 2 +, (1.19) where η is a Poisso radom variable with the parameter λ = σ 2. The right had side of (1.19) is give as a explicit fuctio of λ ad x i BKZ Remark 1.7. A law of trasformatio {σ1, 2..., σ} 2 σ 2 i (1.1), (1.3) ad (1.4) is a liear fuctio of the variables σ1, 2..., σ. 2 I bouds ivolvig skewess ad kurtosis the correspodig trasformatios are o-liear, see, for example, (1.12), where the trasformatio {γ 1,..., γ } γ is give explicitly. Ackowledgemet. We thak R. Norvaiša for umerous remarks which helped to improve the expositio. 2. A aalytic descriptio of trasforms G 2 of biomial survival fuctios G. I this sectio we recall a explicit aalytical descriptio of the right had side of (1.4) G 2 (x) def = if h<x (x h) 2 E (T h) 2 +,

6 6 V. BENTKUS where T is a sum of idepedet copies of the Beroulli radom variable (1.2). The descriptio is take from BKZ Let G(x) = P {T x} be the survival fuctio of T. The probabilities p, q ad the variace σ 2 are defied i (1.2). Write λ = p. The sum T = ε ε assumes the values d s = σ 2 + s(1 + σ 2 ) s λ q, s = 0, 1,...,. The related probabilities satisfy p,s = P {T = d s } = ( ) s q s p s. The values G(d s ) of the survival fuctio of the radom variable T are give by Write G(d s ) = p,s + + p,, s = 0, 1,...,. ν,s = sp,s G(d s ). Now we ca describe the trasform G 2. Cosider a sequece 0 = r 0 < r 1 <... < r 1 < r = of poits which divide the iterval [0, ] ito subitervals [r s, r s+1 ], such that The G 2 (x) = r s = λ pν,s, s = 0, 1,..., 1. qν,s + λ s λ + ν,s (s λ p) qν 2,s qx 2 2qν,s + λ + ν,s (s λ p) G(d s), r s x r s Proofs Proof of Theorem 1.2. This theorem is a special case of Theorem 1.4. Ideed, choosig σ 2 k =, κ k =, k = 1, 2,...,, we have v(κ k ) =. Hece α 2 k from the coditio of Theorem 1.4 have to satisfy α2 k u2 (γ k ). We choose α 2 k = u2 (γ k ). The σ 2 = (u 2 (γ 1 ) + + u 2 (γ ))/. A elemetary calculatio shows that with such σ 2 the skewess γ = σ 1 σ with the expressio give i (1.12) of Beroulli radom variables ε = ε(σ2 ) coicides Proof of Theorem 1.3. This theorem is a special case of Theorem 1.4. Ideed, choosig σ 2 k =, γ k =, k = 1, 2,...,, we have u(γ k ) =. Hece α 2 k from the coditio of Theorem 1.4 have to satisfy α2 k v(κ k). We choose α 2 k = v(κ k). The σ 2 = (v(κ 1 ) + + v(κ ))/. I the proof of Theorem 1.4 we use the ext two lemmas.

7 SKEWNESS AND CURTOSIS 7 Lemma 3.1. Assume that a radom variable X 1 has mea E X = 0, variace s 2 = E X 2, ad skewess such that Proof. It is clear that E X 3 s 3 g with some g R. The s 2 u 2 (g), u(x) def = 1 + x2 4 x 2. (3.1) (t + s 2 ) 2 (1 t) 0 for t 1. (3.2) Replacig i (3.2) the variable t by X ad takig the expectatio, we get s 2 s 4 E X 3. E X3 Dividig by s 3 ad usig g, we derive 1 s 3 s that the latter iequality implies (3.1). s g. Elemetary cosideratios show Lemma 3.2. Assume that a radom variable X 1 has mea E X = 0, variace s 2 = E X 2, ad kurtosis such that E X 4 s 4 c with some c 1. The s 2 v(c), 2v(t) def = t (t + 1) 2 4, t 1. (3.3) Proof. By Hölder s iequality we have E X4 s 4 1. Hece, the coditio c 1 is atural. The fuctio v satisfies v(c) 1 for c 1. Therefore i cases where s 2 1, iequality (3.3) turs to the trivial s 2 1 v(c). Excludig this trivial case from further cosideratios, we assume that s 2 > 1. Write a = 2s 2 1. The a 1. It is clear that (t + s 2 ) 2 (1 t)(a t) 0 for t 1. (3.4) Replacig i (3.4) the variable t by X ad takig the expectatio, we get E X 4 s 2 s 4 + s 6. E X4 Dividig by s 4 ad usig c, we derive s 4 show that the latter iequality implies (3.3). 1 s s2 c. Elemetary cosideratios Proof of Theorem 1.4. The proof starts with a applicatio of the Chebyshev iequality similar to (1.7). This reduces the estimatio of P {M x} to estimatio of expectatios E exp{hm }, E (M h) 2 +. As it is oted i the proof of Lemma 4.4 i B 2004, it suffices to estimate E (M h) 2 + sice the desired boud for the other expectatio is implied by E (M h) 2 + E (T h) 2 +. (3.5) Let us prove (3.5). By Lemma 3.1 the coditio g k γ k implies s 2 u 2 (γ k ). While applyig Lemma 3.1 oe has to replace X by X k, etc. I a similar way, by Lemma 3.2 the coditio c k κ k implies s 2 k v(κ k). Combiig the iequalities ad the assumptio s 2 k σ2 k, we have s 2 k mi{σk, 2 u 2 (γ k ), v(κ k )}, k = 1, 2,...,. (3.6) The iequality (3.6) together with the coditio of the theorem yields s 2 k α2 k. As it is show i the proof of Theorem 1.1 i B 2004, the latter iequality implies (3.5).

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