Alea 7, 257 276 200 Diagoal approximatios by martigales Jaa Klicarová ad Dalibor Volý Faculty of Ecoomics, Uiversity of South Bohemia, Studetsa 3, 370 05, Cese Budejovice, Czech Republic E-mail address: jaalic@efjcucz LMRS, Uiversité de Roue, Aveue de l Uiversité F 7680 Sait Etiee du Rouvray, Frace E-mail address: daliborvoly@uiv-rouefr Abstract the paper, we study ad compare several types of approximatios of statioary processes by martigales troductio Let Ω, A, µ be a probability space with a bijective, bimeasurable ad measure preservig trasformatio T For a measurable fuctio f o Ω, f T i i is a strictly statioary process ad reciprocally, ay strictly statioary process ca be represeted i this way We shall deote Uf = f T; U is a uitary operator i L 2 The fuctio f will be supposed to have a zero mea F i i Z, where T F i = F i+, is a icreasig filtratio of sub-σ-fields of A H i deotes the space of F i -measurable ad square itegrable radom variables P i f is the projectio of f oto the space H i H i, ie P i f = Ef T i F 0 Ef T i+ F 0 We suppose the fuctio f to be regular, ie f = + i= P if Let σ = S f, where is the L 2 -orm ad S f = i= f T i For simplicity of otatio we write Q 0 S f for ES f F 0 ad R S f for S f ES f F Defiitio We say that a fuctio f has a martigale approximatio if there exists a martigale differece sequece m T i i Z such that S f m 2 = o t is a classical result that the martigale approximatio implies the cetral limit theorem Gordi 969 his article, Gordi oticed that if f = i=j P if, j, j the for m = P 0 i= Ui f we have f = m + g Ug with g L 2 hece we get a martigale approximatio Received by the editors February 23, 200; accepted May 8, 200 2000 Mathematics Subject Classificatio 60G0, 60F05, 28D05, 60G42 Key words ad phrases Martigale approximatio, wea diagoal approximatio, strog diagoal approximatio, WW approximatio Supported by KONTAKT grat D MEB 020807 257
258 Jaa Klicarová ad Dalibor Volý Defiitio 2 We say that for a fuctio f there exists a wea diagoal approximatio if there is a sequece N, = o of itegers such that i for f = i= P i f we have S f f 0, σ ii for f = m + g Ug with m = g = i= P 0 U i f we have g = oσ i P i U j f i= i j=0 P i U j f f for the array U i m,, of martigale differeces a CLT holds, we get it for S f/σ as well cf Volý, 2006; this geeralises Gordi s result to processes with oliear growth of variaces of partial sums Haa 973 has oticed that if the series of P i f = P 0 U i f is summable the the series of m = i Z P 0U i f coverges ad gives a martigale approximatio We, moreover, get that for ay ǫ > 0 there is a such that for m = i= P 0U i f, limsup S f m 2 / ǫ; this holds if we replace m by f = i= P if as well Defiitio 3 We say that for a fuctio f there exists a strog diagoal approximatio if there is a iteger sequece of d N, d = o, such that for all iteger sequeces N which satisfy d for all ad = o: i for f = i= P i f we have S f f 0, σ ii for f = m + g Ug with m = g = i= P 0 U i f we have g = oσ i P i U j f i= i j=0 P i U j f The Haa s coditio thus implies the strog diagoal approximatio The Haa s coditio implies the wea ivariace priciple; i full geerality the result was proved i Dedecer et al 2007, the martigale approximatio for o adapted processes ca be foud i Volý 993 Heyde has show that if the series m = i Z P 0U i f coverges i L 2 ad m 2 limif S f 2 the a martigale approximatio exists cf Hall ad Heyde, 980 Volý 993 has show that the covergece i L 2 is ot sufficiet for the martigale approximatio ad that a ucoditioal covergece is a sufficiet coditio
Diagoal approximatios by martigales 259 Figure Diagoal ad Wu-Woodroofe approximatio f f T f T 2 f T f 0 T i S f f U m R S f P 2 P P 0 P P 2 0 m 0 g S f f 0 Q 0 S f Defiitio 4 We say that a fuctio f has Wu-Woodroofe approximatio if Q 0 S f = oσ ad R S f = oσ This is equivalet to the existece of a array U i m, of martigale differeces such that S f m = oσ ; for m we ca, i such a case, choose m = i P 0U i f + i= i P 0U i f = =0 i= P 0U i f The Wu-Woodroofe approximatio was itroduced i Wu ad Woodroofe 2004 for adapted processes The oadapted case ca bee foud i Volý 2006 The Wu-Woodroofe approximatio together with liear growth of variaces S f 2 are ecessary but ot sufficiet, cf Klicarová ad Volý, 2009 coditios for the martigale approximatio A coditio which, for processes with liear growth of variace, is ecessary ad sufficiet for the existece of a martigale approximatio, has bee foud by Zhao ad Woodroofe 2008 This coditio, valid for adapted processes, ca be formulated i the followig way: For f = P if 2 Results = lim sup j j S j f f 0 Theorem 2 The wea diagoal approximatio ad the Wu-Woodroofe approximatio are equivalet ad both are implied by the strog diagoal approximatio Now, we shall deal with adapted processes, ie f will be supposed F 0 measurable Theorem 22 There exists a statioary liear process f T i i such that f has a martigale approximatio but o strog diagoal approximatio More precisely, for ay sequece of d = o there exists a sequece such that d, = o ad for f = i= P i f, S f f 0
260 Jaa Klicarová ad Dalibor Volý The Propositio shows that i the Zhao-Woodroofe s coditio, the limit of averages caot be replaced by a limit of f + = limsup j j S j f f 3 Proofs 3 Proof of Theorem 2 Let us prove the first part of the theorem At first we prove that the wea diagoal approximatio implies the Wu-Woodroofe approximatio We have Q 0 S f = 0 P i U j f i= i= + 0 P i U j f + i= + j=i+ + = + 2 + 3 0 i= + P i U j f P i U j f + i+ 0 2 3 ad + 3 S f f 2 Ug = g Therefore, uder the assumptios of the wea diagoal approximatio, The proof for R S f is similar Q 0 S f = oσ Now, we show that the Wu-Woodroofe approximatio implies the wea diagoal approximatio We ca suppose that f is adapted to the filtratio ad regular, ie f = P if; the o adapted case ca be treated i the same maer cf Volý, 2006 ad the o regular part of f does ot affect the limit behaviour We have σ 2 = S f 2 2 = h where h is a slowly varyig fuctio i the sese of Karamata cf Kalleberg, 997 ad ES f F 0 2 = oσ Recall that the Wu-Woodroofe coditio implies that there is a martigale differece array U i m such that S f m 2 = oσ ; we the have m 2 = σ / + oσ /
Diagoal approximatios by martigales 26 Note that S f m 2 2 = ES f F 0 2 2 + U j m P j = ES f F 0 2 2 + i=j j m P 0 U i f 2 2 U i f 2 2 = by the assumptios, f is adapted hece P j U i f = 0 for j > i We thus have that for ay ǫ > 0 ad big eough, for all but ǫ j {0,,}, m P 0 j Ui f 2 2 < ǫh Let ǫ be oe of such j We the have P 0 U i f P 0 U i f 2 2 < 2ǫh hece for m = P 0 U i f, i=j S f m 2 2 = oσ2 Deote f = P i f To show that S f f 2 2 = oσ 2 it suffices to prove ES f F 0 2 = oσ By the assumptios U j m P j i=j U i f 2 = oσ hece recall that P j U j m = U j m so that j= + j=0 j m P 0 U i f 2 = oσ j m P 0 U i f 2 = oσ Because Uj m 2 2 σ2 = oσ2, we thus get j=0 j P 0 U i f 2 = oσ hece f = m + g Ug where g 2 = oσ ad S f f 2 = oσ So, the f has a wea diagoal approximatio t remais to prove the secod part of the theorem This part follows immediately from the defiitios of the strog diagoal approximatio ad the wea diagoal approximatio
262 Jaa Klicarová ad Dalibor Volý 32 Proof of Theorem 22 32 Costructio of the process Let e H 0 H ad e = U i e i is thus a martigale differece sequece For i, j N N deotes the set of all positive itegers we deote K j = exp j 2 j N j = 2 K i +, b j = j 2 ad put K 0, N 0 = 0, b 0 = We defie fuctios f i, f i for i N: f i := K i = f i := f i T Ni Ki K i e T + =0 + e T, ad put where f 0 = b 0 e f = + i= b i fi + f 0, 3 322 Mai idea of the proof We will prove that Q0Sf S f 0 Remar that this is a ecessary ad sufficiet coditio for the Wu-Woodroofe approximatio cf Wu ad Woodroofe, 2004 The we will deduce that there is a martigale approximatio the secod part of the proof we will show that for ay sequece of d = o there exists a sequece such that d, = o ad for f = i= P i f, S f f 0 the proof, we shall eed both upper ad lower estimatios of the term S f, a upper estimatio of Q 0 S f ad a lower estimatio of the term S f f We have S f S f Q 0 S f, S f S f Q 0 S f + Q 0 S f, hece for the estimatio of the term S f it suffices to estimate the terms Q 0 S f ad S f Q 0 S f
Diagoal approximatios by martigales 263 Figure 32 The costructio of the process f f T 2 f T f f T 2 f T f T f T f 0 P j P j f 0 N N 2 + N 2 f 2 f 0 b K b K b K b K b2 K 2 b2 K 2 b2 K 2 2 323 Projectios P j S f Sice Q 0 S f = 0 j= P js f ad S f Q 0 S f = + P js f, i order to estimate the terms Q 0 S f ad S f Q 0 S f we eed to ow estimatios of P j S f From the costructio 3 of the fuctio f we have P j S f = P j + S b i fi + f 0 32 Let us study the fuctios f i From the costructio of f i we ca see that i= ad t follows that f i H Ni H Ni 2K i f i T H Ni+ H Ni 2K i+ S f i H Ni+ H Ni 2K i+ 33 Now, let us fix i ad study P j S b i fi We ca write P j S b i fi = y,j e j The part y,j o the right had side of the equatio depeds i fact o i, ad j Let us split all possibilities ito the followig three cases Recall that N i+ = N i + 2K i
264 Jaa Klicarová ad Dalibor Volý The case of > N i+ f we suppose N i+, the the term y,j is as follows: j : j > N i or j N i+ + such case cf 33 y,j = 0 2 j : N i K i + j N i K i y,j = b i =j +N i+k i 3 j : N i 2K i + 2 j N i K i y,j = b i K i =+2 N i K i j = 4 j : N i + j N i 2K i + t is easily see from the costructio of the fuctio f i that Ki P j S b i fi = b i K i + = 0, =N i+k i+j = so y,j = 0 5 j : N i K i + 2 j N i Alie as i 3 K i y,j = b i 6 j : N i 2K i + 2 j N i K i + t is similar case to 2, we have K i y,j = b i = N i K i j+2 The case of N i Remar that P j S f = 0 for j > sice f is adapted So, if have the case of < N i, the we eed to express the projectio P j S b i fi for egative j oly, especially for j [ N, N 2K ] Now, let us have N i Sice S f i H Ni+ H Ni 2K i+, we are iterested oly i projectios P j for egative j The expressio of the term y,j is as follows 2 j : j > N i or j N i+ + such case cf 33 y,j = 0 22 j : N i + j N i K i y,j = b i =j +N i+k i 23 j : N i K i + j N i y,j = b i K i+j+n i =j +N i+k i 24 j : + 2 Ni K i + j N i K i y,j = b i K i+j+n i = j+ N i K i+
Diagoal approximatios by martigales 265 Figure 33 The case of 2K i f f T 2 f T f T N i 2 N i K i 0 N i+ + N i 3 4 5 N i K i 6 N i+ + 25 j : N i K i + j 2 Ni K i K i j N i+ y,j = b i =j+n i+k i+ 26 j : N i 2K i + j N i K i K i j N i+ y,j = b i = j N i K i+ 27 j : N i 2K i + j N i 2K i y,j = b i K i = j N i K i+ Observe that if is eve, the the term P Ni K i++ 2 S f i is equal to zero The case of N i < < N i+ We ca split this case ito the followig three parts f 2K i, the we ca apply formulas 6 the case of K i, the formulas 2 27 ca be used Whe is such that K i < < 2K i the we ca use the formulas 2 25 ad 27, where the upper boud i 25 is N i K i such a case the set of j i the formula 26 is empty
266 Jaa Klicarová ad Dalibor Volý Figure 34 The case of N i f f T 0 2 N i 22 N i N i K i 23 24 25 N i K i N i+ + 26 27 N i+ + 2 Geeral expressio for positive j As we saw above, the projectio for fixed i P j S b i fi depeds o, j Let us deote P j S b i fi = y,j e j For geeral j, N, we ca thus express the term y,j i,j is such i that N i+ + 2 j N i ; recall that i,j is the oly i for which the projectio
Diagoal approximatios by martigales 267 P j S f i is ozero as y,j = b i,j K i,j =j +N i,j +K i,j = b i,j for j : N i,j K i,j + j N i,j K i,j =+2 N i,j K i,j j 34 = 0 i other case for j : N i,j+ + 2 j N i,j K i,j From the previous calculatios we ca deduce that still for positive j P j S f 2 = P j S f 0 + b i,j fi,j 2 = + y,j 2 35 where i,j is such that j belogs to the iterval [ N i,j+ +2, N i,j ] ad y,j is as above Remar that y,j [, 0] The upper estimatio of the term S f Q 0 S f Now, let us estimate the term S f Q 0 S f A upper estimatio of this term ca, usig the previous remars, be easily see the orm of each term P j S f is less or equal to oe, cf 35 to be S f Q 0 S f 2 = P j S f 2 36 The lower estimatio of the term S f Q 0 S f Sice cf 35 P j S f 2 = P j S f 0 + b i,j fi,j 2, for the lower estimatio of S f Q 0 S f 2 we eed to estimate P j S f 0 + b i,j fi,j 2 for 0 j Let us fix N The, there exists a iteger such that N < N + Usig 35 we have S f Q 0 S f 2 = P j S f 2 = P j S f 0 + b i,j fi,j 2 We ca write the term o the right had side as P j S f 2 = N i j=max, N i++ P j S f 0 + b i fi 2 Recall that for positive j, the projectios P j S b i fi are equal to zero for every i except oe of them i,j We get S f Q 0 S f 2 = A i
268 Jaa Klicarová ad Dalibor Volý where for i A i = ad N i j= N i++ P j S f 2 = A = N N i j= N i 2K i+ P j S f 0 + b f 2 Remar that all fuctios f i,j cotaied i A i are the same P j S f 0 + b i,j fi,j 2 Let us study the case of i, usig 3,4 recall that N i+ = N i +2K i : A i = Sice = = 2 N i j= N i 2K i+ N i j= N i 2K i+2 N i j= N i K i+2 P j S f 0 + b i,j fi,j 2 P j S f 0 + b i,j fi,j 2 + P Ni 2K i+s f 0 2 + y,j 2 + + y, Ni K i+ 2 + we have A i 2 N i j= N i K i+2 + y,j 2 + 2y,j + 2y,j + + 2y, Ni K i+ + Usig the expressio of y,j as a sum of the fractios, see 34, puttig l := j + N i +K i, we obtai recall that m =l log m l ad m =l log m log xdx: l K i A i 2 2 K i i 2 l=2 =l 2 K i 2 K i i 2 + 2i K i 2 = l=2 log K i l + + 2 2 i 2 + logk i = 2K i 2 K i 2K i 2 i logk i 2 logl + 2K i 2 i 2 2K i logk i 2[xlog x x] Ki + l=2 = 2K i 2 i 2 2K i 2logK i logk i + logk i + 2K i 2 = 2K i 2 K i 2K i 2 i 2 log K i + + logk i Usig log x x = log + x x for x >, we have
Diagoal approximatios by martigales 269 A i 2K i 2 2K i 2 i 2 K i + + logk i = 2K i 2 i 2 2K i + logk i = 2K i 4K i + 2 log K i 2 i 2 We have derived recall that K i = exp i 2 Deote C = i= A i We have C = A i = i= A i 2K i + O i= Ki log K i 2K i 4K i + 2 log K i 2 i 2 N N + O i= K 4K i + 2 log K i 2 log K Now, for fiishig of the estimatio of the term S f Q 0 S f, we eed to estimate the term A The estimatio ca be divided ito followig three parts Let = N + K The, cf 34 P j S f 2 = P j S f 0 + b f 2, for j [, N ] Hece, i this case, we estimate the terms P j S f 0 + b f 2 usig the same methods as above ad we obtai usig N K + = ad N = K A = K P j S f 0 + b f 2 K 2K + log K 2 For S f Q 0 S f 2 we have deduced that recall that i such a case K ad log K 2 S f Q 0 S f 2 C + K 2K + log K 2 K + O 2K + log K log K 2 = + o 2 Let N < < N + K We eed to estimate the term N A 2 = P j S f 0 + b f 2 i 2
270 Jaa Klicarová ad Dalibor Volý Usig similar argumets as before N < K we get A 2 N 2 2 N 2 2 K K j=n +K + =j K j=n +K + log K j N 2 2 N logk [xlog x x] K K +N = N 2 K 2 N log K +N K K log K +N + N N 2 2 N K +log N 3 2 N We thus have got K K +N + N log K K + N A 2 N 3 2 N So, for S f Q 0 S f 2, we have S f Q 0 S f 2 C + N 3 2 N K + O 3 log K 2 N = + o 3 Let N +K < < N +2K such a case we eed to estimate the term N A 3 = P j S f 0 + b f 2 Sice N > K we divide the term A 3 ito two parts as follows N P j S f 0 + b f 2 = N j= N K + P j S f 0 + b f 2 + N K + P j S f 0 + b f 2 where we ow the estimatio of the first term o the right-had side t is the same estimatio as for the term A So, let us deote by A 32 the secod term o the right-had side This term ca be expressed as
Diagoal approximatios by martigales 27 A 32 N K N K A 32 = 2 2 2 N K 2 2 K =j+ N K 2 2 = N K 2 2 N K K =j+ log K j 2 N K logk [xlog x x] N K N K log K N K + Now, let us tae a costat c such that 0 < c < ad N K = ck ; we thus have A 32 ck 2 ck 2 log K + ck ck 2 2 ck log c So, for the term A 3 A 3 c + K 2 log K we have derived : 2 = c + K 2 2 2 ck ck log c + K + 2K c logc 2 Usig the iequality log c c, we get A 3 c + K 4K 2 We have obtaied for the term S f Q 0 S f 2 recall that C = N + O K log K ad = N + K c + : S f Q 0 S f 2 C + K c + 4 2 K K + O 4 log K 2 K = + o We thus have deduced recall the upper estimate that for each N, S f Q 0 S f 2 = + o 37
272 Jaa Klicarová ad Dalibor Volý Estimatio of the term Q 0 S f Now, we will study the term Q 0 S f Recall that 0 0 Q 0 S f 2 = P j S f 2 + 2 = P j S f 0 + b i fi j= j= Usig 35 ad a iequality x y 2 x 2 + y 2 for x, y 0 We thus get Q 0 S f 2 where i = 0 j= P js b i fi 2 The estimatio of the terms i ca be divided ito three parts t depeds o the relatio betwee i ad Let i be such that i we still suppose : N < N + ; this meas that N i + 2K i such a case we ca use 5,6 some egative j are also i the part 4 but the projectios i 4 are equal to zero ad puttig l := j + N i + K i i 5 ad l := j + N i + K i 2 i 6 we obtai l= =l + i, i= K i i K i 2 2 i 2 38 i 2 K i i 4 + log K i 2 + log 2 K i j j=2 2 K i i 4 + 2 logk i + log 2 K i + K i log 2 K i log 2 j 2 i 4 + 2 logk i + K i log 2 K i [xlog 2 x 2xlog x + 2x] Ki 2 i 4 2 + 2K i + log K i 2K i 5K i i 2 The sum of such i ca be estimated by recall that K j = exp j 2 i= 5K i i 2 5K 2 2 + i= 5K i i 2 6K 2 6 2 39 2 Let i +, ie N i The, usig expressios 2 27 we get i = N i K i j= N i 2K i P j S b i fi 2 + + 0 j= N i+ P j S b i fi 2 N i j= N i K i+ P j S b i fi 2 +
Diagoal approximatios by martigales 273 hece i 3 K i i 4 j+ =j 2 Usig the iequality i= a i 2 i= a2 i we get K i j+ =j 2 K i j+ =j 2 K i j j + j=2 Ki+ = j + j 3 + log j=k i + 2 2 hece i 3 3 + log i4 Now, we ca sum the terms i ad usig the iequality log i 2 we get + i=+ i + i=+ 3 + i 4 3 + log 3 i=+ 3 + i 2 i 4 4 i = b 2 i +b 2 i +b 2 i +b 2 i K i =K i +j K i 2b 2 i j= + 2 = j+2 K i 2 + b 2 i j+ =K i j+ 2 j=0 j+ =j+ j+ =j+ 2 2 + b 2 i 2 2 + 2b 2 i 2 j=0 K i K i j+ =j+ +j =j+ 2 K i j = +K i+ j 2, 2
274 Jaa Klicarová ad Dalibor Volý i 2 K i i 4 4 K i i 4 j+ =j+ j+ =j+ 2 + 2i4 2 2 j=0 j+ =j+ 2 Usig the iequality i= a i 2 i= a2 i we get i 4 i 4 4 i 4 4 i 4 K i j+ =j+ K i j + + j j 4 + log i4 3 t remais the case of i = Accordig to paragraph The case of : N i < < N i+ i the subsectio Projectios P j S f, this estimatio ca be split ito three parts the case of 2K we have For K we have 5K 2 33 + log 4 For K < N + 2K we have, similarly as i the previous case, K Q 0 S b f 2 3 K 2 2 =j 38 we get Q 0 S b f 2 0K 2
Diagoal approximatios by martigales 275 t remais to estimate Q 0 S f 2 for N < < K We have Q 0 S b f 2 + + N 2K + P j S b f 2 + j= N 2K 0 P j S b f 2 + j= N + N j= N K + 4 3 P j S b f 2 K =K j 2 + 2 N j= N + N K j= N 2K + K P j S b f 2 + P j S b f 2 + K j+ =K j +2 2 which, lie i 38, ca be estimated by For all it is thus 25K 2 Q 0 S f 2 25 Wu Woodroofe approximatio of the fuctio f Usig the estimatios for terms S f Q 0 S f ad Q 0 S f we obtai Q 0 S f 2 S f 2 25 + o O Sice + recall that log + as +, we have proved Wu-Woodrofe approximatio the Wu-Woodroofe approximatio, we ca tae D,0 = i P 0U i f + i= i P 0U i f the case of a statioary liear process P 0 U i f is a multiple of e hece we get D,0 = c e From S f 2 / cost cf 37 it follows that the c coverge to a costat c ad we get a martigale approximatio with m = ce Strog diagoal approximatio of the fuctio f Now, we show that the fuctio f does ot satisfy the strog diagoal approximatio Tae a sequece d such that d = o The, there exists a sequece of j = l j N j + K j, where N j + K j d j, N j + K j = o ad l j teds to ifiity Defie h j = 0 i= N j K j P i f The S j h j 2 = o j As we proved above S j f 2 = j + o j, hece S j f h j 2 = j o j This shows that for arbitrary d = o the coditio i of Defiitio 3 is ot satisfied
276 Jaa Klicarová ad Dalibor Volý Refereces J Dedecer, F Merlevède ad D Volý O the wea ivariace priciple for oadapted sequeces uder projective criteria J Theoret Probab 20 4, 97 004 2007 MR2359065 M Gordi A cetral limit theorem for statioary processes Soviet Math Dol 0, 74 76 969 MR025785 P Hall ad CC Heyde Martigale limit theory ad its applicatio Academic Press, New Yor 980 J Haa, E Cetral limit theorem for time series regressio Z Wahrscheilicheitstheorie verw Geb 26, 57 70 973 MR033683 O Kalleberg Foudatios of Moder Probability Spriger, New Yor 997 MR464694 J Klicarová ad D Volý O the exactess of the Wu-Woodroofe approximatio Stochastic Proccesses ad their Applicatios 9, 258 265 2009 MR253087 D Volý Approximatig martigales ad the cetral limit theorem for strictly statioary processes Stochastic Processes ad their Applicatios 44, 4 74 993 MR98662 D Volý Martigale approximatios of o adapted stochastic processes with oliear growth of variace Pe Bertail, P Douha ad P Soulier, editors, Depedece i Probability ad Statistics, volume 87 of Lecture otes i Statistics, pages 4 56 Spriger 2006 MR2283254 WB Wu ad M Woodroofe Martigale approximatios for sums of statioary processes The Aals of Probability 32 2, 674 690 2004 O Zhao ad M Woodroofe O martigale approximatios The Aals of Applied Probability 8 5, 83 847 2008 MR2462550