Diagonal approximations by martingales

Transcription

1 Alea 7, Diagoal approximatios by martigales Jaa Klicarová ad Dalibor Volý Faculty of Ecoomics, Uiversity of South Bohemia, Studetsa 3, , Cese Budejovice, Czech Republic address: LMRS, Uiversité de Roue, Aveue de l Uiversité F 7680 Sait Etiee du Rouvray, Frace 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, 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

2 258 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

3 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

4 260 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+ + = i= + P i U j f P i U j f + i 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 /

5 Diagoal approximatios by martigales 26 Note that S f m 2 2 = ES f F U j m P j = ES f F 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

6 262 Jaa Klicarová ad Dalibor Volý 32 Proof of Theorem 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, 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

7 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 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

8 264 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+

9 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 N i K i 6 N i 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

10 266 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 N i K i N i 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

11 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

12 268 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 y,j 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 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

13 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

14 270 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

15 Diagoal approximatios by martigales 27 A 32 N K N K A 32 = 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 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 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

16 272 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 = 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 logk i + log 2 K i + K i log 2 K i log 2 j 2 i logk i + K i log 2 K i [xlog 2 x 2xlog x + 2x] Ki 2 i K 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 i= 5K i i 2 6K 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 j= N i+ P j S b i fi 2 N i j= N i K i+ P j S b i fi 2 +

17 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 hece i log i4 Now, we ca sum the terms i ad usig the iequality log i 2 we get + i=+ i + i=+ 3 + i 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 b 2 i b 2 i 2 j=0 K i K i j+ =j+ +j =j+ 2 K i j = +K i+ j 2, 2

18 274 Jaa Klicarová ad Dalibor Volý i 2 K i i 4 4 K i i 4 j+ =j+ j+ =j i4 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 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

19 Diagoal approximatios by martigales 275 t remais to estimate Q 0 S f 2 for N < < K We have Q 0 S b f N 2K + P j S b f 2 + j= N 2K 0 P j S b f 2 + j= N + N j= N K P j S b f 2 K =K j 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 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

20 276 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, MR M Gordi A cetral limit theorem for statioary processes Soviet Math Dol 0, MR 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, MR O Kalleberg Foudatios of Moder Probability Spriger, New Yor 997 MR J Klicarová ad D Volý O the exactess of the Wu-Woodroofe approximatio Stochastic Proccesses ad their Applicatios 9, MR D Volý Approximatig martigales ad the cetral limit theorem for strictly statioary processes Stochastic Processes ad their Applicatios 44, 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 MR WB Wu ad M Woodroofe Martigale approximatios for sums of statioary processes The Aals of Probability 32 2, O Zhao ad M Woodroofe O martigale approximatios The Aals of Applied Probability 8 5, MR