Number vrince from probbilisic perspecive: infinie sysems of independen Brownin moions nd symmeric α-sble processes. Ben Hmbly nd Liz Jones Mhemicl Insiue, Universiy of Oxford. July 26 Absrc Some probbilisic specs of he number vrince sisic re invesiged. Infinie sysems of independen Brownin moions nd symmeric α-sble processes re used o consruc new exmples of processes which exhibi boh divergen nd suring number vrince behviour. We derive generl expression for he number vrince for he spil pricle configurions rising from hese sysems nd his enbles us o deduce vrious limiing disribuion resuls for he flucuions of he ssocied couning funcions. In priculr, knowledge of he number vrince llows us o inroduce nd chrcerize novel fmily of cenered, long memory Gussin processes. We obin frcionl Brownin moion s wek limi of hese consruced processes. 2 MSC Primry 6G52, 6G15; Secondry 6F17, 15A52. Key words nd phrses: Number vrince, symmeric α-sble processes, Gussin flucuions, funcionl limis, long memory, Gussin processes, frcionl Brownin moion. Inroducion Le X, F, P) be poin process on R, h is, collecion X := {x i ) i= : x i R, i nd #x i : x i [b, b + L]) < b R, L R + }, wih F he miniml σ-lgebr genered by hese poin configurions nd P some probbiliy mesure on X, F). The ssocied number vrince is defined s V rn[b, b + L]) := Vrince P [#x i : x i [b, b + L])]. More generlly, in order o del wih non-spilly homogeneous cses, i is more convenien o work wih he verged number vrince which we define s V [L] := E b [V rn[b, b + L])] E-Mil: hmbly@mhs.ox.c.uk E-Mil: jones@mhs.ox.c.uk. Reserch suppored by he EPSRC hrough he Docorl Trining Accoun scheme 1
king n pproprie uniform verge of he number vrince over differen inervls of he sme lengh. As we increse he lengh L of he underlying inervl he number of poins conined in h inervl will lso increse. However, in mny siuions, i is no immediely cler wh will hppen o he vrince of his number s he inervl lengh grows. One of he min quesions considered in his pper will be he behviour of V [L] s L. We shll see h, somewh couner-inuiively, in some insnces we hve lim V [L] = κ <, L in which cse we will sy h he number vrince sures o he level κ R +. Unil recenly, ineres in he number vrince sisic hd been confined o he fields of rndom mrix heory see e.g. [19, 12]) nd qunum heory e.g [3, 17]) where i is commonly used s n indicor of specrl rigidiy. In he sudy of qunum specr, he mnner in which he number vrince of eigenvlues) grows wih inervl lengh provides n indicion of wheher he underlying clssicl rjecories re inegrble, choic or mixed. In he lrge energy limi, he specrl sisics of qunum sysem wih srongly choic clssicl dynmics, should, ccording o he conjecure of Bohigs, Ginnoni nd Schmid [7], gree wih he corresponding rndom mrix sisics which re independen of he specific sysem. However, in reliy, for mny such sysems, when i comes o he long rnge globl sisics, his rndom mrix universliy breks down. In hese cses, he number vrince ypifies his rnsiion o non-conformiy in h following n iniil rndom mrix consisen logrihmic growh wih inervl lengh, he number vrince hen sures. Aemps o improve he undersnding of he deviions from rndom mrix predicions hve led o convincing explnions for number vrince surion behviour in erms of periodic orbi heory, see for exmple [4, 5, 1]. In [4] here is heurisic derivion of n explici formul for he empiricl number vrince of he zeros of he Riemnn ze funcion on he criicl line which is consisen wih numericl evidence. In he ls few yers, number vrince hs been considered from slighly differen viewpoin in relion o deerminnl poin processes see [24] for bckground on his opic). Resuls on he growh of he number vrince for hese processes re given in e.g. [24, 25]. The emphsis in hese cses is on scerining he divergence of he number vrince s his, i urns ou, is he key ingredien needed o prove Gussin flucuion resuls for he couning funcions of lrge clss of deerminnl processes, including hose rising nurlly in he conex of rndom mrix heory. Moived by he Riemnn ze exmple, Johnsson [14] recenly inroduced n exmple of poin process wih deerminnl srucure which demonsres he sme ype of number vrince surion behviour s conjecured by Berry [4]) for he Riemnn zeroes. This process is consruced from sysem of n noncolliding Brownin pricles sred from equidisn poins u j = Υ + n j) wih Υ R, R + j = 1,..., n. There re number of pproches o describing such sysem, see [13, 15] for deils. In ny cse, i cn be shown h he configurion of pricles in spce formed by he process fixed ime, is deerminnl process nd s such is correlion funcions or join 2
inensiies) ke he form R m n) x 1, x 2,..., x m ) = de K n) x i, x j ) ) m i,j=1..1) Here nd for deerminnl processes in generl, he deerminn my be inerpreed s he densiy of he probbiliy of finding m of he poins in he infiniesiml inervls round x 1, x 2,..., x m. As he number of pricles n, Johnsson shows h he correlion kernel K n) converges uniformly o kernel wih leding erm he correcion is of order e dβ, β > ) K x, y) = 1 sin πx y) πx y) + d cos πx + y) + y x) sin πy + x) ) πd 2 + y x) 2,.2) ) where d = 2π 2. This limiing kernel defines, ech ime, limiing, nonspilly homogeneous deerminnl process which my loosely be hough of s he configurion of pricles in spce formed by n infinie sysem of noncolliding Brownin pricles sred from n equispced iniil configurion. When he inervl lengh L is smll relive o d, he verged number vrince for his process compued from he leding pr of he correlion kernel.2) hs leding erm 1 π 2 log2πl/) + γ Euler + 1)..3) However, if d is held consn, while L is incresed, i is deduced h he number vrince sures o he level 1 π 2 log2πd) + γ Euler + 1)..4) The smll L expression.3) grees wih he number vrince of he deerminnl poin process ssocied wih he sine kernel of densiy Kx i, x j ) = sin πx i x j )/..5) πx i x j ) This deerminnl process is he universl scling limi obined for he eigenvlues of rndom mrices from e.g. he Gussin Uniry Ensemble nd Un), s mrix size ends o infiniy see for exmple, [19]). For our purposes i will be convenien o hink of he bove verged number vrince s he number vrince of he spil pricle configurions rising from n verged model which we choose o inerpre s n infinie sysem of non-colliding Brownin moions sred from he iniil posiions u j = j ɛ), j Z, R +, ɛ Uniform[, 1]..6) The wo key feures of Johnsson s model s jus described re he equispced sring posiions nd he fc h he Brownin pricles re condiioned o no collide. In his work we consider he number vrince sisic in he independen process nlogues of Johnsson s model. Since hese independen versions do no hve deerminnl srucure, exising number vrince resuls no longer pply. 3
The pper is orgnized s follows. We begin by deriving generl formul for he number vrince for he spil pricle configurions rising from infinie sysems of independen Brownin moions nd symmeric α-sble processes in R sred from he iniil configurion.6). This enbles us o deduce he sympoic behviour of he sisic s he lengh of he inervl over which i is defined goes o infiniy. We give n explici formul for he surion level for he cses in which surion does occur. Once his is chieved we re ble o explin he number vrince surion phenomenon in erms of he il disribuion behviours of he underlying processes. We provide wo specific illusrive exmples s corollries nd briefly discuss how our resuls compre o hose lredy exising in he lierure. We conclude he firs secion by demonsring he close relionship beween he number vrince nd he re of convergence of he disribuion of he ssocied couning funcion o Poisson lw. In he second secion we use he number vrince o prove Gussin flucuion resuls for he couning funcions of our pricle configurions in wo differen sclings. In he hird nd finl secion we dd some dynmics o he flucuions of he couning funcions o consruc collecion of processes, ech of which is shown o converge wekly in C[, ) o cenered Gussin process wih covrince srucure similr in form o h of frcionl Brownin moion. Our erlier resuls on he behviour of he number vrince llow us o beer chrcerize hese limiing processes. In priculr, he long-rnge dependence propery exhibied by he covrince of heir incremens is direcly deermined by he re of growh of he ssocied number vrince. In he cses corresponding o α, 1), furher rescling of he limiing Gussin processes llow us o recover frcionl Brownin moions of Hurs prmeers 1 α 2 s wek limis. 1 The independen pricle cses 1.1 A Poisson process iniil configurion We begin by illusring he effec of he iniil posiions on he number vrince of he spil pricle configurions rising from such infinie sysems of processes s hose considered in his pper. The following heorem is he well known resul see for exmple, [9], Chper VIII, secion 5) h he Poisson process is invrin for n infinie sysem of independen pricles wih he sme evoluion. Theorem 1. Consider n infinie collecion of independen, idenicl in lw, rnslion invrin rel-vlued sochsic processes {X j ), ); j Z}. Suppose h {X j )} j= is Poisson process of inensiy θ on R. Then {X j )} j= is Poisson process of inensiy θ for every. Consequenly, if we begin wih mixed up Poisson process iniil configurion nd llow ech pricle o move s sy) Lévy process, independenly of he ohers hen we observe non-suring, linerly growing number vrince V [L] = θl) he sr nd for ll ime. 4
1.2 The Brownin nd symmeric α-sble cses This ls heorem served o highligh he impornce of he regulriy or rigidiy of he sring configurion.6) in deermining he number vrince behviour. However, i is resonble o suppose, h in Johnsson s model, he srong resricions plced on he movemen of he Brownin pricles mus lso conribue significnly o he surion behviour h is number vrince demonsres. This leds us o sk; wh would hppen if we sred wih n iniil configurion such s.6) bu did no plce such srong resricions on he movemen of he pricles? We nswer his quesion for he cses in which ech pricle moves independenly s one-dimensionl Brownin moion or s symmeric α-sble process on R. Recll see e.g. [21]) h symmeric α-sble process is Lévy process X α,c ), ) wih chrcerisic funcion, for ech, given by φ X α,c )θ) := E [ e iθxα,c ) ] = exp c θ α ), c >, α, 2). 1.1) Some of he properies enjoyed by his clss of processes re; {X α,c ), } wih ssocied rnsiion densiy p x, y), x, y R is emporlly nd spilly homogeneous. {X α,c )} is symmeric nd self-similr in he sense h {X α,c )} dis = { X α,c )} nd {λ 1/α X α,c λ ) } dis = {X α,c )} for consns λ R. The rgumens h follow lso pply o he α = 2 Gussin cses. Noe h we hve sndrd Brownin moion when α = 2, c = 1 2. Theorem 2. Fix symmeric α-sble process X α,c ), ) on R wih properies s described bove. Suppose we sr n independen copy of his process from ech of he sring posiions u j := j ɛ), j Z, where R + nd ɛ Uniform[, 1]. 1.2) The configurion of pricles in spce formed by his infinie sysem of independen symmeric α-sble processes fixed ime hs number vrince V α,c, [L] = L + 2 e [ 2cθ/)α Lθ ) ] π θ 2 cos 1 dθ. 1.3) Proof. Le {X α,c j symmeric α-sble process indexed by j Z. ), ), j Z} be he independen copies of he chosen Denoe he lw of ech X α,c j sred from x R by P x, nd wrie P := P. Now he number of symmeric α-sble pricles in n inervl [, L] R ime is given by he sum of indicor rndom vribles N α,c, [, L] = j= I[X α,c j ) + u j [, L]], 1.4) where u j ) j= is given by 1.2). Noe h by consrucion, for his verged model, for ll b R we hve N α,c, [, L] dis = N α,c, [b, b + L]. Thus he number 5
vrince is given by V α,c, [L] := Vr [ N α,c, [, L] ] = P[X α,c ) + u j [, L]] P[X α,c j ) + u j / [, L]]. 1.5) j= j We cn use he self-similriy propery nd he independence of ɛ nd he X j o wrie he probbiliies under considerion s convoluions which hen llows us o deduce L/ P[X α,c j ) + j ɛ) [, L]] = p / αx, y) dy dx. Hence = j= V α,c, L/ [L] 1.6) p / αx, y) dy dx } {{ } T 1 L/ L/ By Fubini s Theorem nd symmery we hve T 1 = L/ p / αx, y)p / αx, z) dz dy dx. } {{ } T 2 p / αy, x) dx dy = L. 1.7) For he oher erm we mke use of he Chpmn-Kolmogorov ideniy nd he spil homogeneiy before performing n inegrl swich o obin T 2 = L/ L/) y = L y L/ L/ p 2/ α, z) dz dy p 2/ α, z) dz + L/ z p 2/ α, z) dz L/ z p 2/ α, z) dz. From he symmery propery we know h, for ech, p, z) is n even funcion in z nd gz) := z p, z) is n odd funcion in z. These fcs llow us o conclude, bringing he wo erms ogeher, V α,c, [L] = 2L L/ p 2/ α, z) dz + 2 L/ z p 2/ α, z) dz. 1.8) Applying Fourier inversion o he chrcerisic funcion φ X α,c )θ) given 1.1), we deduce h he rnsiion densiy cn be expressed s p, z α, c) = 1 π coszθ)e cθα dθ. Using his densiy, he symmery propery nd he expression 1.8) we obin s required. V α,c, [L] = L + 2 π e 2cθ/)α θ 2 [ cos Lθ ) ] 1 dθ, 6
Hving found generl expression for he number vrince we re ble o consider is behviour s he inervl lengh L is incresed. Noe h given posiive rel vlued funcions g nd h we le g h signify h lim g h = 1. Theorem 3. Consider he number vrince V α,c, [L] for he sysem of symmeric α-sble processes considered bove. We clim h V α,c, [L] k α,c,, L 1 α, for α, 1), k α,c,, logl), for α = 1, k α,c,, L 1 α + κ s α, c,, ), for α 1, 2), k α,c,, e L2 /8c + κ s α, c,, ), for α = 2, s L, where k α,c,, is consn depending on α, c, nd, nd 1.9) κ s α, c,, ) = 2 π 2c)1/α Γ 1 1 ), 1.1) α where Γx) := s x 1 e s ds is he usul Gmm funcion. Proof. From 1.8) noe h we my re-wrie he expression for he number vrince s V α,c, [L] = L [ X P α,c 2 ) ] [ >L/ X α,c 2 ) X α +E P ; α,c 2 ) ] <L/ α α, 1.11) Now he behviour of he firs erm in his expression is well known see [2], pge 16). For α, 2) we hve L [ X P α,c 2 ) ] 2c > L/ α k α L1 α s L, 1. where k α = x α sin x dx) We use logl) when α = 1. In he Gussin cse α = 2) we hve insed L [ X 2 ) ] P > L/ 2 e L2 /8c 2cπ s L. To del wih he second erm of 1.11) observe h we hve [ ] [ ] X E α,c P 2/ α ) ; X α,c 2/ α L X ) < L/ α,c EP 2/ α ) ] X = P[ α,c 2/ α ) > λ dλ. Thus i is cler h he re of divergence or surion of he number vrince is deermined by he upper il of he disribuion of he underlying symmeric α-sble process. This my lso be seen by differeniing he inegrl expression for V α,c, [L] wih respec o L V α,c, [L] = 1 [ P X 2 ) L ] L α. 7
Consequenly, for α 1, 2] he suring cses) [ ] lim V α,c, [L] = E P X α,c 2 ) L α =: κ s α, c,, ) <. The exc expression for κ s cn be obined from he momens. By [22], if X is symmeric α-sble rndom vrible wih < α 2 nd scle σ, hen for 1 < δ < α we hve [ E X δ] = σδ/α 2 δ Γ ) ) 1+δ 2 Γ 1 δ/α. Γ1/2) Γ1 δ/2) Applying his heorem wih δ = 1, σ = 2c gives 1.1). To see how his fis in α wih he inegrl expression for V α,c, [L], i my be verified h for α 1, 2] κ s α, c,, ) = 2 π e 2cθ/)α θ 2 dθ. The surion resul is now consequence of Theorem 3. Corollry 1. L. 1. If α, 1], ech ime >, V α,c, [L] diverges s 2. If α 1, 2], ech ime <, V α,c, [L] sures o he level κ s α, c,, ) s L. Remrk 1. Even from he simples equion 1.5) i is cler h he lrges conribuions o he number vrince come from he civiy he edges of he inervl under considerion. Thus inuiively, he fer he ils of he disribuions concerned he greer he number of pricles h my be in he viciniy of hese edges mking hese subsnil conribuions, he slower he decline in his number s he inervl lengh increses nd consequenly he slower he decy in he growh of he number vrince. We now pply Theorems 2 nd 3 o wo well known exmples. Corollry 2 Brownin cse). Consider n infinie sysem of Brownin pricles sred from he iniil configurion u j ) j= s given 1.2). The number vrince for his process is [ V 2, 1 2, [L] = 2 L ) )] L Φ + 1 e L2 /4, 2 π where Φx) := 1 2π x e y2 /2 dy. As L his number vrince sures exponenilly quickly o he level 2 π. 8
Corollry 3 Symmeric Cuchy cse). Consider n infinie sysem of symmeric Cuchy processes sred from he iniil configurion u j ) j= s given 1.2). The number vrince for his process is V 1,1, [L] = L [ 1 2 π rcn [ L 2 For lrge relive o 2) L, we hve V 1,1, [L] 4 L ) π log 2 ] ] + 2 [ L 2 )] log 1 +. 1.12) π 2) nd so he number vrince diverges logrihmic re s L. Remrk 2. Coincidenlly, in he symmeric Cuchy cse, if we se = 1, = 1 4π we hve V 1,1,1 1 [L] 1 4π π 2 log2πl) nd so we see similr number vrince behviour s in he sine kernel cse.3). Remrk 3. In generl, he number vrince for he sine kernel deerminnl process nd for Johnsson s deerminnl process described in he inroducion, cn be expressed in he form 1.8) wih p 2/ α, z) replced by he funcions sin 2 πz) π 2 z nd sin2 πz) 2 π 2 z + d2 z 2 2 2π 2 d 2 +z 2 ) respecively. In he ler cse he surion 2 level is gin given by he expecion erm. Remrk 4. The Cuchy α = 1) cse hs he slowes growing non-suring number vrince mongs ll hose considered here. Anlogously, he sine kernel deerminnl process hs he slowes growing number vrince mongs ll rnslion invrin deerminnl processes whose kernels correspond o projecions i.e. he Fourier rnsform of he kernel is n indicor) see [24]. Remrk 5. A he oher exreme noe h s α we recover Poisson behviour in h lim V α,c, [L] = L α 1 e 2c ). 1.3 A Poisson pproximion for he couning funcion A he beginning of his secion we reclled h sysem of independen processes sisfying firly generl condiions) sred from Poisson process on R remins in Poisson process configurion nd hence demonsres number vrince liner in L, for ll ime. Now from 1.6) nd 1.7) we deduce h V α,c, [L] L/, α, c,,. 1.13) From he inegrl expression for he number vrince 1.3), we see h for fixed L nd for ech, s L is decresed V α,c, [L] L/ s, V α,c, [L] L L/. 9
So in boh hese limiing cses s well s in he α cse c.f. Remrk 5) he mximl liner number vrince is ined. The blnce of he prmeers L,, α nd, encpsuled by he number vrince, deermines how fr wy from being Poisson he disribuion of N α,c, [, L] cully is. Below we mke his observion precise. Recll h given mesurble spce Ω, F) we define he ol vriion disnce d T V, ) beween wo probbiliy mesures µ 1, µ 2 on Ω by d T V µ 1, µ 2 ) := sup µ 1 F ) µ 2 F ). F F Proposiion 1. Le LN α,c, [, L]) denoe he lw of he rndom vrible [, L] defined 1.4). Le PoL/) denoe he lw of Poisson rndom N α,c, vrible wih men L/. Then, for L 1, L V α,c, 32L [L] d T V LN α,c, [, L]), PoL/) ) 1 e L L V α,c, ) [L]. L Proof. The resul is n pplicion of heorem of Brbour nd Hll [2]. Their heorem ses h if A := j I j is he sum of independen indicor rndom vribles indexed by j nd q j L) = E[I j ], λ = j q jl) hen if we denoe he lw of A by LA) we hve 1 32 min1, 1/λ) j q j L) 2 d T V LA), PoL/) ) 1 e λ λ q j L) 2. In our specific cse we hve N α,c, [, L] s he sum of independen indicor rndom vribles given by 1.4), λ = L/ nd j q jl) 2 = L V α,c, [L]. Remrk 6. For fixed, he Poisson pproximion becomes less ccure s L. The greer he vlue of α he fser he quliy of he pproximion deeriores. For α > 1, due o he fc h he number vrince sures, he pproximion of he lw of N α,c, [, L] by Poisson disribuion of men L/ becomes very poor wih he ol vriion disnce clerly bounded wy from zero. 2 Gussin flucuions of he couning funcion Thus fr we hve been concerned wih he vrince of he couning funcion N α,c, [, L] 1.4).Of course his vrince is, by definiion, descripion of he flucuion of N α,c, [, L] round is men L. In his secion we will furher chrcerize hese flucuions. Proposiion 2. Le N α,c, [, L], V α,c, [L] denoe, s usul, he couning funcion nd number vrince. For he cses wih α, 1] we hve h N α,c, [, L] L/ V α,c, [L] converges in disribuion o sndrd norml rndom vrible s L. j 2.1) 1
Proof. Recll h he cumulns c k, k N of rel vlued rndom vrible Y re defined by log E[expiθY )] = k=1 iθ) k c k. k! Using he independence of he individul symmeric α-sble processes nd hen pplying he Tylor expnsion of log1 + x) bou zero, we hve log E P [expiθn α,c, [, L])] = = j= [ e log iθ 1 ) ] q j L) + 1 e iθ 1) m m=1 m 1) m+1 j= q j L) m), where q j L) := P[X α,c j )+u j [, L]] nd X α,c j ) denoes s usul he underlying symmeric α-sble process lbelled by j wih u j he corresponding sring posiion. Hence, he cumulns of N α,c, [, L] re given by c k = dk dθ k e iθ 1) m m=1 m I is srighforwrd o see h c 1 = L/, c 2 = 1) m+1 j= j= q j L) m) θ=. 2.2) q j L) q j L) 2 = V α,c, [L] give he men nd number vrince respecively. More generlly from he equion 2.2) i is possible o deduce he following recursive relion c k = k 1 m=2 β k,m c m + 1) k k 1)! j= q j L) q j L) k, 2.3) where β k,m re consn, finie, combinoril coefficiens which will no be needed here. Now le Y α,c, := N α,c, [, L] L/. V α,c, [L] I is esily deduced h he cumulns c k, k N of Y α,c, c 1 =, c k = c k V α,c,, [L]) k/2 for k 2. re given by To prove he Proposiion i is sufficien o show h in he limi s L, he cumulns correspond hose of Gussin rndom vrible. Th is, we hve c 3 = c 4 = c 5 = =. Equivlenly, we need o show c k = ov α,c, [L]) k/2 ) = oc k/2 2 ), for k 3. 11
We use n inducion rgumen. Suppose h c m = oc m/2 2 ) for m = 3,..., k 1. Assume, wihou loss of generliy, h k is even. We use he inequliy q j L) q j L) k = k 1 q j L) l q j L) l+1 l=1 k 1) q j L) q j L) 2 ) 2.4) in conjuncion wih he recursive relion for c k given by 2.3) o deduce k 1 m=2 β k,m c m c k k 1)!k 1)c 2 + From our inducion supposiion his implies h k 1 m=2 β k,m c m. oc k/2 2 ) c k oc k/2 2 ) + k 1)!k 1) c 2. 2.5) However, from he resuls of he previous secion, we know h, for hese cses wih α, 1], for k 3, c k 2 2 2 = V α,c, [L] k 2 2 s L. Thus from 2.5) c k = oc k/2 2 ) lso. Now using he sme rgumens s for he inequliy 2.4) we find 1 c2 c3 c 3/2 2 1 c2. Thus we hve c 3 = oc 3/2 2 ). By he inducion rgumen we cn deduce h c k c 2) s L for ll k 3 which concludes he proof. k/2 Remrk 7. The divergence of he number vrince is relied upon in similr wy o prove he nlogous Gussin flucuion resuls for lrge clss of deerminnl processes, see [8, 25, 24]. In ll cses he overll srucure of he proof is he sme. We noe h he Proposiion could lso hve been proven by pplying he Lindberg-Feller Cenrl Limi Theorem see e.g. [1]). Proposiion 2 pplies o he cses wih α, 1]. The following convergence in disribuion resul pplies o ll cses wih α, 2] nd is obined by llowing boh inervl lengh nd ime end o infiniy ogeher in n pproprie wy. Proposiion 3. For ny fixed s [, ) we hve h N α,c, [, s 1/α ] s 1/α / 1/2α 2.6) converges in disribuion s, o norml rndom vrible wih zero men nd vrince f α,c, s), where f α,c, s) := V α,c, 1 [s] = 4s π sin 2 u/2) u 2 1 e 2cu/s) α) du. 2.7) 12
Proof. Since V α,c, [s 1/α ] s, similr rgumen s for he proof of Proposiion 2 llows us o conclude h N α,c, [, s 1/α ] s 1/α / 2.8) V α,c, [s 1/α ] converges in disribuion s o sndrd norml rndom vrible. Mking he chnge of vrible u = s 1/α θ/ in he inegrl expression for he number vrince 1.3), using sin 2 u)/u 2 du = π/2 nd double ngle formul yields V α,c, [s 1/α ] = 4s sin 2 u/2) 1 e 2cu/s) α) du =: f α,c, 1/α π u 2 s). Noe h from 1.13) we know f α,c, s) < for ll s < nd so he resul follows from he scling propery of he Gussin disribuion. 3 The flucuion process We proceed by dding some dynmics o he flucuions of he couning funcion nd define, for ech α, 2], c >, R +, he process Z α,c, s) := N α,c, [, s 1/α ] s 1/α /, s [, ). 1/2α 3.1 The covrince srucure We begin o chrcerize hese processes by idenifying he covrince srucure. Lemm 1. {Z α,c, s); s [, )} hs covrince srucure Cov Z α,c, r), Z α,c, s) ) = 1 f 2 α,c, s) + f α,c, r) f α,c, r s ) ). Proof. By consrucion N α,c, [, r s) 1/α ] N α,c, [, r s) 1/α ] dis = N α,c, [, r s 1/α ]. Hence, from he definiion of Z α,c, which implies h Z α,c, r s ) dis = Z α,c, r s) Z α,c, r s), Vr Z α,c, r s ) ) =Vr Z α,c, r s) ) + Vr Z α,c, r s) ) 2Cov Z α,c, r s), Z α,c, r s) ). Rerrnging gives Cov Z α,c, s), Z α,c, r) ) = 1 Vr Z α,c, r s) ) + Vr Z α,c, r s) ) Vr Z α,c, r s ) )) 2 = 1 ) V α,c, 2 1/α [s 1/α ] + V α,c, [r 1/α ] V α,c, [ r s 1/α ]. On referring bck o he definiion of f α,c, ) we see h his ls semen is equivlen o he resul of he Lemm. Noe h he covrince does no depend on. 13
3.2 Convergence of finie dimensionl disribuions Given he covrince srucure of Z α,c, nd he idenificion of is Gussin one dimensionl mrginl disribuions he nurl nex sep is o consider he finie dimensionl disribuions. Proposiion 4. Le {G α,c, s) : s [, )} be cenered Gussin process wih covrince srucure Cov G α,c, s i ), G α,c, s j ) ) = 1 2 f α,c, s i ) + f α,c, s j ) f α,c, s i s j ) ). 3.1) For ny s 1 s 2 s n < we hve Z α,c, s. s 1 ), Z α,c, s 2 ),..., Z α,c, s n )) G α,c, s 1 ), G α,c, s 2 ),..., G α,c, s n )) Proof. As previously noed, he men nd covrince srucure of Z α,c, s) re no dependen on. Therefore, ll h remins is o show h, in he limi s, he join disribuions re Gussin. We gin mke use of he cumulns. Recll h given rndom vecor Y := Y 1, Y 2,..., Y n ) R n, he join cumulns of Y denoed C m1,m 2,...,m n Y) re defined vi he m j h pril derivives of he logrihm of he chrcerisic funcion of Y. Th is, ) m1 C m1,m 2,...,m n Y) := ) iθ 1 ) iθ 2 If nd in priculr C,..., 1 }{{} i h C,,..., 1 }{{} i h C,,..., 2 }{{} i h,...,,..., 1 }{{} j h ) m2 ) mn ) log E [exp n ) ] iθ j Y j θ=. iθ n,..., Y) = E[Y i] =,..., Y) = Vr[Y i] = Σ ii,..., Y) = Cov[Y i, Y j ] = Σ ij j=1 C m1,m 2,...,m n Y) =, whenever n m i 3, i=1 hen Y hs mulivrie norml, Σ) disribuion. To prove he Proposiion i is enough o show h Z α,c, s 1 ), Z α,c, s 2 )) MulivrieNorml, Σ α,c, ) 3.2) in disribuion s, where Σ α,c, is he 2 2 covrince mrix f α,c, 1 s 1 ) 2 f α,c, s 1 )+f α,c, s 2 ) f α,c, s 1 s 2 ) ) ) 1 2 f α,c, s 1 )+f α,c, s 2 ) f α,c, s 1 s 2 ) ) f α,c,. s 2 ) 14
We begin by compuing he chrcerisic funcion of N α,c, [, s 1 1/α ], N α,c, [, s 2 1/α ] ). Using he independence of he individul pricles we hve [ ] E P expiθ 1 N α,c, [, s 1 1/α ] + iθ 2 N α,c, [, s 2 1/α ]) [ )] = E P exp i θ 1 + θ 2 )N α,c, [, s 1 1/α ] + θ 2 N α,c, [s 1 1/α, s 2 1/α ] = e iθ1+θ2) P [ X α,c j )+u j [, s 1 1/α ] ] j= + e iθ2 P [ X α,c j ) + u j [s 1 1/α, s 2 1/α ] ] + P [ X α,c j )+u j / [, s 2 1/α ] ]). For ese of noion we will henceforh le q j s l, s r ) := P[X α,c j ) + u j [s l 1/α, s r 1/α ]], s l s r <. The join cumulns re given by C m1,m 2 N α,c, [, s 1 1/α ], N α,c, [, s 2 1/α ] ) ) m1 ) m2log = e iθ1+θ2) q j, s 1 )+e iθ2 θ= q j s 1, s 2 )+1 q j, s 2 )) iθ 1 ) iθ 2 ). j= Now using he fc h ) m1 ) m2e iθ 1+θ 2) q j, s 1 ) + e iθ2 q j s 1, s 2 ) + 1 q j, s 2 ) iθ 1 ) iθ 2 ) = e iθ1+θ2) q j, s 1 ) m 1, m 2 s.. m 1 1 nd ) m2e iθ 1+θ 2) q j, s 1 ) + e iθ2 q j s 1, s 2 ) + 1 q j, s 2 ) iθ 2 ) = e iθ1+θ2) q j, s 1 ) + e iθ2 q j s 1, s 2 ) m 2, long wih e iθ1+θ2) q j, s 1 ) + e iθ2 q j s 1, s 2 ) + 1 q j, s 2 )) θ= = 1, we deduce he following generlizion of he recursive relion 2.3), wih obvious shor-hnd noion for he join cumulns C m1,m 2 = β k,l,m1,m 2 C k,l k,l: 2 k+l m 1+m 2 1 + 1) m1+m2 m 1 + m 2 1)! j= q j, s 1 ) q j, s 1 ) m1 q j, s 2 ) m2. 15
Now suppose h C k,l = o k+l 2α ) for ll k, l such h k + l {3, 4,..., m 1 + m 2 1} nd wihou loss of generliy ssume h m 1 + m 2 is even. Since q j, s 1 ) q j, s 1 ) m1 q j, s 2 ) m2 q j, s 1 ), if we ssume h he bove inducion hypohesis holds, hen we hve o m 1 +m 2 1 2α ) C m1,m 2 o m 1 +m 2 1 2α ) + m 1 + m 2 1)! s 1 1/α, 3.3) which implies h C m1,m 2 = o m 1 +m 2 2α ) lso. We check he hird order join cumulns direcly nd deduce h o 3 2α ) Ck,l o 3 s2 1/α ) 2α ) + 2 when k + l = 3, since he vrinces nd covrince of N α,c, [, s 1 1/α ], N α,c, [, s 2 1/α ]) i.e. C 2,, C,2, C 1,1 ) grow mos like 1/α s. Therefore, by inducion, whenever m 1 + m 2 3 we hve C m1,m 2 N α,c, In erms of he join cumulns of Z α,c, [, s 1 1/α ], N α,c, [, s 2 1/α ]) s. m1+m2)/2α his implies C m1,m 2 Z α,c, s 1 ), Z α,c, s 2 )) s whenever m 1 + m 2 3, from which he clim 3.2) nd he Proposiion follow. 3.3 A funcionl limi for he flucuion process In order o give funcionl limi resul we consider coninuous pproximion s) : s [, )} o he process {Z α,c, s) : s [, )}. Le {Ẑα,c, Ẑ α,c, s) := N α,c, [, s 1/α ] s 1/α / 1/2α, 3.4) N α,c, where [, s 1/α ] is defined o be equl o N α,c, [, s 1/α ] excep he poins of disconinuiy where we linerly inerpole. Le C[, 1] be he spce of coninuous rel vlued funcions on [, 1] equipped wih he uniform opology. We shll denoe he mesure induced by {Ẑα,c, s) : s [, 1]} on he spce C[, 1], BC[, 1])) by Q α,c,. To simplify noion we resric enion o he inervl [, 1] bu noe h he ensuing funcionl limi heorem exends rivilly o ny finie rel indexing se. The reminder of his secion is devoed o esblishing he following wek convergence resul. Theorem 4. Le Q α,c, be he lw of he cenered Gussin process {G α,c, s) : s [, 1]} inroduced in he semen of Proposiion 4. Then s. Q α,c, Q α,c, 16
Proof. Noe h by definiion Ẑα,c, s) Z α,c, s) 1. 3.5) 1/2α Thus, s, he finie dimensionl disribuions of Ẑα,c, s) mus converge o he finie dimensionl disribuions of he limiing process G α,c, s) o which hose of Z α,c, converge. Hence, immediely from Proposiion 4 we hve h for ny s 1 s 2 s n <, Ẑα,c, s 1 ), Ẑα,c, s 2 ),..., Ẑα,c, s n )) G α,c, s 1 ), G α,c, s 2 ),..., G α,c, s n )) s. Therefore, by well known resul of Prohorov s see e.g. [6]), he proposed funcionl limi heorem holds if he sequence of mesures {Q α,c, } is igh. Indeed his ighness requiremen follows from Proposiion 5 given below. The sufficien condiions for ighness verified below re sed in erms of he disribuions PẐα,c, A) = Q α,c, A) A BC[, 1]), nd he modulus of coninuiy, which in his cse is defined by wẑα,c,, δ) := sup Ẑα,c, s) Ẑα,c, r), δ, 1]. s r δ Proposiion 5. Given ɛ, λ > δ >, N such h P[wẐα,c,, δ) λ] ɛ, for. Proposiion 5 is proven vi he following series of Lemms. Lemm 2. Suppose u r s v 1, hen Ẑα,c, s) Ẑα,c, r) Ẑα,c, v) Ẑα,c, u) + v u) 1/2α. Proof. Clerly, by consrucion α,c, N [, s 1/α ] α,c, N [, r 1/α ] α,c, N [, v 1/α ] N α,c, [, u 1/α ]. Therefore, using he definiion of Ẑα,c,, we hve Ẑα,c, s) Ẑα,c, r) + s r) 1/2α Ẑα,c, v) Ẑα,c, u) + v u) 1/2α. The resul follows by rerrnging, using he fcs R +, v u) s r) nd hen considering seprely ech cse Ẑ α,c, s) Ẑα,c, r) or Ẑ α,c, s) Ẑα,c, r) <. Lemm 3. Ẑα,c, s) Ẑα,c, r) 2 + Zα,c, 1/2α s) Z α,c, r). 3.6) 17
Proof. Follows from 3.5) nd n pplicion of he ringle inequliy. To obin resuls on he disribuion of he modulus of coninuiy for our sequence of processes {Ẑα,c, } we divide he inervl [, 1] ino m disjoin subinervls of lengh pproximely δ s follows. Le where we define = r < r k1 < < r km 1 < r km = 1, 3.7) r i := i, 1/α i N k j := j δ 1/α, j {, 1, 2,..., m 1} nd denoes he ceiling funcion. We hve δ r kj r kj 1 δ + 1 1/α j {1, 2,..., m 1}, wih he subinervls [r i 1, r i ], i N ypiclly being shorer in lengh. Lemm 4. [ P wẑα,c, ] m 1 [, δ) λ P mx Z α,c, r i ) Z α,c, r kj ) λ k j i k j+1 9 7 ]. 3 1/2α j= Proof. Given he priion 3.7), sndrd mehods see Theorem 7.4 of [6]) yield [ P wẑα,c, ], δ) λ m 1 j= [ P By he ringle inequliy we hve sup Ẑα,c, r kj s r kj+1 s) Ẑα,c, r kj ) λ 3 ]. 3.8) Ẑα,c, s) Ẑα,c, r kj ) Ẑα,c, s) Ẑα,c, r i ) + Ẑα,c, r i ) Ẑα,c, r kj ). 3.9) Now if s [r kj, r kj+1 ], hen eiher Ẑ α,c, s) = Ẑα,c, r i ) for some i N immediely simplifying 3.9), or i N such h r kj r i 1 < s < r i r kj+1 in which cse from Lemm 2 we hve Ẑα,c, s) Ẑα,c, r i ) Ẑα,c, r i ) Ẑα,c, r i 1 ) + r i r i 1 ) 1/2α = Ẑα,c, r i ) Ẑα,c, r i 1 ) + 1 1/2α Ẑα,c, r i ) Ẑα,c, r kj ) + Ẑα,c, r i 1 ) Ẑα,c, r kj ) + 1. 3.1) 1/2α 18
Therefore, using he inequliy 3.9) in conjuncion wih 3.1) nd Lemm 3, we hve h for s, r i [r kj, r kj+1 ] Ẑα,c, s) Ẑα,c, r kj ) Ẑα,c, r i ) Ẑα,c, r kj ) + Ẑα,c, r i 1 ) Ẑα,c, r kj ) + 1 1/2α + Ẑα,c, r i ) Ẑα,c, r kj ) 3 mx Ẑ α,c, r i ) k Ẑα,c, r kj ) + 1 j i k j+1 1/2α [ Z α,c, 3 mx r i ) Z α,c, r kj ) + 7 ]. k j i k j+1 3 1/2α Thus [ P sup Ẑ α,c, s) Ẑα,c, r kj ) λ ] r kj s r kj+1 3 [ Z α,c, P mx r i ) Z α,c, r kj ) + 7 k j i k j+1 3 1/2α λ 9 Subsiuing his ls inequliy ino 3.8) gives he semen of he Lemm. Now h we hve reduced he sudy of he disribuion of he modulus of coninuiy o h of he mximum flucuion over our consruced subinervls we cn progress by inroducing mximl inequliy. In order o do his we use he following known resul ken from [6] nd prphrsed for use here. Theorem 5. Consider sequence of rndom vribles {ξ q } q 1 nd he ssocied pril sums u S u := ξ q S :=. Le If q=1 M n := mx 1 u n S u. P [ S v S u γ ] 1 γ 2 u<l v for γ > nd some b 1, b 2,..., b n R +, hen where κ is consn. Proof. See [6], Theorem 1.2 Lemm 5. P [ mx 1 u n Zα,c, where κ is consn nd γ >. b l ) 2 P[M n γ] κ γ 2 <l n b l ) 2, u v n r kj+u) Z α,c, r kj ) γ] κ n 1/2α ) 2, γ 2 ]. 19
Proof. Le Then S u := ξ q := Z α,c, u q=1 r kj+q) Z α,c, r kj+q 1). ξ q = Z α,c, r kj+u) Z α,c, r kj ) S :=. In his cse, for v, u N, pplying Chebyshev s inequliy nd using he definiion of f α,c, ) nd he upper bound given 1.13), we hve P [ S v S u γ ] = P [ Z α,c, r kj+v) Z α,c, r kj+u) γ ] f α,c, r kj+v r kj+u ) γ 2 v u) 1/α γ 2 [ ] v u) 1/2α 2 γ 2. Thus we cn ke b l = 1/2α for l = 1, 2,..., n nd pply Theorem 5 which gives he mximl inequliy of he Lemm. Concluding he proof of Proposiion 5 is now srighforwrd. Proof of Proposiion 5. Tking n = k j+1 k j in he semen of Lemm 5 gives P [ mx Z r i ) Z r kj ) γ ] k j i k j+1 κ ) 2 γ 2 rkj+1 r kj κ δ + 1/α ) 2. γ 2 Subsiuing his ls inequliy wih γ = λ 9 ) 7 3 which is sricly posiive 1/2α for sufficienly lrge ) ino he inequliy given by Lemm 4 gives P [ wz α,c,, δ) λ ] 81mκ δ 1/2α + 1/2α) 2 λ 1/2α 21 ) 2. On solving he pproprie qudric equion we find h we cn mke his upper bound less hn ɛ by choosing δ from 1/2α c ɛ λ 21 1/2α ), 1/2α + c ɛ λ 21 1/2α )) ), 1, where c ɛ = ɛ 81mκ. Since his inervl is non-empy for sufficienly lrge his complees he proof. 3.4 Properies of he limiing process G α,c, s) We hve consruced fmily {G α,c, s), s [, )), α, 2], c >, R + } of cenered, coninuous, rel-vlued Gussin processes wih he inheried covrince srucure CovG α,c, s), G α,c, r)) = 1 2 f α,c, s) + f α,c, r) f α,c, s r ) ). 3.11) 2
I is cler h he processes re recurren for 1 < α 2 s he number vrince sures nd he sionry disribuion is norml wih men zero nd vrince κ s α, c,, 1). We re ble o deduce furher properies of hese limi processes by using our erlier resuls on he number vrince for he sysems of symmeric α-sble processes from which hey re consruced. Proposiion 6. G α,c, s) is non-mrkovin. Proof. Recll see e.g. [16], Chper 13) h Gussin process wih indexing se T R nd covrince funcion ρ : T R is Mrkov if nd only if ρs, u) = ρs, r)ρr, u) ρr, r) s, u, r T. I is cler h his relionship does no hold in generl for he covrince funcion 3.11). From he resuls of previous secions, since f α,c, s) = V α,c, 1 [s], we know h f α,c, s) 1 e 2c ) s s α nd f α,c, s) s k α,c, s. Therefore, G α,c, ppers o sr ou, for smll ime s scled Brownin moion nd s α his iniil Brownin chrcer previls for longer. We cpure his more precisely in he following esily verified proposiion. Proposiion 7. 1. {G α,c, s) : s [, 1]} converges wekly o Brownin moion {B 1 e 2c s) : s [, 1]} s α. 2. Le G α,c, ɛ s) = ɛ 1/2 G α,c, ɛs). Then {G α,c, ɛ s) : s [, 1]} converges wekly o Brownin moion {B s ) : s [, 1]} s ɛ. Remrk 8. The covrince srucure of G α,c, is similr o h of Brownin bridge. Recll h he sndrd Brownin bridge B br s), s [, ]) of lengh, is cenered Gussin process wih covrince srucure CovB br s), B br r)) = s r sr nd rises s wek limi of mny empiricl processes. In priculr, i my be obined from he ppropriely scled couning funcions of Poisson process on R see e.g. [16]). We cn re-wrie he covrince 3.1) in he lernive form CovG α,c, s), G α,c, r)) = s r r s p 2/ αy, z) dy dz, bu we see h precise mch would require p 2/ αy, z) = 1. Proposiion 8. The process G α,c, hs sionry incremens which re negively correled. 21
Proof. I is srighforwrd o see h he incremens hve zero men nd h for ny s, r [, ) In ddiion, for u we hve VrG α,c, s) G α,c, r)) = f α,c, s r ). CovG α,c, s) G α,c, ), G α,c, r + s + u) G α,c, s + u)) = 1 f α,c, s + r + u) f α,c, r + u) [ f α,c, s + u) f α,c, u) ]). 3.12) 2 Since f α,c, is concve funcion f α,c, s + r + u) f α,c, r + u) f α,c, s + u) f α,c, u), for ll s, r, u [, ) so i follows h his covrince is non-posiive. Proposiion 9. G α,c, is no in generl self-similr. For ny consn b R we hve he relionship G α,c, bs) dis = b 1/2 G α, c b α, s). Proof. Boh sides of he proposed equion hve zero men nd Gussin disribuion. I is cler from he expression given for f α,c, 2.7) h he vrinces/covrinces lso gree. Proposiion 1. G α,c, is long memory or long rnge dependen) process in he sense h he covrince beween incremens decys s power lw s he seprion beween hem is incresed. More precisely, for α, 2) we hve Cov G α,c, s) G α,c, ), G α,c, r + s + u) G α,c, s + u) ) u k u α+1), where k is consn depending on α, c,, s nd r. Proof. The covrince in quesion is expressed in erms of he funcion f α,c, 3.12). Noe from 1.9) h we lredy know he sympoic behviour of he individul componens of his expression. Applying l Hopil s rule wice in succession yields he given power lw. We hve lredy menioned similriy beween he covrince srucure of G α,c, nd h of Brownin moion. More generlly we cn drw prllels beween our limiing process nd frcionl Brownin moion. Recll see for exmple [18]) h frcionl Brownin moion W H s), s ) wih Hurs prmeer H, 1) is cenered, self-similr Gussin process wih covrince funcion CovW H s), W H r)) = 1 s 2H + r 2H s r 2H), 3.13) 2 The cse H = 1 2 corresponds o sndrd Brownin moion. Noe he resemblnce beween he form of he covrince funcions 3.11) nd 3.13). Heurisiclly, we cn deduce h f α,c, s) my be pproximed by funcion of he form κ α,c, s 2Hα,c,s) where H α,c, : [, ) [, 1 2 ] is monooniclly decresing funcion wih iniil vlue H α,c, ) = 1 2. Thus 22
loosely speking G α,c, cn be viewed s ype of frcionl Brownin moion wih ime vrying Hurs prmeer. In priculr, he long rnge dependence propery of Proposiion 1 my be compred o he nlogous semen for frcionl Brownin moion: Cov W H s) W H ), W H r + s + u) W H s + u) ) u ku 2H 2, where k is consn depending on H, s nd r. We mke he link beween he process G α,c, nd frcionl Brownin moion precise wih he following semen. Proposiion 11. For α, 1) le Then G α,c, b s) := Gα,c, bs), s, b [, ). b 1 α α,c, { G b s) : s [, )} {k 1/2 α,c, W 1 α 2 s) : s [, )} s b where k α,c, = 4c π Γα 1) sin απ/2 ). Proof. From 2.7), by pplying Tylor expnsion we deduce h f α,c, bs) b 1 α 1 α 8c = s π + 4 π sin 2 u/2) du u 2 α sin 2 u/2) 1) j+1 2c) j s j! j=2 Now by he Domined Convergence Theorem, lim b 4 π = 4 π = b j=2 1 2jα ujα 2 du. jα α sin 2 1) j+1 u/2) 2c) j 1 2jα ujα 2 s j! b jα α du j=2 }{{} h b u) sin 2 1) j+1 u/2) lim 2c) j s j! b b 1 2jα ujα 2 du, jα α since, seing Mu) = sin 2 u/2)1 exp 2cu/s) α ))/u 2 + sin 2 u/2)/u 2 α, we hve posiive inegrble funcion such h h b u) Mu) for ll b R. This implies h which llows us o conclude f α,c, bs) lim b b 1 α = k α,c, s 1 α Cov Gα,c, α,c, b r), G b s) ) = b α 1 Cov G α,c, br), G α,c, bs) ) b k α,c, 2 = Cov k 1/2 s 1 α + r 1 α + s r 1 α) α,c, W 1 α 2 s), k 1/2 α,c, W 1 α 2 r) ). 23
The processes re Gussin herefore he convergence of finie dimensionl disribuions is implied by he convergence of he covrince funcions nd ighness follows esily from, for exmple, [16] Corollry 16.9 nd well known expressions for he even momens. Remrk 9. We menion h vrious oher long-rnge dependen Gussin processes hve recenly been found o rise from he flucuions of spilly disribued pricle sysems, see [11] nd references wihin. Nobly, in his conex, he spil pricle configurions of infinie sysems of symmeric α- sble processes sred from Poisson process on R hve been considered. In hese cses, frcionl Brownin moion wih Hurs prmeer H = 2 1 α, α 1, 2] ws obined s scling limi of he occupion ime process essenilly scling he couning funcion in ime rher hn in ime nd spce s in his pper). Remrk 1. I seems nurl o sk wheher, in he sme fshion s we creed G α,c,, similr limiing processes could be consruced from Johnsson s sysems of non-colliding Brownin moions. Unforunely, he formul for he verged number vrince given for hese processes in [14] does no scle in ime nd spce in he sme convenien wy s V α,c, [L] in his cse. However, s noed by Johnsson, leing in his model, one obins he sine kernel deerminnl process, from which limiing Gussin process prmeerized nd scled in compleely differen wy) ws consruced in [23]. References [1] R. Aurich nd F. Seiner. Periodic-orbi heory of he number vrince Σ 2 L) of srongly choic sysems. Phys. D, 823):266 287, 1995. [2] A. D. Brbour nd P. Hll. On he re of Poisson convergence. Mh. Proc. Cmbridge Philos. Soc., 953):473 48, 1984. [3] M. V. Berry. The Bkerin lecure, 1987. Qunum chology. Proc. Roy. Soc. London Ser. A, 4131844):183 198, 1987. [4] M. V. Berry. Semiclssicl formul for he number vrince of he Riemnn zeros. Nonlineriy, 13):399 47, 1988. [5] M.V. Berry nd J.P. Keing. The Riemnn zeros nd eigenvlue sympoics. SIAM review, 41:236 266, 1999. [6] P. Billingsley. Convergence of probbiliy mesures. Wiley Series in Probbiliy nd Sisics: Probbiliy nd Sisics. John Wiley & Sons Inc., New York, 1999. [7] O. Bohigs, M.J. Ginnoni, nd C. Schmid. Chrcerizon of choic qunum specr nd universliy of level flucuion lws. Phys.Rev.Le, 52:1 4, 1984. [8] O. Cosin nd J. Lebowiz. Gussin flucuions in rndom mrices. Phys.Rev.Le, 75:69 72, 1995. [9] J. L. Doob. Sochsic processes. John Wiley & Sons Inc., New York, 1953. 24
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