Hölderian Version of Donsker-Prohorov s Invariance Principle

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1 Hölderia Versio of oser-rohorov s Ivariace riciple Haadouche jael ad Taleb Youcef Abstract The wea covergece of a sequece of stochastic processes is classically studied i the Sorohod space [0, ] or C[0, ] but the wea Hölder covergece offers ore cotious fuctioals tha C[0, ] for statistical applicatios We study the wea covergece of stochastic processes i Hölder spaces ad usig soe results of tightess proved i these spaces, we obtai a Hölderia versio of oser-rohorov s ivariace priciple First for the polygoal iterpolatio of the partial sus process, geeralizig Laperti s ivariace priciple to the o-statioary case ad siilar results are proved for the covolutio soothig of the partial sus process Keywords: stochastic processes, tightess, Hölder space, ivariace priciple, Browia otio AMS classificatios: 60B0, 60F05, 60F7, 6G30 Itroductio I paraetric ad o paraetric statistics, ay statistical applicatios estiatio, testig hypothesis, are based o cotiuous fuctioals of paths of processes ad solved usig the wea covergece of stochastic processes The wea covergece of a sequece ξ, of stochastic processes i soe fuctioal space provides results about the asyptotic distributio of cotiuous fuctioals of the paths Sice the Hölder spaces are topologically ebedded i the spaces C[0, ] of cotiuous fuctios ad i the Sorohod space [0, ], they support ore cotiuous fuctioals Fro this poit of view, the alterative fraewor of Hölder spaces gives fuctioal liit theores of a wider scope This choice ay be relevat as soo as the paths of ξ ad the liit process lie eg the Browia otio ad the Browia bridge share soe Hölder regularity The first result i this directio sees to be Laperti s Hölderia ivariace priciple [6] for the cetered ad oralized polygoal partial sus processes This result was copleted i recet cotributios by ačausas ad Suquet [8, 0] who exteded it to the case of adaptive self-oralized Laboratory of Matheatics, Faculty of Scieces, Uiversity M Maeri Tizi-Ouzou, 5000 Algeria, ail: djhaad@yahoofr, talebyoucef@yahoofr partial sus processes ad proposed a ecessary ad sufficiet coditio for a geeralized for of Laperti s ivariace priciple Soe statistical applicatios of wea Hölder covergece are proposed by the sae authors [9, ] We cosider a sequece X j j of idepedet rado variableot ecessarily idetically distributed with X j = 0 ad Xj = σj We deote ξ the rado polygoal lies obtaied by liear iterpolatio betwee the poits j, Sj where S j = j X j ad = σ + + σ Whe the X j are idetically distributed with X j = σ, the oser-rohorov s ivariace priciple establishes the the C [0, ] wea covergece of ξ to the Browia otio W The ivariace priciple i the Baach Hölder space H α [0, ] has bee established by Laperti Keryacharia ad oyette have derived it agai usig the Faber-Schauder basis of triagular fuctios Theore Laperti [6] Let X j j be a sequece of idepedet idetically distributed rado variables with X j = 0 ad X j = σ Suppose that for soe costat >, X j < For all N, 0 j <, defie ξ t, ω = =j X ω+ t j X j+ ω, j t < j + The the sequece ξ coverges wealy to the Browia otio W i H 0 α for all α < Usig soe results of tightess proved i these spaces, Haadouche [3] has exteded this result to depedet rado variablesα-ixig ad associatio ad has proved the wea covergece i Hα 0 of the covolutio soothed process to the Browia otio Our ai is to exted Laperti s theore to polygoal ad covolutio soothed partial sus process with a sequece of idepedet rado variables X j j ot ecessarily idetically distributed The polygoal soothig

2 is soeties rough, lie i the study of wea covergece of epirical ad quatile processes i [, 4] Thus it is iterestig to study the covolutio soothig which is useful i statistical applicatios, estiatio of desity, etc Also we use the classical defiitio of the partial sus process, which is useful i the literature, for a ostatioary sequece of idepedet rado variables istead of the adaptative costructio used i [0] because for the geeral case of triagular array of rado variables with ueve variaces, the covergece of fiitediesioal distributios to Browia otio doeot iediately follow fro the cetral liit theore I Sectio, we recall the Baach Hölder space H α [0, ] ad its closed subspace Hα 0 [0, ] We cosider stochastic processes with paths i Hα 0 [0, ] ad treat the as rado eleets of Hα 0 [0, ] We give soe results of the wea covergece ad tightess Our ivariace priciples are preseted i Sectio 3 We exted the oser- rohorov s theore for the idepedet rado variables ot ecessarily idetically distributed ad prove the wea covergece i Hα 0 [0, ] of the polygoal soothed of the partial sus process to the Browia otio Siilar result is proved for the covolutio soothed process ado eleets i Hölder space We study stochastic processes with Hölderia paths as rado eleets of the fuctioal space H α [0, ] We observe directly the whole path, which correspods to select at rado a fuctio ξ with distributio ξ efiitios We defie the Hölder space H α [0, ] 0 < α as the space of fuctios f vaishig at 0 such that f α = f t f s sup 0< t s t s α < efie the Hölderia odulus of cotiuity of f by w α f, δ = f t f s sup 0< t s δ t s α ad the subspace H 0 α [0, ] of H α [0, ] by f H 0 α f H α ad li δ 0 w α f, δ = 0 3 H α, α is a o-separable Baach space H 0 α, α is a separable Baach space H α, β is separable for 0 < β < α ad is topologically ebedded i H β Wea covergece i Hölder space The cocept of wea covergece of probability easures ca be forulated for the geeral etric space We use this theory to obtai a whole class of liit theores for fuctios of the partial sus S,, S, 3 Tightess For the tightess, it is ore coveiet to wor with H 0 α [0, ] which is separable istead of H α [0, ] As the caoical ijectio of H 0 α [0, ] i H α [0, ] is cotiuous, wea covergece i the forer iplies wea covergece i the later A first sufficiet coditio for the tightess i H 0 α [0, ] is give by Theore Keryacharia-oyette [5] Let ξ be a sequece of processes vaishig at 0 ad suppose there are > 0, δ > 0 ad c > 0 such that λ > 0, ξ t ξ s c λ t s +δ 4 The the sequece ξ is tight i Hα 0 [0, ] for 0 < α < δ The followig corollary gives the oets versio of the precedet theore Corollary 3 Laperti [6] Let ξ be a sequece of processes vaishig at 0 Suppose there are > 0, δ > 0 ad c > 0 such that ξ t ξ s c t s +δ 5 The the sequece ξ is tight i H 0 α [0, ] for 0 < α < δ The sufficiet ad ecessary coditio which ca be useful to test the optially of certai results is give by the Hölder versio of Ascoli s theore Theore 4 ačausas, Suquet [7] Let ξ be a sequece of rado eleets of Hα 0 [0, ] ξ is tight if ad oly if ε > 0, li sup δ 0 w α ξ, δ ε = 0 6

3 For ore flexibility i the hadlig of oet iequalities, we use this followig result for all j t < j+ Theore 5 Haadouche [3] Let ξ be a sequece of rado eleets of Hα 0 [0, ], satisfyig the followig coditios a there exist costats a >, b >, c > 0 ad a sequece of positive ubers a 0 such that ξ t ξ s a c t s b, 7 for all t s a, 0 s, t ad b For ay ε > 0, li ω α ξ, a > ε = 0 8 The for all α < a i a, b, ξ is tight i H 0 α [0, ] 3 Ivariace priciples i Hölder space We cosider the sequece X j j of idepedet rado variables, ot ecessarily idetically distributed with X j = 0 ad σj = X j eote = σ + + σ, S i = i X ad S 0 = 0 We suppose that there exist >, > 0 ad M > 0 such that j X j M < ad σ j = X j j σ 9 3 olygoal soothig of partial sus process We exted here the oser-rohorov s theore for idepedet rado variableot ecessarily idetically distributed which is the first extesio of Laperti s ivariace priciple Theore 6 Let X j j be a sequece of idepedet rado variables, ot ecessarily idetically distributed with X j = 0 ad Xj = σ j ad satisfyig 9 efie for all N, 0 j <, ξ t, ω = =j X ω + t j X j+ ω, 0 The the sequece ξ coverges wealy to the Browia otio W i H 0 α [0, ] for all α < roof We apply Theore 5, with a = Tightess of distributios = ξ By Corollary 3, it is sufficiet, to prove that uder the assuptios of Theore 6 ξ t ξ s K t s +δ with + δ = > First case: j s t j+ We have ξ t ξ s = t s X j+ ω Uder the assuptio 9 it is easy to see that there exists a costat M = M such that < X j M, j By the last iequalities, we deduce ξ t ξ s t s X j+ = t s X j+ The ξ t ξ s t s X j+ M t s, sice t s ad X j+ < M Fially ξ t ξ s K t s +δ, with K = M ad + δ = > j Secod case: s j j+ 0,,, j t j++, = By triagular iequality, ξ t ξ s ξ t ξ j + + ξ j + ξ j

4 + ξ j ξ s By Jese s iequality we have ξ t ξ s 3 ξ t ξ j + + ξ j + ξ j + ξ j ξ s The first ad the third ters ca be treated as i the precedet case, thus there exist soe costats K ad K 3 such that ξ t ξ j + K t s +δ 3 ad ξ j ξ s For the iddle ter, we have ξ j + ξ j K 3 t s +δ 4 = S j+ S j Sj+ S j By the Marciiewicz-Zygud s iequality, it follows ξ j + ξ j C C M X i i= C M = Sice t s j+ j =, ξ j + ξ j M C t s We deduce that there exist costats K = C ad δ > 0 such that ξ j + ξ j K t s +δ 5 M With the three iequalities 3, 4 ad 5 we obtai ξ t ξ s 3 K + K + K 3 t s +δ So, there exists a costat K = 3 K + K + K 3 such that ξ t ξ s K t s +δ with + δ = > Thus by Corollary 3, the sequece of distributios of processes ξ is tight i Hα 0 [0, ] for ay 0 < α < δ = Covergece of the fiite-diesioal distributios To show that the fiite-diesioal distributios of the ξ coverges to those of W, we cosider first a poit s ad ust prove that ξ s W s With the defiitio of ξ, we have ξ s S [s] = s [s] X [s]+ X [s]+ It suffices to prove that W s X [s]+ 0 ad S [s] For ε > 0, the Bieayé-Tchebychev s iequality iplies that X [s]+ ε M ε 0 sice s ad 0 X [s]+ V A X j M, j The We ow that by the Lidberg s theore, S coverges i distributio to the oral law N if li X d = 0 X ε s With the assuptio X ε, we have X δ s X ε X d s M ε δ +δ ε δ +δ X X ε ε δ δ 0 The S [s] N Usig the idepedece of rado variables ad the assuptio 9, it is easy to prove that [ ] li V A S[s] S[s] = s ad = 0, d

5 so S [s] Ws Cosider ow two poits s ad t with s < t By the sae arguets applied to the triagular array X i, [s] < i [t], we obtai S [t] S [s] The two copoets of S[s] Wt W s, S [t] S [s] are idepedet by the idepedece of X j Sice is separable, it follows that S[s], S [t] O the other had ξ t ξ s S [s] S[t] W s, W t W s S [s] X [t]+ + X [s]+ We have show that X [s]+ 0 cosequetly X [t]+ 0 so it follows that S[t] ξ t ξ s S [s] 0 ad hece ξ t ξ s W t W s Sice ξ s ad ξ t ξ s are idepedet by the idepedece of X j, we deduce that ξ s, ξ t ξ s W s, W t W s We coclude that ξ s, ξ t W s, W t sice the fuctio h defied by x, y x, x + y is cotiuous We treat a set of three or ore poits i the sae way, ad hece the fiite-diesioal distributios coverges properly This achieves the proof of Theore 6 3 Covolutio soothig of partial sus process We recall here soe results ad soe assuptios used by Haadouche [3] for the covolutio soothed process i H α [0, ] We cosider the oser-rohorov oralized partial sus process ξ t = S [t] t, t [0,] 6 For the sae of coveiece, we shall use both followig expressios of ξ ξ t = i= S i [ i, i+ [ t, 7 ξ t = X [,] t 8 Let K be a probability desity o the real lie such that u K u du < 9 ad b a sequece of positive ubers such that li b = 0 ad = O τ, 0 < τ < b 0 We defie the sequece K of covolutio erels by K t = t K, t b b Lea 7 Haadouche [3] Let f be a bouded easurable fuctio with support i [0, ] ad K a covolutio erel satisfyig K L [, ] L [, ], K x K y α K x y, x, y [, ], 3 for soe costat α K The the restrictio to [0,] of f K f K 0 is i H [0, ] We cosider the soothed partial sus process defied by ζ t = ξ K t ξ K 0 t [0, ] 4 The ter ξ K 0 is subtracted i order to have a process with paths vaishig at zero Theore 8 Let X j j be a sequece of idepedet rado variables, ot ecessarily idetically distributed with X j = 0 ad Xj = σ j ad satisfyig 9 Suppose that the covolutio erels K satisfy 9,, ad 3 The the sequece of soothed partial sus process ζ defied by 4 coverges wealy to the Browia otio W i Hα 0 [0,] for all α < τ, ax roof By lea 7, ζ is i Hα 0 [0, ] for all α < We apply Theore 5 with a = We recall that ξ K t = ξ t u K u du =

6 ξ u K t u du Tightess Usig Theore 5 with a =, we study separately the cases t s ad t s < Without loss of geerality we ca assue that t > s First case: t s ζ t ζ s = ξ K t ξ K s = [t u] s X K u du =[s u]+ = [t u] K s X =[s u]+ Applyig the Jese s iequality whith respect to K u du we obtai ζ t ζ s [t u] K By the Fubii s theore we have =[s u]+ X ζ t ζ s K [t u] [t u] s X =[s u]+ =[s u]+ X K u du Usig the Marciiewicz-Zygud s iequality for the oets of sus of the idepedet rado variables we obtai ζ t ζ s [t u] C X K u du =[s u]+ Usig the assuptios 9 ad, we deduce that C [t u] M =[s u]+ ζ t ζ s K u du C M [ t u] [ s u] K u du Sice [ t u] [ s u] t s + ad t s, ζ t ζ s C M t s + K u du C M t s + K u du C M t s + t s K u du C M 3 t s K u du C M 3 t s K u du C M 3 t s Hece there exists a costat C = C M 3 that ζ t ζ s C t s Secod case: 0 t s < such ζ t ζ s = ξ K t ξ K s = X K t u K t s [,] u du X b X K t u K t s [,]udu t u K b K s u Usig Assuptio 3 of Lea 7, it follows ζ t ζ s X αk t s b b [,] u du X t s αk b b [,] u du X b α K t s b sice < ad hece αk ζ t ζ s t s α b α K b [,] u du X t s, X t s α

7 As a result, w α ζ, ζ t ζ s = sup t s t s α α b α K X b α α K X Now to prove that w α ζ, 0, it suffices to show that b α K X α 0 By the Marov s iequality we have [ αk b α αk δ b α ] X > δ X The assuptio 9 ad the Schwartz s iequality give X M so [ αk ] b α X > δ αk M δ b α 0 Usig the assuptio 0 we deduce that b a = α b 0 if α < τ We coclude about tightess by Theore 5, otig that its hypothesis are satisfied for a =, b =, c = C ad a = We obtai the the tightess of ζ i Hα 0 [0, ] for all α < τ ad α < so for all α < τ, ax Covergece of the fiite-diesioal distributios of {ζ, } By Theore 6, the fiite-diesioal distributios of ξ coverge to those of the Browia otio, it will be the sae for those of ζ if we prove for istace the covergece to zero of ζ t ξ t for all t [0, ] ζ ξ t = ξ t u K u du ξ t = S[t u] S [t] K u du [t u] = X i K u du i=[t]+ = [t u] X i i=[t]+ Applyig the Jese s iequality with respect to K, we obtai [t u] ξ K t ξ t X i By the Fubii s theore we have ξ K t ξ t K [t u] X i i=[t]+ i=[t]+ [t u] X i i=[t]+ K u du Applyig agai the Marciiewiez-Zygud s iequality, it follows ξ K t ξ t [t u] c X s i K udu c i=[t]+ c M u + K udu sice [t u] [t] u + c M u + u + K udu K udu with c = c M By ad otig v = u b, it follows that c ξ K t ξ t b v + Kvb dv b c b v Kvdv +

8 Sice v Kvdv < ad b goes to zero a goes to ifiity, we deduce that ξ t K t ξ t goes to 0 i L Ω for all t [0, ] I particular for t = 0, ξ K 0 goes to 0 a goes to ifiity We have fially, for all t [0, ] ζ t ξ t = ξ K t ξ K 0 ξ t ξ K t ξ t + ξ K 0 0 Hece [8] ačausas A, Suquet Ch 00, Ivariace priciples for adaptive self-oralized partial sus processes Stochastic rocess Appl, N95, 63 8 [9] ačausas A, Suquet Ch 00, Hölder covergeces of ultivariate epirical characteristic fuctios Matheatical Methods of Statistics, vol, N3, [0] ačausas A, Suquet Ch 003, Hölderia ivariace priciple for triagular arrays of rado variables Lithuaia Matheatical Joural, vol 43, N4, so ζ t ξ t L 0 ad iplies ζ t ξ t 0 [] ačausas A, Suquet Ch 004, Hölder or test statistics for epideic chage Joural of Statistical laig ad Iferece, vol 6, Issue, i= ζ t i ξ t i = ζ ξ 0 This achieves the proof of the covergece of the fiitediesioal distributio of ζ The Theore 8 is the proved efereces [] ricso V 98, Lipschitz soothess ad covergece with applicatios to the cetral liit theore for suatio processes A robab, N9, [] Haadouche 998, Wea covergece of soothed epirical process i Hölder spaces Stat robab Letters, N36, [3] Haadouche 000, Ivariace priciples i Hölder spaces ortugal Math, N57, 7 5 [4] Haadouche, Suquet Ch 006, Optial Hölderia fuctioal cetral liit theores for uifor epirical ad quatile processes Math Meth Statist, Vol 5, N, 07-3 [5] Keryacharia G, oyette B 99, Ue déostratio siple des théorèes de Kologorov, oser et Ito-Nisio C Acad Sci aris Sér Math, Issue 3, [6] Laperti J 96, O covergece of stochastic processes Tras Aer Math Soc, N04, [7] ačausas A, Suquet Ch 999, Cetral liit theore i Hölder spaces robab ad Math Statist, N9, 33 5