POBABILITY AD MATEMATICAL STATISTICS Vol., Fasc., pp. SELFSIMILA POCESSES WIT STATIOAY ICEMETS I TE SECOD WIEE CAOS BY M. M A E J I M A YOKOAMA AD C. A. T U D O LILLE Absrac. We sudy selfsmlar processes wh saonary ncremens n he second Wener chaos. We show ha, n conras wh he frs Wener chaos whch conans only one such process he fraconal Brownan moon, here s an nfny of selfsmlar processes wh saonary ncremens lvng n he Wener chaos of order wo. We prove some lm heorems whch provde a mechansm o consruc such processes. AMS Mahemacs Subec Classfcaon: Prmary: 6F5. ; Secondary: 65, 9G7. Key words and phrases: Selfsmlar processes, saonary ncremens, second Wener chaos, lm heorems, mulple sochasc negrals, weak convergence.. Inroducon The selfsmlar processes wh saonary ncremens have been wdely suded. Le >. A sochasc process Y = Y s -selfsmlar f for any c > he processes Y c and c Y have he same fne-dmensonal dsrbuons. ere s called he selfsmlary parameer of Y. The process Assocae member of SAMM, Unversé de Pars Panhéon-Sorbonne. The auhor acknowledges suppor from Japan Socey for he promoon of Scence whch made possble a research say a Keo Unversy. Parally suppored by he A gran Masere BLA.
M. Maema and C. A. Tudor Y has saonary ncremens f Y and Y +h Y h have he same fne dmensonal dsrbuons for every h >. Le, ]. All -selfsmlar processes wh saonary ncremens and wh fne varances have he same covarance funcon gven by, s = C + s s, for all s,, where C s he second momen of he process a me one. We refer o he monographs [4] and [] for a complee exposon on selfsmlar processes. Snce he Gaussan processes are characerzed by her covarance, here s only one Gaussan selfsmlar process wh saonary ncremens and wh un varance a me one. Ths s he fraconal Brownan moon. The Gaussan processes lve n he frs Wener chaos, ha s, hey can bascally be expressed as sngle negrals, wh a deermnsc negrand, wh respec o he Wener process. The purpose of hs paper s o dscuss selfsmlar processes n he second Wener chaos. The elemens of he second Wener chaos are double eraed sochasc negrals wh respec o he Wener process. The law of such processes s no gven anymore by her covarance funcon, herefore he fac ha wo selfsmlar processes wh saonary ncremens n he second Wener chaos have he same covarance does no mply he equvalence of fne dmensonal dsrbuon of hese processes. I s hen expeced o have more han one selfsmlar process n he second Wener chaos. We wll acually show ha here exss an nfny of such processes. Ths paper s organzed as follows. In Secon we sudy he so-called nonsymmerc osenbla process, whch depends on wo parameers, and by suably choosng hese parameers, we oban an nfny of selfsmlar processes wh saonary ncremens n he second Wener chaos. The analyss of he laws of hese processes are based on he cumulans and hs s done n Secon 3. Secons 4 and 5 conan he proofs of some non-cenral lm heorems n whch selfsmlar pro-
Ss-s processes n he second Wener chaos 3 cesses wh saonary ncremens appear as lms. Our resuls exend hose from [], [3] or []. We fnsh our paper wh some houghs abou how many selfsmlar processes wh saonary ncremens are n he second Wener chaos and how hey can be obaned as lms n non-cenral-ype lm heorems.. A class of selfsmlar processes wh saonary ncremens n he second Wener chaos The purpose of hs secon s o dscuss a parcular class of selfsmlar processes wh saonary ncremens lvng n he second Wener chaos. Ths class conans an nfne number of elemens and all of hem have dfferen fne dmensonal dsrbuons. We nroduce our se as follows. Le,, such ha. + >. Consder he sochasc process Y, =. Y, = c, Y, gven by, for every, u y + u y + du db y db y, where he negral above s a mulple Wener-Iô sochasc negral of order wo. We refer o [8] for he defnon and he basc properes of mulple Wener-Iô negrals. Le us denoe by f he kernel of Y,.3 f y, y = c,, ha s, u y + u y + du for every y, y. The kernel f s n general no symmerc wh respec o he varables y, y excep he case =. We denoe by f s symmerzaon f y, y = f y, y + f y, y. In hs way, usng he usual noaon for mulple negrals, we can wre Y, = I f for every. The condon. assures ha he kernel f belongs o
4 M. Maema and C. A. Tudor L [, for every hs can be seen n he sequel of hs secon and hus he double negral n. s well-defned. The consan c, wll be chosen such ha E [ Y ] =. Ths consan plays acually an mporan role n our paper. I wll be explcly calculaed laer. POPOSITIO.. Assume.. Then he process Y, s + selfsmlar and has saonary ncremens. Proof: Le c >. We have Y, c = c, = c, c = c, c d = c + Y c u y + u y + du db y db y cu y + cu y + du db y db y cu cy + cu cy + du db cy db cy where we have used he - selfsmlary of he Wener process B. ere d = means equvalence of all fne dmensonal dsrbuons. I s obvous ha he process Y, has saonary ncremens snce for every h > and we have Y, +h Y, d= h Y,. EMAK.. The parcular case = = corresponds o he osenbla process as defned n [3], []. In our paper, we wll call hs process n our paper as he symmerc osenbla process. The process Y, wh wll be called non-symmerc osenbla process. Also noe ha he selfsmlar parameer of Y, s always conaned n he nerval,. v We wll need he followng lemma hroughou he paper. LEMMA.. Le v < u and,,. Then u y v y dy = +, u v +, where a, b = xa x b dx denoes he bea funcon wh parameers
a, b >. Therefore, for every u, v > u v = Ss-s processes n he second Wener chaos 5 u y v y dy +, + u v + + +, Proof: Ths follows by makng he change of varables z = v y u y wh dy = v u z dz and from he fac ha x + = x. EMAK.. Usng he well-known properes of he bea and gamma funcons recall ha Γa = x a e x dx for a >,.e., a, b = ΓaΓb Γa + b we can gve a varan of he above lemma: u v = Γ u y v y dy + = Γ + Γ Γ and ΓaΓ a = Γ Γ + Γ π snπa u v a,, where a, = snπ / f v < u and a, = snπ / f u < v. Or oherwse u v = Γ u y v y dy + Γ Γ + u v. Γ u v u v snπ / + snπ /. We wll now compue he renormalzng consan appearng n.. LEMMA.. Assume ha,, and ha. s sasfed. The normalzng consan c, appearng n he defnon of Y, n. s gven by c, = where = +.,, +,,
6 M. Maema and C. A. Tudor Proof: Snce Y, = I f = I f for every wh f gven by.3, we have from he somery propery of mulple sochasc negrals see [8] [ ] E Y, = f L = f y, y dy dy = c, u y + u y + du + u y + u y + du v y + v y + dv + v y + v y + dv dy dy = [ c, u y + u y + du v y + v y + dv + u y + u y + du v y + v y + dv + u y + u y + du v y + v y + dv + u y ] + u y + du v y + v y + dv dy dy and, by nerchangng he order of negraon and nocng ha he frs and hrd summands, and he second and fourh, concde, we oban [ ] c, E Y, [ u v u v = dy u y v y u v u v + dy u y v y Observe ha he funcon nsde he negral dudv s symmerc. Therefore [ ] c, E Y, [ v = u du v + v dy u y v y dy u y v y v dy u y v y dy u y v y ] dudv. dy u y v y ] dy u y v y dv.
We oban, usng Lemma., [ ] c, E Y, = =, Ss-s processes n he second Wener chaos 7 du,, u + u v + dv, + + +. If + =, hen [ ] c, E Y, = whch mples, +, +,, +,, +, +,,.4 c, = +,,,,. EMAK.3. In he parcular case = = we have c, := c = and concdes wh he consan used n e.g. []., 3. Cumulans of he non-symmerc osenbla process We wll prove n hs secon ha he processes Y, gven by. have dfferen laws upon he values of he selfsmlar parameers and. We wll
8 M. Maema and C. A. Tudor use he concep of cumulan. The cumulans of a random varable X havng all momens appear as he coeffcens n he Maclaurn seres of g = log Ee X,. The frs cumulan c s he expecaon of X whle he second one s he varance of X. Generally, he nh cumulan s gven by g n. The key fac s ha for random varables n he second Wener chaos he cumulans characerzes he law. Le us consder a mulple negral I f wh f L symmerc. Then he mh cumulan of he random varable I f are gven by see [7] or [5] 3. c m I f = m m! fy, y fy, y 3...fy m, y m fy m, y dy...dy m m For he follwng resul we refer o [5]: EMAK 3.. I s known ha he law of a mulple negral of order wo s compleely deermned by s cumulans n he sense ha, f wo mulple negrals of order have he same cumulans, hen her dsrbuons are he same. We can sae he man resul of hs secon. POPOSITIO 3.. Le us consder he process Y, gven by.. There exs couples,,,, wh + = + = > such ha,, and for any >, he laws of he random varables Y, and Y, are dfferen. Proof: I suffces o show ha for fxed he wo random varables Y, and Y, have a leas one dfferen cumulan. The frs wo cumulans ha s, he expecaon and he varance of hese wo random varables are he same snce Y, s an -selfsmlar process wh saonary ncremens. Le us compue he hrd cumulan. Le us consder he case m = 3. Then, usng 3., he expresson of f and changng he order of negraon, we ge c 3 I f
=: c, 3 Ss-s processes n he second Wener chaos 9 [g, u, u, u 3 + g, u 3, u, u + f, u, u, u 3 +f, u, u 3, u + f, u, u, u 3 + f, u, u 3, u +f, u 3, u, u + f, u 3, u, u ] du du du 3, where we have denoed by g, u, u, u 3 = and f, u, u, u 3 = u y + u 3 y + dy + u y + dy + u 3 y + dy u y u y u y + u 3 y + dy + u y + dy + u 3 y + dy. u y u y Therefore, he funcon under he negral du du du 3 s symmerc wh respec o he varables u, u, u 3. The negral du du du 3 s hen equal o 3! du du 3 du 3. u 3 <u <u,u,u,u 3 [,] Also, from Lemma holds ha, for u 3 < u 3 < u g, u, u, u 3 = +, u u 3 + +, u u + +, u u 3 + and f, u, u, u 3 =, u u 3 +
M. Maema and C. A. Tudor Thus we have, +, u u + u u 3 +. c 3 I f = 3!c, 3 [ +, +,, + u 3 <u <u,u,u,u 3 [,] = 3!c, 3 +,, u 3 <u <u,u,u,u 3 [,] = 3!c, 3,, + u 3 <u <u,u,u,u 3 [,] +, + +,, +, + +, u u 3 + u u + u u 3 + du du du 3 + +, +, +, +, u u 3 + u u + u u 3 + du du du 3,, Furher, usng gamma negrals we ge +,, u u 3 u u u u 3 du du du 3. ]
Ss-s processes n he second Wener chaos c 3 I f = 3!c, 3 Γ Γ Γ Γ Γ Γ Γ + Γ Γ Γ Γ + Γ u 3 <u <u,u,u,u 3 [,] u u 3 u u u u 3 du du du 3. I s obvous, gven he expresson of he normalzng consan c,, ha here exs,, wh c 3I f, c 3 I f,. For example hs happens when = =.4 and =.3, expresson of he gamma funcon can be compued numercally. =.5 because hen he EXAMPLE 3.. There are oher classes of selfsmlar process wh saonary ncremens. For hs example we refer o [9] and [6]. Consder α, such ha < α < 3 4 and < α <. Defne for every X = u y α + u y α + u u du db y db y. The process X = X defned above s -selfsmlar wh saonary ncremens where = α. The proof s mmedae and follows he lnes of Proposon.. I can also be proved ha for suable choces of α,, he law of he process X defned above s dfferen from he law of he process Y gven by.. We wll come back o hs process X defned above n he las secon. 4. Lm heorem for non-symmerc osenbla process Le B, B be wo fraconal Brownan moons wh urs parameers, respecvely. We wll assume ha he selfsmlar parameers and are boh bgger han. We wll also assume ha he wo fraconal Brownan moons can be expressed as Wener negrals wh respec o he same Wener process B.
M. Maema and C. A. Tudor Ths mples ha B and B are no ndependen. We have 4. B = c db y u y 3 + du, B = c db y u y 3 + du where he consans c, c are such ha E [ ] [ ] B = E B =. Acually, by applyng Lemma. wh, replaced by, respecvely, we ge 4. c = and an analogous expresson for c., Defne, for every, he sequence 4.3 V = [] = [ E B + B + B B B + B + B B ]. I s well-known ha, n he case = = 3 4,, he renormalzed sequence V converges, as, n he sense of fne dmensonal dsrbuon, o a symmerc osenbla process wh selfsmlar parameer. Our am s o exend hs resul o he suaon when. We wll acually prove ha, afer suable normalzaon, he sequence 4.3 converges n he sense of fne dmensonal dsrbuons o he process Y, n.. Frs, we need o undersand he correlaons srucure of he fraconal Brownan moons B and B. where LEMMA 4.. Le > s. Then [ ] E B B s B B s = b, s b, = c c, +,
Ss-s processes n he second Wener chaos 3 where c, c are gven by 4. and = +. Proof: Snce B B s = c db y u y 3 + du we oban, usng he somery of Wener negrals and Lemma., [ E B B s = c c = c c B s s ] B s dudv s u du +c c = c c s u v s u y 3 + v y 3 + dy, u v dv v dv, v u du s s, +, s.. EMAK 4.. The above consan b, s equal o one f = = The followng resul consues an exenson o he non-symmerc case of he non-cenral lm heorem proved n [], [3], []. TEOEM 4.. Le V be gven by 4.3 and assume ha + = > 3. Then, when, c, c c b, V converges n L Ω o he random varable Y, gven by.. Proof: Usng he produc formula for mulple negrals see [8], Proposon.., we can express V as V = b, c c + + = db y db y u y 3 + v y 3 + dudv.
4 M. Maema and C. A. Tudor I suffces o show ha he sequence = + db y db y + u y 3 + v y 3 + dudv converges n L Ω, as, o db y db y u y 3 + u y 3 + du or equvalenly, by he somery formula for mulple negrals, ha he sequence a y, y = = + + converges n L as o he funcon ay, y = u y 3 + v y 3 + dudv u y 3 + u y 3 + du whch represens he kernel of he non-symmerc osenbla process. Le us esmae he L norm of he dfference a a. We have a a L = a L a, a L + a L. We compue separaely he hree quanes above. Frs, + + a L = =,= + +,,= + u y 3 + v y 3 + dudv u y 3 + v y 3 + du dv dy dy +, + dudv + where we have used Fubn heorem and Lemma.. In he same way a, a L =,, u u v v du dv,
Ss-s processes n he second Wener chaos 5 = + + dudv u u v u du and a L =,, To summarze, we ge u v 4 4 dudv. a a L =,,= +,= = +, + du + + + + + dudv + dudv ] u v 4 4 dv + u u v v du dv u u v u du and usng he change of varables ũ = u and smlarly for he oher varables u, v, v we ge a a L =,, 4,= [ +, [ k Z u u + dudu v v + dvdv u u + v u + dudvdu ] u v + 4 4 dudv, 3 4 u u + k dudu v v + k dvdv
6 M. Maema and C. A. Tudor + u u + k v u + k dudvdu ] u v + k 4 4 dudv. As n [], proof of Proposon 3.. we can prove ha he sum over k Z s fne. Indeed, hs sum can be wren as where [ F x = + k 4 4 F k Z k u ux + dudu v vx + dvdv u u x + v u x + dudvdu ] u vx + 4 4 dudv. The concluson follows snce 3 4 < < we can see ha F x behaves as x for x close o zero. EMAK 4.. The condon + > 3 s naural snce exends he classcal condon > 3 4 necessary o oban non-gaussan lm f V n he symmerc case. Followng exacly he lnes of he above proof, he followng corollary s mmedae. COOLLAY 4.. Le V be as n 4.3 and assume = + > 3. The sequence of sochasc processes c, c c b, V converges n he sense of fne dmensonal dsrbuons as o he sochasc process Y,.
Ss-s processes n he second Wener chaos 7 COOLLAY 4.. Consder B, B as before and assume = + > 3. Se, for every, S = = [] = [] = { [ ]} B + B B + B E B + B B + B B + B B + B b,. Then he sequence of sochasc processes converges as o dsrbuons. c, c c S Y, Proof: Snce for every we have S = I g wh g y, y = + + n he sense of fne dmensonal u y 3 + u y 3 + dudv, y, y by usng he change of varable as n he proof of Proposon., can be seen ha V has he same law as b, S. 5. Generalzaon and houghs: how many selfsmlar processes wh saonary ncremens are n he second Wener chaos? I s well-known ha n he case = = he resul n Corollary 4. s sll rue f B + B s he erme polynomal wh degree wo, see below he defnon s replaced by h B + B where h s funcon wh erme rank equal o wo. We propose here a more general verson of Corollary 4. n he non-symmerc case. Le us defne, for every [] 5. W = [ where c = E B + B = g [ ] B + B g B + B c ] B + B and where g s a deermnsc funcon wh erme rank equal o one whch has a fne expanson no erme
8 M. Maema and C. A. Tudor polynomals of he form 5. gx = M c q q x where M, c and n denoes he nh erme polynomal n x = n x d n exp n! dx n exp x, x. q= TEOEM 5.. Consder wo fraconal Brownan moons B and B gven by 4. wh + = > 3. Le g : be a deermnsc funcon gven by 5. such ha for every q 5.3 q <. Then he sequence of sochasc processes W converges n he sense of fne dmensonal dsrbuons as o he process c c, c c b, Y, n.. EMAK 5.. Assumpon 5.3 excludes he exsence of erms wh q = n he expanson of g. Proof: Agan we assume =. We have, snce q I ϕ = q! I qϕ q see, e.g. [8], W = = B + B M c q q! I qf q,, c where f q,, y,..., y q = u y 3 +...u q y q 3 + du...du q. [,+] q Thus W = = I f,, M q= c q q! I qf q,, q=
= c = Ss-s processes n he second Wener chaos 9 I f,, I f,, + M q= = I f,, c q q! I qf q,,. From Theorem 4. holds ha he frs summand above converges n L Ω o he desred lm. Le us show ha he remanng erm := = I f,, I q f q,, converges o zero n L Ω for every q. By he produc formula for mulple negrals see [8], Proposon.., we can wre = = = = =:, +, I q+ f,, f q,, + q = I q f,, f q,, [I q+ f,, f q,, + c,, qi q f q,, ] here c,, q denoes a generc consan dependng on,, q ha may change from lne o lne, where we used he fac ha, by Lemma., f,, f q,, y,.., y q = dxf,, xf,, x I q f q,, = c,, q + + u v + dudv = c,, q. We frs rea he erm,. More precsely we show ha hs erm converges o zero n L Ω as. Snce for any square negrable funcon f f below a b means ha he sequences a and b have he same lm as E [, ] c,, q 4 + +,= + + u v dudv u v dudv q
M. Maema and C. A. Tudor c,, q 4,=; + q = c,, q 4 kk + q k= = c,, q 3 4 k= + c,, q 4 k + q k= k + q. The sequence 3 4 k= k + q converges o zero when. Indeed, when he seres k= k + q s convergen and he sequence 3 4 k= k + q converges o zero snce > 3 4. When he same seres s dvergen, behaves as + q + and he summand goes o zero because 3 4 + + q + = q <. The second summand can be reaed smlarly. We have Le us prove now ha he erm, converges o zero n L Ω as. E, = c,, q 4 c,, q 4,=,=; = c,, q 3 4 k= + c,, q 4 + + q k q k= k q + q u v dudv where we used agan he change of summaon = k and we noced ha he dagonal erm, whch behaves as 4 converges o zero. The fac ha q <
Ss-s processes n he second Wener chaos mples ha he seres k= k q s convergen and snce > 3 4 he sequence 3 4 k= k q goes o zero as. The second seres s bounded by he frs one snce k and hus converges o zero. In prncple, Theorem 5. can be exended o funcons g havng an nfne seres expanson no erme polynomals. Bu n hs case, W s gven by an nfne sum of mulple negrals and s much more dffcul o conrol he L norm of he res. EMAK 5.. a Le + = > 3 and, >. Corollary 4. shows ha ξ and ξ are wo saonary Gaussan sequences wh zero mean and un varance, and wh correlaon funcon r n c n, r n c n such ha [ ] E ξ ξ c, + hen [] k= { fξ k, ξ k E [ fξ k, ξ k ]} wh he funcon f gven by fx, y = xy = x y converges n he sense of fne dmensonal dsrbuon o, modulo a consan, a non-symmerc osenbla process wh urs parameers and. Theorem 5. shows ha he resul can be exended o funcon f of he form M fx, y = x c q q y wh suable assumpons on q, and. b Le us dscuss he selfsmlar process wh saonary ncremens from Example 3.. Ths process, denoed by X = X can be also obaned as a lm n a non-cenral lm heorem n he followng way see [9], page 7-3. Defne ξ k k Z a saonary Gaussan sequence wh zero mean and un varance and q=
M. Maema and C. A. Tudor wh covarance r k = E [ξ ξ k ] k α. Se X k = ξ k and U m = k Z a k X m k wh a k = f k =, a k = k f k > and a k = k f k <, >. Assume ha α, sasfy he assumpons from Example 3.. Then α m= U m converges o he process X from Example 3.. For he proof of hs fac, we refer o [9]. c Takng no accoun he wo pons above, we can fnd a mechansm o consruc more selfsmlar processes wh saonary ncremen n he second Wener chaos. For example, consder he sequence V gven by 4.3 and from consruc a lnear process as U m above wh suable wegh a k. I s expeced o fnd a new selfsmlar process wh saonary ncremens as a lm. eferences [] J.-C. Breon and I. ourdn 8: Error bounds on he non-normal approxmaon of erme power varaons of fraconal Brownan moon. Elecron. Commun. Probab. 3, 48-493. [] P. Breuer and P. Maor 983: Cenral lm heorems for nonlnear funconals of Gaussan felds. J. Mulvarae Analyss 3, 45-44. [3].L. Dobrushn and P. Maor 979: on-cenral lm heorems for non-lnear funconals of Gaussan felds. Z. Wahrsch. Verw. Gebee 5, 7-5. [4] P. Embrechs and M. Maema : Selfsmlar processes. Prnceon Unversy Press, Prnceon, ew York. [5]. Fox and M.S. Taqqu 987. Mulple sochasc negrals wh dependen negraors. J. Mulvarae Analyss, 5-7. [6] T. Mor and. Oodara 986: The law of he eraed logarhm for self-smlar processes represened by mulple Wener negrals. Probab. Th. el. Felds 7, 367-39. [7] I. ourdn and G. Pecca : Cumulans on Wener space. Journal of Funconal Analyss 58, 3775-379. [8] D. ualar 6: Mallavn calculus and relaed opcs, nd ed. Sprnger. [9] M. osenbla 979: Some lm heorems for paral sums of quadrac forms n saonary Gaussan varables. Z. Wahrsch. Verw. Gebee 49, 5-3.
Ss-s processes n he second Wener chaos 3 [] G. Samorodnsky and M. Taqqu 994: Sable on-gaussan random varables. Chapman and all, London. [] M.S. Taqqu 975: Weak convergence o he fraconal Brownan moon and o he osenbla process. Z. Wahrsch. Verw. Gebee 3, 87-3. [] C.A. Tudor 8: Analyss of he osenbla process. ESAIM Probably and Sascs, 3-57. Deparmen of Mahemacs Faculy of Scence and Technology Keo Unversy 3-4-, yosh, Kohoku-ku, Yokohama 3-85, Japan E-mal: maema@mah.keo.ac.p Laboraore Paul Panlevé U.F.. Mahémaques Unversé de Llle F-59655 Vlleneuve d Ascq, France E-mal: udor@mah.unv-llle.fr eceved on ; las revsed verson on xx.xx.xxxx