Fluctuation inequalities with applications to convergence and regularity of stochastic processes indexed by [0,1] [superscript] q

Pub . Ma . UAB Vol . 29 Ns 1 Ab il 1985 FLUCTUATION INEQUALITIESMITH APPLICATIONS TO CONVERGENCE AND REGULARI'TY OF STOCHASTIC PROCESSES INDEXED BY [0,1] q . An oniSin es Blanc Uni e si a Au ónoma de Ba celona Spain Abs ac . Ex ending esul s o Billingsley and Chen so , Bickel & Wichu a p o ed some luc ua ion inequali ies o p o- cesseswi h mul i-dimensional ime pa ame e . In he same o de o ideaswe gi ehe e an ex ension o he case ha he ma ginals o he con ol measu e a e no necessa ily con inuous . Applica ions o his esul s o ge some use ul con e gen ce c i e ia o [0,1] 5 indexed p ocesses a e gi en, as well as a heo em on egula i y o igh s ochas ically con inuous p oce sses . AMS subjec classi ica ion (1 .983) . 60F05, 60005 Key wo ds and ph ases : Fluc ua ion inequali ies, weakcon e gen ce, D[0,1] a - alued andom a iables, egula i y . 0 . In oduc ion . In (1) P . Bickel & M . Wichu a p o e luc ua ion inequali ies o p ocesses indexed by a q-dimensional pa ame e se , ex ending esul s o Chen so and,Billingsley, (2), (3) . He e we ex end hei heo em 3 o he casewhe e he ma ginals o m a e no necessa ily con inuous . Bickel & Wichu a (op . ci . pg . 1665, inal) announce apossible ex ension o he case ha m dependson n, and he measu esmn con e ge weakly o a measu e wi h con inuous ma ginals . Ou ex ension has a di e en cha ac e : m will be ixed (independen o n), we will suppose ins ead ha p ocesses in ques ion ha e independen inc emen s, and he cons án s ha appea in hei heo em 1 will depend on m,q,y and P, in ou case . This is he con- en o poin 2 . Poin 3 is de o ed o gi e applica ions o he luc ua ion inequali ies o he con e gence o p ocesses indexed by [0,1] q . A poin 4 we see an applica ion o he egula i y o p ocesses wi h independen inc emen s o e [0,1] ° - . On his la e esul i is~wo hy o say ha R . Mo - k enas (6), using Dynkin-Kinney's ype condi ions, p o es ha all p ocesses wi h independen inc emen s ands ochas ically con inuous ha e e sions in  D [0,1] q .  Ou Thm . (4 .1) is no enclosed in his esul because we onlyimpose igh s ochas ic con inui y . 1 . De ini ions and p e ious esul s . No a ion is much as in  (1) .  Le  q  be a posi i ein ege and  T1,TZ, . . . . T q 84 subse s o (0,1] each o which .con ains 0 and 1, and is ini e o  [0,11 .  Le  {X } E T  be a s ochas ic p ocess indexed by T = T 1 x T Z x . . .x T q , wi h alues in a no med space (E, I .I) . We suppose X is sepa able and anish on he lowe bounda y o T, ain T, i .e- . he poin s o T ha ing some coo dina e equal o 0 . Fo each  p,  1 <p <q,  and each  E T_  we de ine P n X P) : T 1 x . . .x Tp x . . .x T q - E  by (P) X ( l ,. . . .  l ,  , . . .  )  = X( l~ ... . p-l, , p+l . . ., q) . P' P+1 q I  s 6 9 u  in  T ,  we de ine P m  (s, , u) (x)  =  min(11x (P) -  x ( P ) II,IIx ( P ) -  x(P)II) p  su Whe e II . II is he sup emum no m . De ini ion (1 .1) : M" (X)  = sup {mp (s, ,u) (X)  :  s< <u ;  s, ,uET P } M' , (X)  =  máx  Mp (X) 1 <p Qq M (X)  =  sup  { 1 X( ) 1  :  E T}  0 men a y . The ollowing p oposi ion is e yuse ul and qui e ele- q P oposi ion (1 .2) : I l q = (1,  l), hen q M (X)  <  E  M" (X)  +  I X(1  )I p =1P  q 5  q .M" (X)  +  I X (1 q ) I  0 We say ha  B C T  is a block i q B = II (s ] p=1 p p we also w i e B = (s, ] whe e s= (s 1 ,.. .,S ) and 4 Deno e X(B) he ec angula inc einen o X o e he block B, i .e . : q 1  1  1  q- El p X (B)  =  E  E  . . .  E  (-1)  P~  X(s +e  ( -s  ), . . .,S + e  ( -s  )) el=0 e 2 =0  e q =0  1  1  1  1  q  q  q  q TIe say ha X has independen inc emen s i X(B) and X(C) a e independen andom a iableswhene e B and C a e disjoin blocks . De ini ion - (1 .3) :  We w i e  X EC m (P,y)  i  X  has i independen inc emen s and P { I X (B)  (m (B) )p,  a x > 0 o all B C T, block o T, whe e y and p a e ixed posi i e eals, and m is a ini e measu eo e T . anishing o e  a in T E iden ly i  XEC m (p ,y)  hen he pai  (X,m)  belongs o C(2p,2y) in he sense o Bickel & Wichu a (1) . Theo em (1 .4) :  I  (X,m) E C(P .,'Y) , i.e .  i o all pai o disjoin blocks B,C o T we ha e hen  d X > 0 P{IX(B)I >X,  IX(C)I  X} : Q X -y (m (BUC)) p ,  yx> 0 o all  p,  1 Sp <q,  and P{MB (X)  }6K q (P Y)  -y (m(T) ) R P {M" (X) > T}6L q (p,y)  X -y (m(T)) p  O This is heo em 1 o Bickel and Wichu a(1) . In oduc ion o he ollowingmoduli is sugges ed by he iden i ica ion D q =  D (I 0, l) q ;  R)  =  D([0,11 ;  D q_1 ) n 0 De ini ion  (1 .5) :  I  x E D q and  S > 0  we de ine w" (P)  (S)  =  sup  min (II x (p) -x (p) II ,II x (P)-x(P) II ) x  G u s, ,uETp s S -<u, u-s 5 S w" (S) = máx  w ., (P) (S) x  x 1-<P ! ~q In wha ollows we shall also need he ollowing esul on igh ness in he space (D[O,l] q ; D q ), whosep oo may be ound in Neuhaus (7) . Theo em (1 .6) : A sequence {Pn}n=1  o p obabili y measu es on (D[0,1 q, Dq ) is igh i and only i : i)  Fo all  i? >O,  he eexis s  aER  such ha Pn {x : sup 1x( )I >a}-< 1,  o all n>1 . ii) Fo all  e >O,  77 >O,  he e exis  S, 0 < S<l,  and such ha o  n >_n . Pn {x : w , (S) >e } S1 2 . Fluc ua ionineguali ies Theo em (2 .1) :  The e exis s a cons an K, K = K(q,R,y,  m(T)),  such ha o all p ocess  XE Cm(0 y), (see De . (1 .3)), is J p [ 0,11  Q m P [ 0,1]1 o all  p, 1 < p -<q,  whe e  Jp[0,1]  is he maximum jumpo P{M" (X) > ñ}<K(?~ 47  V  X 27 )  (m P [ 0,1] )2R  I1  - p he dis ibu ion unc ion Fm o he p- h ma ginal, m p , o P m, and means " he g ea es o " . P oo : S ep 1 . g = 1  and  T  ini e .  Le  0 = o < 1 < . . . < m= 1 be he poin s o T . De ine he p ocess m-1 Y (U)  =  iE0  X ( i ) I[ i  i+1) (u)  +  X ( m ) I{ m} (u) o e 10,11 . Then, i i-1 <8< i< h< < h+l< k<u< k+l = 7~ -27 ( 2 :  m{ j }m{ j ,}) R < j =i .j = h +1 h <X -27 [( E m{ . } ( E  m{ .~})) 0 A ( E  m{ .~} (  m{ M)01 5 j=i  7  j=h+l  7  j=i  7  j =1  7 < x -2'Y  [(m (T)  -  J m(T) ) k E m{ .}l R j=i 7 k-1 C  27 (m (T) - Jm (T) )~  (  E  2 m { j } + m{ k}-m{ i})R  = j=i -- a -27 (m (T) -Jm (T) )l)  ( k E 1 2m{ j } +m{ k }  -1 2 ; 1 2m{ j }  - m{ i} )Q  = _ X-27 (m (T)  -  Jm (T) )0  (F ( k )  -  F ( i ) )Q < < X-27 (m (T) -  J  (T) )R (F (u)  -  F (s) )Q m whe e F, con inuous,is de ined by he ela ions F(0)  = 0,  F( j )  - F( j-1)  = m{ j }  + m{ j_1}  and is linea o e he in e als  [ j . ~-1 ,  j Hence, he p o es Y, oge he wi h he measu e, m', associa ed o he dis ibu ion unc ion F' = (m(T) - Jm(T))F, belongs o C(R,27) . By heo em (1 .4) we ha e 90 P{ML (X)  } = P{M1 (Y) T  }  KX -2^i (m' (T) =  K ~ 27  (m (T)  -  J  (T) )R  (F (1) )R 6 m < 20  K X -27 (m (T) )R  (m (T)  - J  (T) 1R m whe eweha eused  F(1) < 2m (T) .  Thisp o es he heo em in his case . S ep 2 . g = 1,  T = [0,11, m  a bi a .  Le 0 = o < 1 < . . . < m = 1,  and  Y  he p ocess  X  es ic ed o { , . . ., } . o m De ine u as in s ep 4, p oo o heo em 1, in Bickel & Wichu a (1) : u  { J .}  =  m( J . -l , 7 .]  i  j ól, u { o } = 0 . Then  YE Ci  as  a p ocess o e  { o , i . . . m } . S ep 1 now implies J{ , , . . ., } R P{ML (Y)>X}<  X-2yK(u{ o . l, . . . . m }) 2R  1 -  u  o  i  m í  u ( o .. . . . m} K  X-27 (m (T) ) 20  1  Ju[0 .1]  R í - m(0,1] I now we ake limi when  m -- " hese { o, l, .. ., m}  inc easing o a dense subse o [0,1]  ha con ains he poin so discon inui y o F , we ob ain (by se- pa abili y) : (  ] P{M1 (Y)  ] X}  K X- 2'y (m(T) )2R 11-  J m ( 0, 1 m (T) S ep 3) q >-2, T  and  m  a bi a y .  We know ha he heo em is ue o q = 1 . We now will show ou esul o be ue o p = 1 ; o o he p he a gumen is he same . Like in s ep 5 o Bickel & Víichu a's p oo o heo- em 1, he key poin is ha he e sion o  q= 1 o ou heo em wo ks o he unc ion alued p oces  {X (l)} E T  To 1 Wi h espec o ii) : a) is a consequence o hm_ (3 .1), c)  is he hypo hesis iii) . Le us see i) . 98 Hence : P{ sup ET  I x ( ) I  > a  } =  P{ sup II x ( P ) II > a} E[0 .11 P{sup ET  I (Xn) l > a } = P {  sup  II (X n ) l .) II > a } E[ 0,11 <P{w" (1) (S) > 1 } +P{máx  sup  I (X  ) (1)  ( * ) I > a  } x  n  o n  l IQ i <, k  * E T 2x . . . x Tq (1)  k SP{w"  (8) >1 }  + E  P {  sup  I  (x  I> a o } . n  i=1  * ET 2 x . . . xT q Now becauseo P{w«X n ) (1) [ 1 - ó,l) > e,  o some p' ,  2 <p' <q } 6 . 1 6P{wXP ) [1 - 8,1)> e, o some p, 16p<q } n he p ocesses . (Xn) l) sa is y i), ii) and iii) o ou i heo em, i = 1,2, . . . .k . By induc ion hypo hesis he e exis s { a i } k such ha P{sup *I (X,) (1) I > a i } 6 n/2k . Gi en  n >O,  le  S > 0  be such ha P{WX(1)  (S ) > 1 }  X1/2 . n choose  0 = 0 < I < . . .  k = 1  such ha  i - i-1 <  ó  and a = máx  a . o  i . .k Then i a = a + 1 0 P{sup ET I (X n ) 1 >a } < n Thisp o es ha i), ii) and iii) imply i) o hm . (3 .2), by induc ion on q . 0 E < S0 I only es s o e i y condi ion b) o ii) . By induc ion hypo hesis (Xn )S P) .W ; (X)á P) . Hence, II (X n ) 6 ( p ) II -~ II (X) s (P)  also .  Now obse e ha as a consequence o he igh con inui y o  (X) (P) and Gi en posi i e  n  and  e,  le  S 0 > 0  be such ha i Then II (X)8(P)II ó 0 P{II(X)(P)I1 : e}<n/2 . limsup  P{II (xn ) ( P ) II ~e } <P{II (X) (P) II >e }  n/2 . n -, o0 {x : wXP ) I 0,6) > 4E } C {X : w " (P) (S } ~ E } U {x : II x8(p) - X 0 (p) II ~lE } ppose : y>0 . Now om we ge : lim  sup __  P n {x : wXP)( 0,8 )  4e } : !Z 17 Thisp o es b) and he heo em . In applica ions qui e equen ly we don' know ha X E D .  I is henuse ul  o ha e he ollowing a ian o he q p e ious hm . ., whosep oo equi es he same a gumen as abo e . Theo em (3 .4) : Le 00 be as in hm  , (3 .3) . S i) The ini e dimensional dis ibu ions o X n a e weakly con e gen and lim  lim sup  P {x : IIx ( P ) II >E} =0 Slo  n-' oo n o all  E > 0  andall  p,  1 <p<q . ii)  X n E Ci (Q y) ,  n = 1, 2, . ..,  o some  Q >112  and iii) lim  lim sup  P n {X :W ( P )[ 1-5,1) >e, o sume p, 1<p<q} = 0 640  n--~ o all e >O . Then  {P n = L(Xn)}n=1  is .weakly con e gen  O 4 . Regula i y o p oce sses wi h i ndlQenden in c emen s . Theo em (4 .1) :  I  XEC ° ' (R ,Y) ,  whe e  R > 1/2,  y>0, i . hen  X  has a e sion wi h sample pa hs in  D[ 0,1]q . P oo :  Le  8 0 < 1/2 .  Fo  E=- [ 0,1] q  de ine : 6 0 ( )  _  ( 1 ,... , i-1 , iI [S o , 1-So] ( i )  +S o I [ 0  So] ( i) + o all  i,  1 <i <q . ( 1-8 0) 1 (1 -50,1] ( i ),  i+l ,.... q ) S ( ) = ((1 -250 ) 1 ( 1 0), ". ., (1-2 ó) -1 ( i - S0 ), . . ., (1-25 0 ) 1 ( q_80) ) 0 8 ( )  _  ( 8  0  S 0 0  a  0 . . .a  s  ) ( ) . o  0  0  0 We i s p o e ha he p oces  Y = X S ( ) has a e sion wi h sample pa hs in  DIO, llq .  0 Obse e ha  YE Cm  i 1 m ( .) = m( 8 (. n[8o,1-8o]q)) 0 on  10, 1] q , Fo each n we de ine a p oces Y n on [0,1,q, cons- an o e each ec angle o he dyadic ne o o de n, and equal o he alue o Y a "sou h-wes " e ex, i .e . : o all E[ (i 1 - 1)2 -n ,i1 2 -n )x . . .x[(i q - 1)2-n,ig2-n), 1 . < i 1 6 2 n , . . .,  15  i g  <2n . a gumen like ha in he p oo o hm . (3 .1) shows ha (4 .1 .1)  lim  limsup  P{w"  (ó ) > e } = 0 64 0  n -  0o  Yn lo  - . l l  1  I1 a i  n-  c  ~  . In ac : I Z  is de ined o e T * = [0,1] P-l x [a, ] x n x [0,1]g - P  om Y n , as in hm_ (3 .1) Y n is de ined om X , Z m(2)  ep esen s he es ic ion o Z  o he dyadic ne , n n  n Tm(2) , o T*, and (m)  is de inedo e Tm*(2)  like he o s ep 2, in he p oo o hm . (2 .1), hen : Y n ( )=Y((i 1 - 1)2 -n ,. . . . (i  - 1)2n) g 00 We show ha {Yn}n=1 is a igh sequence . Fi s , an P{M" (Z ) > X} = lim  P{M" (Zm(2) ) >ñ } S P  n  m -s 00  P  n , T ] -IR J m (a ' lim  K X-4y ( m (a~ 1) 2 01 -  P  = m -~  00  P  m(a,T ] j ] -_ K X_47 m 2Q (a ]  1  -JP(a, P  mp(a .7] whe e Hence, {Y n } sa is ies (3 .1 .3), and now all ollows as in hm . (3 .1)'s p oo . I  1-k <b  and  T k(2)  deno e hese o poin s o he .2-k -dyadic ne in  T = [ 0,1] q,  hen sup ETIYn( )I < max ET  IYn( )I+qw'Y (s) k(2)  n Mo eo e , obse e ha he a iables max E  I Y  ( ) I ,  n = k,  k +  1, ... Tk(2) n a e iden icallydis ibu ed . This, oge he wi h (4 .1 .1) gi es condi ion i) o ou hm . (3 .2) . Besides, {Yn}n=1sa is ies b) and c) o ii), hm . (3 .2), by cons uc ion . Hence, {Y }°°  is igh . I W is he weak limi o n n=1 some subsequence, heni is easy o see ha W is a e sion o Y, looking i s a dyadicpoin s, and app oaching hen . any poin by dyadics . The applica ion g ^ being bijec i e and con inuous be ween [S . ,l -S o ] q and [O,l]q, and X = Y(Póo)-1( ) , he heo em is p o ed . 0 Rema ks and commen s . a) I willbe e y in e es ing o ge a esul like hm . (2 .1) o p ocesses whoseinc emen s a e no necessa ily independen . I don' IMow a p esen how o do his . b) All p e ious esul sex endeasily o [0,_) q -indexed p ocesses usina well known esul s on D[0,-) q (see  B .G . I ano (5)) . c) Using abo e esul s and someo he s, (which cons i u- e my Doc o al Thesis, as p esen ed a he Uni e si a Au ónoma de Ba celona, Spain),we can p o e he Cen al Limi Theo em o p ocesses ha admi a ep esen a ion as s ochas icin e- g als w . . . .Lé y p ocesses wi h mul idimensional ime pa ame- e . Thiswillappea elsewhe e . d) Finally I wan o exp ess my indeb ness and g a i ude o P o esso E . Giné, ha sugges ed his p oblems o me and has gi en e icien help, whene e needed . REFERENCES (1) Bickel, P .J . & Wichu a, M .J . (1971) . Con e gence c i e ia o mul ipa ame e s ochas ic p ocesses and some applica ions . The Annals o Ma hema ical S a is ics ol . 42 . Ns 5, 1656-1670 . (2) Billingsley, P . (1968), "Con e gen e o P obabili y Measu- es" . John-Wiley & Sons . 104 (3) Chen so , N .N . (1956) . Weak con e gence o s ochas ic p o- cesses whose ajec o iesha eno discon inui ies o he second kind and he "heu is ic" , app oach o he Kolmogo o - Smi no es " . Theo . P obabili y Appl . 1, 140-144 . (4) Giné, E . & Ma cus, M .B . (1983) . The Cen al Limi Theo em o . s ochas ic in eg als wi h espec o Lé y p ocesses . The Annals o P obabili y, ol .ll, Ns 1, 58-77 . (5) I ano , B . (1980) . The unc ion space D((o,oo)q ;E) . The Canadian Jou nal o S a is ics . Vol . 8, Ná 2, 179-191 . (6) Mo k enas, R . (1980) . On weak compac nesso he se s o mul ipa ame e s ochas ic p ocesses . "S ochas ic Di e en ial Sys ems" . Lec u e No esin Con ol and In o ma ion Science, 25 . (7) Neuhaus, G . (1971) . On weak con e gence o s ochas ic p o- cesses wi h mul idimensional ime pa ame e . The Annalso Ma hema ical S a is ics, ol . 42, Ne 4 1285-1295 . Rebu el 11 de duembne del 1984 Uni e si a Au ónoma de Ba celona Facul a de Ciéncies Depa amen de Ma emá iques Bella e a - Ba celona SPAIN