Estimation of safety areas for epidemic spread

Estimatio n of s afet y areas f or epidemic spread Beatriz Marr´ on ∗ , 1 , Ana T ablar ∗ Ma y 10, 2001 Abstract In this w ork w e study safet y a reas in epidemic spred. The aim of this work is, giv en th e ev olution of epidemic at time t , fin d a safet y set at time t + h . This is, a random set K t + h suc h that th e probabilit y that infection reac hes K t + h at time t + h is small. More precisely , in spired on the study of epidemic spread, we consider a mo del i n wh ic h the measure µ n ( A ) is the incidence - density of infective s individuals- in th e set A , at time n and µ n +1 ( A )( ω ) = Z S π n +1 ( A ; s )( ω ) µ n ( ds )( ω ) , for any Borel set A, with r andom tr an s ition kernels of the form π n ( . ; . )( ω ) = Π( . ; . )( ξ n ( ω ) , Y n ( ω )) , where ξ , Y satisfy some ergo dic conditions. T he supp ort of µ n is called S n . W e also assume th at S 0 is compact with regular b order and that for any x, y the k ernel Π( . ; . )( x, y ) has c ompact supp ort. A random set K n +1 is a safet y area of lev el α if: i ) K n +1 is a fun ction of S 0 , S 1 , · · · , S n . ii ) P ( K n +1 ∩ S n +1 6 = ∅ ) ≤ α. W e p r esen t a metho d to fin d these safet y areas and some r elated resu lts. Keywor ds: T ransition k ernel, Epidemic sp read, Safety area. AMS subje ct classific ations: 60F05. ∗ Departamento de Matem´ atica. Universidad Nacional del Sur. 1 Corresponding au- thor: b eatriz.ma rron@uns.edu.ar 1 1 In tro duction Mathematica l mo delling for some t yp e of epidemic sp read, like those that af- fect the w hole planet called “pand emic”, shoud repro duce tw o basic asp ects observ ed in real data: a) A su sceptible ind ivid ual may b e infected by individual who is usually lo cated at a ve ry distant p oin t. F or instance: an inf ected tourist infects someone h e visits. b) The temp oral evo lution is t yp ically non-mark o vian. Given the present, if the p ast indicate s that the spread is d iffu siv e or if the past sho ws that spread is in co ntrac tion, w e will prob ab ly not make th e same prediction for the futur e. That means that giv en the present , p ast and future may not b e indep enden t, whic h is a clear argumen t against Mark o v mo dels. In ord er to d escrib e more p recisely our model we in tro du ce some basic notation: w e denote by S a space of sites (typica lly S = Z d , R d or a fi nite subset of R d ), b y B ( S ) its Borel σ -alge bra and b y P ( S ) the set of probabilit y measure on ( S , B ( S )). A transition probabilit y ke rnel (TPK) on S , π ( A ; s ), is a fu nction π : B ( S ) × S − → [0 , 1] , suc h that: • F or any fixed s ∈ S , π ( . ; s ) ∈ P ( S ) . • F or any fixed A ∈ B ( S ), π ( A ; s ) is a B ( S )-measurable function. The set of T PK on S is denoted by K ( S ), and in this work we deal with random TPK th at describ e pr obabilities for infection from one p oin t to another. Lo osely sp eaking, π ( A ; s ) giv es th e probability of a transition from s to an elemen t of A , in this conte x, “transition” means “infection”. The state of th e epidemic pr op agation at time t = 0 is describ ed by a n on- random probability measure µ 0 , so µ 0 ( A ) giv es the infection den sit y on A for any Borel set A , this is the num b er of infected p eople living in A /total n umber of infected p eople. If µ n describ es the state at time t = n , then the description at time t = n + 1 is giv en by µ n +1 ( A )( ω ) = Z S π n +1 (A; s)( ω ) µ n ( ds )( ω ) , (1) 2 where π 1 ( . ; . )( ω ) , π 2 ( . ; . )( ω ) , · · · is a sequence of random TPK of th e form π n ( . ; . )( ω ) = Π( . ; . )( ξ n ( ω ) , Y n ( ω )) , f or any n and ω , (2) where i ) Π( . ; . )( e, e ′ ) : E × E ′ − → K ( S ) is a measurable function, where E and E ′ are p olish sp aces. W e w ill assume in addition that this f u nction is con tin uous on the second co ord inate e ′ . ii ) ξ = ( ξ n ) n ∈ N is an i. i .d sequence of E -v alued r andom v ariables. iii ) Y = ( Y n ) n ∈ N is an E ′ -v alued pro cess sat isfying that its empirical measure F n Y defined b y F n Y ( B )( ω ) = 1 n P n i =0 1 I { Y i ( ω ) ∈ B } for an y Borel set B of E ′ , conv erge to a random measure λ Y ( ω ) in this w a y: √ n sup x ∈ R + | F n Y ( x )( ω ) − λ Y ( x )( ω ) | − → 0 a.s. iv ) ξ and Y are ind ep endent. Remark 1.1 Let u s note that, with th e exception of µ 0 , µ n are rand om measures. Remark 1.2 F or a random pro cess Y = ( Y n ) n ∈ N denote σ Y m the σ -alg ebra generated by { Y n : n ≥ m } and define σ Y ∞ = ∞ \ m =1 σ Y m , w e will sa y that Y is regular if σ Y ∞ is trivial, in the s en se that for an y A ∈ σ Y ∞ one has P ( A ) = 0 or P ( A ) = 1. It is easy to c hec k that th e limit measure λ Y in ( iii ) is a σ Y ∞ -measurable random measure. Therefore, if Y is regular, λ Y is deterministic, non-random. The in tuitiv e idea b ehind this t yp e of mo dels is th at the evol ution mo d- els of the type (1) allo w to easily mo del trans itions and to consider random transition kernels p ermits to mo del a complex and h ighly v ariable transi- tion d ynamics. F or in stance, in a pandemic s pread, m igration and touristic mobilit y p la y a k ey role on the spread. This curren ts of trans ition ma y h a v e differen t “reg imens” with d efferent inte nsities and d irections. Moreo ver, random TP K mak es measures defined in (2) do not describ e just a non- homogeneous Marko v (as in the case when TP K are determin istic) b u t a 3 non-Mark o v mo del, with dep endence b etw een past and futur e w hen present is giv en. Finally , the idea of descomp osin g r andomness in tw o indep en dien t sources, one of them ( ξ , corresp onding to “pure noise”, in regresion analysis terms) of a ve ry simple probabilistic nature and the other ( Y , corresp onding to some “explicativ e” v ariables, in regresion terms) p ossibly more complex but whose empirical measure con v erges, is a w a y to constru ct very general mo dels where limits th eorems can b e easily obtained. The aim of this pap er is to fi nd safety areas for the mo del (1), this is a random set su c h that the probabilit y of an ind ividual placed in that set to b e infected in th e next step is small. W e denote by S 0 the supp ort of µ 0 , by S n the supp ort of µ n , and by d i = diameter ( S i ), and we will assume that S uppor t ( π ( ξ i , Y i )( ., s )) = B ( s, r i ) , for an y s ∈ S, (3) where B ( s, r i ) is an op en ball cen tered in s of radiu s r i . First, we find a s afet y area for the case that the radius r i dep end s only on ξ i , while the wh ole measure π ( ., s ) dep ends on b oth ξ and Y , then w e extend the d efinition for the case that the radius dep ends on ξ and Y . Some final remarks on general n otation used all along th is pap er: • Z n w − → Z or Z n w − → F denote a sequence of random v ariables ( Z n ) n ∈ N that con v erges in distrib ution to a r an d om v ariable Z with distrib u tion function F . • T o simplify statemen ts of resu lts and defin itions, we do not mak e ex- plicit men tion to obvi ous hyp otheses: for instance, if a result refers to an integ ral, the integral is assumed to exist and b e fin ite. • W e ind icate by “:=” a definition that is stated in the midd le of a form ula. • Let A δ denote the set A δ = { s ∈ S : distance ( s, A ) ≤ δ } , for ev ery A ⊂ S . • Let F n X ( t ) denote the empirical distrib ution function F n X ( t ) = 1 n n P i =1 1 I { X i ≤ t } , where X 0 , X 1 , · · · , X n are r andom v ariables. 2 Main results Let u s consider here the case of S = R d or S = Z d (or, more in general, a subset of R d ) and E ′ = R (or, more in general, a sub s et of R ), µ 0 a deter- ministic elemen t of P ( S ) and the sequence of random p r obabilit y m easures 4 Figure 1: Safet y area. defined b y (1). W e assu me that S 0 is compact and its b order is a regular closed curve, an since µ 0 is deterministic, so is S 0 . As we observe µ 0 , µ 1 , · · · , µ n , then d 0 , d 1 , · · · , d n are data, wh ere d i = diameter ( S i ), and by (3), it is clear that d i +1 = d i + 2 r i +1 . Hence r i +1 = d i +1 − d i 2 . (4) This simple equation is b asic for our p urp oses, b ecause it means that we can compute r i +1 in terms of d i and d i +1 , so we can compute r 1 , r 2 , · · · , r n in terms of µ 0 , µ 1 , · · · , µ n , and w e can base our statisti cal pro cedur e on r 1 , r 2 , · · · , r n . The in tuitiv e concept of safety area giv en in the in tro du ction, can b e written more form ally as follo ws, a s afety area of lev el α is a rand om set K n +1 that s atisfies: i ) K n +1 is a fun ction of S 0 , S 1 , · · · , S n , ii ) P ( K n +1 ∩ S n +1 6 = ∅ ) ≤ α , and the condition ii ) is equiv alen t to P ( K n +1 ∩ S r n +1 n 6 = ∅ ) ≤ α , sin ce S n +1 = S r n +1 n . A simple wa y to find a safet y area, is to c ho ose δ n +1 , based on a sample of r 1 , r 2 , · · · , r n , s u fficien tly large so that the set K n +1 = ( S δ n +1 n ) c satisfies the conditions i ) and ii ). Let u s note that n ( S δ n +1 n ) c T S r n +1 n 6 = ∅ o and { r n +1 > δ n +1 } are equiv a- len t ev en ts, this is sho we d in the grafic in Figure 1. First w e considere the case where eac h r i dep end s only on ξ i and, as ξ i are i.i.d., then r i = r ( ξ 0 ) . W e call F 0 ( x ) the distribu tion f unction of the 5 radius and we assume that F 0 is con tin uous. In this case, we defin e safet y area, as follo ws. Definition 2.1 A r andom set K n +1 = ( S δ n +1 n ) c is a safety ar e a of leve l α if: i ) δ n +1 is a fun ction of r 1 , r 2 , · · · , r n , ii ) P ( r n +1 > δ n +1 ) ≤ α. W e present the follo wing theorem, whic h p ro vides a safet y area und er the conditions ab o v e. Theorem 2.1 The r andom set K n +1 = ( S δ n +1 n ) c , wher e δ n +1 = min t> 0 { 1 − F n r ( t ) < α − C n } an d C n = E  sup t ∈ R + | F n r ( t ) − F 0 ( t ) |  , defines a safety ar e a of level α . Pro of: W e need to prov e P ( r n +1 > δ n +1 ) ≤ α . T aking into accoun t the fact that r n +1 and δ n +1 are in dep end en t, we ha v e P ( r n +1 > δ n +1 ) = E ( P ( r n +1 > δ n +1 / δ n +1 )) = ∞ Z 0 P ( r n +1 > u ) dP δ n +1 ( u ) = ∞ Z 0 P ( r 0 > u ) dP δ n +1 ( u ) = 1 − ∞ Z 0 F 0 ( u ) dP δ n +1 ( u ) = 1 − E ( F 0 ( δ n +1 )) , where P δ n +1 is the d istr ibution f unction of δ n +1 . W e can wr ite the last expresion as 1 − E ( F 0 ( δ n +1 )) = 1 − E ( F n r ( δ n +1 )) + E ( F n r ( δ n +1 ) − F 0 ( δ n +1 )) , and since 0 ≤ 1 − F n r ( δ n +1 ) < α − C n b y d efinition, we only need to show that | E ( F n r ( δ n +1 ) − F 0 ( δ n +1 )) | ≤ C n . 6 Then, | E ( F n r ( δ n +1 ) − F 0 ( δ n +1 )) | ≤ E | F n r ( δ n +1 ) − F 0 ( δ n +1 ) | ≤ E  sup t ∈ R + | F n r ( t ) − F 0 ( t ) |  = C n , and th is completes the p ro of.  Next w e consider that the radius r i dep end s on ξ i and Y i , so r i = r ( ξ i , Y i ). Let F ( . ; Y ) b e the distribu tion of r ( ξ 0 , Y ), this is F ( t ; Y ) = P ( r ( ξ 0 , Y ) ≤ t ) for any t ≥ 0 , and let us sup p ose that F ( . ; Y ) is contin u ou s . W e also assume that there exist a random pr obabilit y distribu tion F ( x ; ω ) suc h that √ n su p t ∈ R +      1 n n X i =1 F ( t ; Y i ( ω )) − F ( t ; ω )      → n →∞ 0 a.s. (5) In this case, we can not calculate the safety area in a s tr aigh t wa y , then w e define it using the limit distrib ution in (5), as follo ws. Definition 2.2 A r ando m set K n +1 = ( S δ n +1 n ) c is a safety ar e a of level ( ǫ, α ) if: i ) δ n +1 is a fun ction of r 1 , r 2 , · · · , r n , ii ) lim sup n P (1 − F ( δ n +1 ; ω ) > ǫ ) ≤ α. Remark 2.1 If U 1 , · · · , U n ar e indep endent r ando m variables, uniformly distribute d on [0 , 1] , it is wel l known that sup t ∈ R + | F n U − t | , has the law of the Kolmo gor ov-Smirnov statistic sup t ∈ R + | F n X − F ( t ) | for any X 1 , · · · , X n indep en- dent with a c ommon distribution function F ( t ) . If F ( t ) is c ontinuous, by Donsker Invarianc e Principle, √ n E  sup t ∈ R + | F n X ( t ) − F ( t ) |  → n →∞ E sup t ∈ [0 , 1] | b ( F ( t )) | ! , donde b ( t ) es e l puente Br owniano. T o pro v e the main result of this rep ort, w e w ill u se the follo wing theorem whic h pro of is in the App endix. 7 Theorem 2.2 Supp ose the r andom variables X 1 , · · · , X n ar e i ndep e ndent with c ontinuous distribution function F 1 , · · · , F n and such that i ) 1 n n P i =1 F i ( t ) → n →∞ F ( t ) . ii ) 1 n n P i =1 F i ( s ) (1 − F i ( t )) → n →∞ G ( s, t ) , p ositive and symmetric al function. iii ) lim sup 1 n n P i =1 w i ( δ ) → δ → 0 + 0 , wher e w i ( δ ) is the mo dulus of c ontinuity of F i ( t ) . Then the r ando m variables U n define d by U n ( t ) = √ n F n X ( t ) − 1 n n X i =1 F i ( t ) ! , satisfy U n w = ⇒ U , wher e U is the c enter e d Gaussian pr o c ess with c ovarienc e given b y E { U ( s ) U ( t ) } = G ( s, t ) . W e assum e in adition, that for a p ath fixed y = ( y 1 , · · · , y n , · · · ) ∈ R N , Z R N C ( δ, y ) dP Y ( y ) → 0 as δ → 0 + , where C ( δ, y ) = lim sup n 1 n n P i =1 w ( F ( . ; y i ) , δ ) . Under the conditions enumerated ab o v e, the next theorem p r o vides a safet y area for this case. Theorem 2.3 The r andom set K n +1 = ( S δ n +1 n ) c is a safety ar e a of level ( α, ǫ ) , taking δ n +1 = min t> 0  1 n n P i =1 1 I { r i >t } < ǫ − C α √ n  , and C α verifying that P sup t ∈ [0 , 1] | U ( t ) | ≥ C α ! = α . Pro of: T o pro ve that K n +1 is a safet y area, we need to sho w ( ii ) in Definition 2.2. As P ( | 1 − F ( δ n +1 , . ) | > ǫ ) ≤ P      1 n n X i =1  1 I { r i ≤ δ n +1 } − F ( δ n +1 , . )       > C α √ n ! 8 + P      1 − 1 n n X i =1 1 I { r i ≤ δ n +1 }      > ǫ − C α √ n ! , from the definition of δ n +1 holds that the second term is equal to 0, then it is enough to p r o v e that lim n P sup t ∈ R +      1 n n X i =1  1 I { r i ≤ t } − F ( t ; ω )       ≥ C α √ n ! ≤ α. (6) By (5) and applying Lemma(3.2) in App end ix, there exits a sequence of p ositiv e num b ers { a n } s u c h th at a n ↓ 0 + and P sup t ∈ R +      1 n n X i =1 F ( t ; y i ) − F ( t ; ω )      > a n √ n ! → n →∞ 0 . (7) The argum en t of the limits in (6) can b e b oun ded in this wa y P sup t ∈ R +      1 n n X i =1  1 I { r i ≤ t } − F ( t ; ω )       ≥ C α √ n ! ≤ P sup t ∈ R +      1 n n X i =1 ≤  1 I { r i ≤ t } − F ( t ; y i )       ≥ C α − a n √ n ! + P sup t ∈ R +      1 n n X i =1 F ( t ; y i ) − F ( t ; ω )      > a n √ n ! , (8) and th e second term in the last equation tends to 0, then we need to show that th e fi rst term tends to α . Let u s fix a p ath y = ( y 1 , · · · , y n , · · · ) ∈ R N , and d enote r ′ i = r ( ξ i , y i ) with i = 1 , · · · , n . Hence the rad iu s r ′ i are indep end en t b ecause they d ep end only on the ξ i , then by an Inv ariance Principle applied to the T heorem 3 √ n su p t ∈ R +      1 n n X i =1  1 I { r ′ i ≤ t } − F ( t ; y i )       w = ⇒ sup t ∈ [0 , 1] | U ( t ) | , where U ( t ) is a Gaussian cen tered pro cess with autocov ariance function G ( s, t ). By the assumption ab out C α , we ha v e lim n P sup t ∈ R +      1 n n X i =1  1 I { r ′ i ≤ t } − F ( t ; y i )       ≥ C α √ n ! = α. 9 This expression can b e written as lim n P sup t ∈ R +      1 n n X i =1  1 I { r i ≤ t } − F ( t ; y i )       ≥ C α √ n / Y = y ! = α, for all y ∈ R N b ecause the d istribution of r ′ 1 , · · · , r ′ n , · · · is ju st the same of r 1 , · · · , r n , · · · conditioned to Y = y . Then P sup t ∈ R +      1 n n X i =1  1 I { r i ≤ t } − F ( t ; y i )       ≥ C α √ n ! = Z R N P sup t ∈ R +      1 n n X i =1  1 I { r i ≤ t } − F ( t ; y i )       ≥ C α √ n / Y = y ! dP Y ( y ) . By Dominated Con v ergence Theorem, as the integ rand tend s to α and it is b ound ed b et w een 0 and 1, we ha v e that lim n P sup t ∈ R +      1 n n X i =1  1 I { r i ≤ t } − F ( t ; y i )       ≥ C α √ n ! = α. (9) As α is an y real num b er b et w een 0 and 1, we ha v e ju st pr o v ed that the equalit y lim n P sup t ∈ R +      1 n n X i =1  1 I { r i ≤ t } − F ( t ; y i )       ≥ v √ n ! = P sup t ∈ [0 , 1] | U t | ≥ v ! , (10) is v alid for an y real v . If G ( t ) is the distribu tion of the sup reme of a Gaussian pr o cess U , by Lemma (3.3) in App endix, sup v ∈ R      P √ n su p t ∈ R +      1 n n X i =1  1 I { r i ≤ t } − F ( t ; ω )       ≥ v ! − (1 − G ( v ))      → n →∞ 0 . Then b y (10) lim n P √ n su p t ∈ R +      1 n n X i =1  1 I { r i ≤ t } − F ( t ; ω )       ≥ C α − a n ! = lim n P sup t ∈ [0 , 1] | U ( t ) | ≥ C α − a n ! = P sup t ∈ [0 , 1] | U ( t ) | ≥ C α ! = α. Applying (7) and the last equalities in (8), (6) follo ws.  10 3 App endix Theorem 3.1 Supp ose the r andom variables X 1 , · · · , X n ar e i ndep e ndent with c ontinuous distribution function F 1 , · · · , F n and such that i ) 1 n n P i =1 F i ( t ) → n →∞ F ( t ) . ii ) 1 n n P i =1 F i ( s ) (1 − F i ( t )) → n →∞ G ( s, t ) , p ositive and symmetric al function. iii ) lim sup 1 n n P i =1 w i ( δ ) → δ → 0 + 0 , wher e w i ( δ ) is the mo dulus of c ontinuity of F i ( t ) . Then the r ando m variables U n define d by U n ( t ) = √ n F n X ( t ) − 1 n n X i =1 F i ( t ) ! , satisfy U n w = ⇒ U , wher e U is the c enter e d Gaussian pr o c ess with c ovarienc e given b y E { U ( s ) U ( t ) } = G ( s, t ) . Pro of: W e will derive this result by using the theory of we ek con v ergence in the space of con tinou os functions C . Although U n ( t ) is a function on [0 , 1] pro du ced at random, it is n ot an elemen t of C , b eing ob viously discont inuous. Here we shall circumv ent the d iscontin u it y p roblems by adop tin g a d iferen t definition of empirical d istribution function. Let us define X (0) = −∞ , X ( n +1) = + ∞ and X (1) , · · · , X ( n ) are the v alues X 1 , · · · , X n ranged in increasing order, and let G n ( t ) b e th e d is- tribution function corresp ond ing to an u niform d istribution of mass ( n + 1) − 1 o v er the interv als ( X ( i − 1) , X ( i ) ], for i = 2 , · · · , n and for the interv als ( −∞ ; X (1) ] y ( X ( n ) ; + ∞ ) w e assign the exp onentia l distribu tion E ( − ln ( n +1) X (1) ) and E ( ln ( n +1) X ( n ) ) resp ectiv ely . Then | F n X ( t ) − G n ( t ) | ≤ 1 n , t ∈ R . (11) No w let Z n b e the element of C with v alue at t Z n ( t ) = √ n G n ( t ) − 1 n n X i =1 F i ( t ) ! , (12) 11 and by (11) we ha v e sup t | U n ( t ) − Z n ( t ) | ≤ 1 √ n . (13) W e really analyze U n , replacing U n b y Z n only in order to sta y in C , so w e will pro v e Z n w − → U. (14) W e will sh ow first that the fi nite-dimensional distribu tions of Z n con v erge to those of U . Consid er a single time p oin t t , w e w rite U n ( t ) = n P i =1 ϕ i ( t ), where ϕ i ( t ) = 1 √ n  1 I { X i ≤ t } − F i ( t )  . Since { ϕ i } are a sequence of centered, indep end en t r an d om v ariables with v ariance σ 2 i = 1 n F i ( t ) (1 − F i ( t )) and suc h that | ϕ i | ≤ 1 √ n , for all i , by Chebyshev’s inequalit y follo ws that X k | ϕ i | ≥ ǫ S n k 2 P ( ϕ i = k ) ≤ 1 n P { | ϕ i | ≥ ǫ S n } ≤ 1 n v ar( ϕ i ) ǫ 2 S 2 n where S 2 n = 1 n n P i =1 F i ( t ) (1 − F i ( t )) that h as a finite limit, G ( t, t ). Hence 1 S 2 n n X i =1 X k | ϕ i | ≥ ǫ S n k 2 P ( ϕ i = k ) ≤ 1 n ǫ 2 S 2 n → 0 as n → ∞ , and th e Lindeb er g cond ition is satisfied, then follo ws th at U n ( t ) w − → N (0 , G ( t, t )) . No w consider t w o time p oin t s and t , w ith s < t . W e must prov e a U n ( s ) + b U n ( t ) w − → N  0 , a 2 G ( s, s ) + b 2 G ( t, t ) + 2 abG ( s, t )  , for all a, b ∈ R . 12 But a U n ( s ) + b U n ( t ) = n P i =1 ( aϕ i ( s ) + bϕ i ( t )), and we denote S 2 n the v ari- ance of a U n ( s ) + b U n ( t ). Using the fact that { aϕ i ( s ) + bϕ i ( t ) } is a sequence of cen tered, indep en- den t random v ariables and suc h that | aϕ i ( s ) + bϕ i ( t ) | ≤ | a | + | b | √ n , for all i , we see th at 1 S 2 n n X i =1 X k | ϕ i | ≥ ǫ S n k 2 P ( ϕ i = k ) ≤ ( | a | + | b | ) 2 n ǫ 2 S 2 n → 0 as n → ∞ , and th e Lindeb er g cond ition is satisfied. Since E { U n ( s ) , U n ( t ) } = 1 n n P i =1 F i ( s ) (1 − F i ( t )), ( U n ( s ) , U n ( t )) conv erge in law to a cen tered, biv ariate n orm al such that E { U ( s ) , U ( t ) } = G ( s, t ) . A s et of thr ee or more p oints can b e tr eated in th e s ame wa y , and hence the finite dimensional distrib u tion of U n and, by (13), the finite-dimensional distributions of th e Z n con v erge prop erly . If we p ro v e that { Z n } is tigh t, (14) w ill f ollo w. T o pro v e the tightness of { Z n } it is en ough to show that f or eac h p ositiv e ǫ and η there exists δ , 0 ≤ δ ≤ 1 and η ≥ η 0 suc h that P ( sup t ≤ s ≤ t + δ | Z n ( s ) − Z n ( t ) | ≥ ǫ ) ≤ δ η , b y (13) we n eed to p ro v e that P ( sup t ≤ s ≤ t + δ | U n ( s ) − U n ( t ) | ≥ ǫ ) ≤ δ η . (15) F or this pur p ose w e find suc h b ound b y findin g b ounds, u nder fairly general conditions, for the distribution of th e maximun of certain partial sums in the follo wing w a y: le t ξ 1 , ξ 2 , · · · , ξ m b e rand om v ariables; they need not b e indep end en t or iden tically d istributed. Let S k = ξ 1 + ξ 2 + · · · + ξ k ( S 0 = 0), and p ut M m = max 0 ≤ k ≤ m | S k | . W e sh all obtain upp er b oun ds for P { M m ≥ ǫ } by an indirect app roac h . If M ′ m = max 0 ≤ k ≤ m min {| S k | , | S m − S k |} , 13 then M ′ m ≤ M m , it is easy to chec k that M m ≤ M ′ m + | S m | , and th erefore P { M m ≥ ǫ } ≤ P n M ′ m ≥ ǫ 2 o + P n | S m | ≥ ǫ 2 o . (16) If we find separate b oun ds for the terms on the right in (16), we sh all hav e a b ound for the term on the left. W e get a b oun d for the first term via the follo wing lemma: Lemma 3.1 L et us sup ose tha t ther e exists nonne gative numb ers u 1 , · · · , u m such that E {| S j − S i | γ | S k − S j | γ } ≤   X i ≤ l ≤ j u l   α   X j ≤ l ≤ k u l   α , 0 ≤ i ≤ j ≤ k ≤ m wher e γ ar e p ositive and α ≥ 1 2 , then, for al l p ositive ǫ , P  M ′ m ≥ λ  ≤ K γ ,α λ 2 γ ( u 1 + u 2 + · · · + u m ) 2 α , wher e K γ ,α is a c onstant dep e nding only on γ and α . F or a pro of of the this lemma, s ee Billingsley (1968, p.89). No w for a fixed δ w e consider the random v ariables ξ i = U n ( t + i m δ ) − U n ( t + i − 1 m δ ), i = 1 , · · · , m and for γ = 2 y α = 1 let us see that: E {| S j − S i | 2 | S k − S j | 2 } ≤   X i ≤ l ≤ j u l     X j ≤ l ≤ k u l   , 0 ≤ i ≤ j ≤ k ≤ m. (17) As | S j − S i | 2 = | U n ( t + j m δ ) − U n ( t + i m δ ) | 2 and | S k − S j | 2 = | U n ( t + k m δ ) − U n ( t + j m δ ) | 2 then E (     U n ( t + j m δ ) − U n ( t + i m δ )     2     U n ( t + k m δ ) − U n ( t + j m δ )     2 ) = 1 n 2 E    n X h =1 α h ! 2 n X h =1 β h ! 2    . 14 F or short we call ∆ h = F h ( t + j m δ ) − F h ( t + i m δ ) and γ h = F h ( t + k m δ ) − F h ( t + j m δ ) then α h tak es th e v alue 1 − ∆ h with probabilit y ∆ h if X h ∈ ( t + i m δ ; t + j m δ ], or − ∆ h with p r obabilit y 1 − ∆ h else. In the same w a y β h tak es the v alue 1 − γ h with p robabilit y γ h if X h ∈ ( t + j m δ ; t + k m δ ], or − γ h with pr obabilit y 1 − γ h else. Since the X h are indep enden t s o are the random v ector ( α h , β h ). Now E ( α h ) = E ( β h ) = 0 so (18) is equiv alen t to 1 n 2    n X h =1 E  α 2 h β 2 h  + X l 6 = h E  α 2 h  E  β 2 l  + X l ≥ h E ( α h β h ) E ( α l β l )    , but E ( α 2 h ) = (1 − ∆ h ) 2 ∆ h + ∆ 2 h (1 − ∆ h ) ≤ ∆ h , E ( β 2 h ) = (1 − γ h ) 2 γ h + γ 2 h (1 − γ h ) ≤ γ h , E ( α h β h ) = (1 − ∆ h )( − γ h )∆ h − ∆ h (1 − γ h ) + ∆ h γ h (1 − γ h − ∆ h ) = − ∆ h γ h , E ( α 2 h β 2 h ) = (1 − ∆ h ) 2 γ 2 h ∆ h + ∆ 2 h (1 − γ h ) 2 γ h + ∆ 2 h γ 2 h (1 − γ h − ∆ h ) ≤ 2∆ h γ h , then 1 n 2 E    n X h =1 α h ! 2 n X h =1 β h ! 2    ≤ 1 n 2    n X h =1 2∆ h γ h + X l 6 = h ∆ h γ l + X l 6 = h ∆ h γ h ∆ l γ l    ≤ 2 n X h =1 ∆ h n n X l =1 γ l n = j X r = i +1 u r k X r = j +1 u r , where u r = √ 2 n n P h =1  F h ( t + r m δ ) − F h ( t + r − 1 m δ )  . Then (17) follo w s and therefore P ( M ′ m ≥ ǫ 2 ) ≤ 2 4 K ǫ 4 " √ 2 n n X h =1 ( F h ( t + δ ) − F h ( t )) # 2 ≤ 2 5 K ǫ 4 " 1 n n X h =1 w h ( δ ) # 2 ≤ 2 5 K ǫ 4 C 2 ( δ ) , 15 where C ( δ ) = lim sup n 1 n n P h =1 w h ( δ ) and K = K 2 , 1 . F r om this and (16) we ha v e P ( M m ≥ ǫ ) ≤ 2 5 K ǫ 4 C 2 ( δ ) + P n | U n ( t + δ ) − U n ( t ) | ≥ ǫ 2 o . (18) No w, for eac h ω , U n ( s, ω ) is right-c on tin uous in s. As m → ∞ , therefore, M m con v erges to sup t ≤ s ≤ t + δ | U n ( s, ω ) − U n ( t, ω ) | for eac h ω . Hence (18) implies P ( sup t ≤ s ≤ t + δ | U n ( s ) − U n ( t ) | ≥ ǫ ) ≤ 2 5 K ǫ 4 C 2 ( δ )+ P n | U n ( t + δ ) − U n ( t ) | ≥ ǫ 2 o . (19) Because of the asymp totic n ormalit y of ( U n ( t + δ ) , U n ( t )) for fixed t and δ , U n ( t + δ ) − U n ( t ) w − → σ 2 U N , as n → ∞ , with σ 2 U = ( F ( t + δ ) − F ( t )) (1 − ( F ( t + δ ) − F ( t ))) , and then P n | U n ( t + δ ) − U n ( t ) | ≥ ǫ 2 o → P  N ≥ ǫ 2 σ U  as n → ∞ . It is easy to c hec k that σ 2 U ≤ C ( δ ), then P  N ≥ ǫ 2 σ U  ≤ 2 4 σ 4 U ǫ 2 E  N 4  ≤ 2 4 3 C 2 ( δ ) ǫ 4 . No w, for n exceding s ome n δ P n | U n ( t + δ ) − U n ( t ) | ≥ ǫ 2 o < 2 4 3 C 2 ( δ ) ǫ 4 . Hence by (19) P ( sup t ≤ s ≤ t + δ | U n ( s ) − U n ( t ) | ≥ ǫ ) ≤ 2 4 3 ǫ 4 ( K + 1) C 2 ( δ ) . Giv en ǫ and η , c hose δ , since lim δ ↓ 0 C ( δ ) = 0, s o that 2 4 3 ǫ 4 ( K + 1) C 2 ( δ ) ≤ δ η . F or n ≥ n δ , follo ws th at P ( sup t ≤ s ≤ t + δ | U n ( s ) − U n ( t ) | ≥ ǫ ) ≤ δ η , this complete the p ro of.  16 Lemma 3.2 L et { Z n } b e a se quenc e of r andom variables that satisfies Z n → n →∞ 0 a.s. , then ther e exists a se q uenc e of numb ers { a n } that satisfies a n ↓ 0 + such that P ( | Z n | > a n ) → n →∞ 0 Pro of: Let a 0 b e an y r eal num b er, and let u s define a n = a 0 . As P ( | Z n | > a 0 2 ) → n →∞ 0. There exists n 0 suc h th at P ( | Z n | > a 0 2 ) < 1 2 , f or all n > n 0 . So no w we defi n e a n = a 0 2 from n 0 on and as P ( | Z n | > a 0 4 ) → n →∞ 0, there exists n 1 suc h that P ( | Z n | > a 0 4 ) < 1 4 , for all n > n 1 . So no w w e define a n = a 0 4 from n 1 on. The s equ ence a n is defined as a n = a 0 if n ≤ n 0 , a n = a 0 2 if n 0 < n ≤ n 1 , a n = a 0 4 if n 1 < n ≤ n 2 , and s o on. It is clear th at a n ↓ 0 + and P ( | Z n | > a n ) → n →∞ 0  Lemma 3.3 L et f n , f b e r e al f unctions su ch that: i ) f n , f ar e monotono us functions for al l n , ii ) f n ( x ) → n f ( x ) , f or al l x ∈ R , iii ) f n (+ ∞ ) → n f (+ ∞ ) , f n ( −∞ ) → n f ( −∞ ) , for al l n , iv ) f is a c ontinuous and b ounde d function in R , 17 then sup x ∈ R + | f n ( x ) − f ( x ) | → n 0 . Pro of: Without loss of generalit y we assume f n and f increasing. Let us denote: I = f ( −∞ ) = inf x ∈ R + f ( x ) and S = f (+ ∞ ) = sup x ∈ R + f ( x ). Then, for all ǫ > 0, there exists a < b su c h that f ( a ) < I + ǫ 2 , f ( b ) > S − ǫ 2 . As f is conti nuous and b oun ded in [ a, b ] then f is absolutely con tin uous in [ a, b ], so ∃ δ > 0 suc h that if x, y ∈ [ a, b ] , | x − y | < δ then | f ( x ) − f ( y ) | < ǫ 2 . Let u s choose k ∈ N such that δ > 1 k and let us defi ne x 0 = −∞ , x 1 = a, · · · , x i = a + i − 1 k ( b − a ) · · · , x k +1 = b, x k +2 = + ∞ . Let u s fi rst n ote that: if x, y ∈ [ x i , x i +1 ] for some i, | f ( x ) − f ( y ) | < ǫ 2 . (20) W e will consider the follo wing three cases: • If i = 0 then x, y ∈ ( −∞ , a ] so, as f is monotonous, we ha v e: I < f ( x ) ≤ f ( a ) < I + ǫ 2 , I < f ( y ) ≤ f ( a ) < I + ǫ 2 . So | f ( x ) − f ( y ) | = max { f ( x ) , f ( y ) }− min { f ( x ) , f ( y ) } < I + ǫ 2 − I = ǫ 2 . • If i = k + 1 then x, y ∈ [ b, + ∞ ) so, as f is monotonous, we ha v e: S > f ( x ) ≥ f ( b ) > S − ǫ 2 , S > f ( y ) ≥ f ( b ) > S − ǫ 2 So | f ( x ) − f ( y ) | = max { f ( x ) , f ( y ) } − min { f ( x ) , f ( y ) } < ǫ 2 . • If 1 ≤ i ≤ k then x, y ∈ [ a + i − 1 k ( b − a ) , a + i k ( b − a )] th en x, y ∈ [ a, b ] , ( x − y ) ≤ 1 k < δ , so | f ( x ) − f ( y ) | < ǫ 2 . By an other h and, let us note that: ∃ n ( ǫ ) ∈ N , suc h that , for all , n ≥ n ( ǫ ) , max 0 ≤ i ≤ k +2 | f n ( x i ) − f ( x i ) | < ǫ 2 . (21) Precisely , by ii ) an d iii ) f n ( x i ) → n f ( x i ), for all n and 0 ≤ i ≤ k + 2, and as k is fin ite we hav e that max 0 ≤ i ≤ k +1 | f n ( x i ) − f ( x i ) | → n 0 . No w, let u s prov e that th e f ollo win g inequalit y holds from (20) and (21): sup x ∈ R | f n ( x ) − f ( x ) | < ǫ, for all n ≥ n ( ǫ ) . (22) 18 F or th at, let us consider any x ∈ R an d let us note that | f n ( x ) − f ( x ) | < ǫ, for all n ≥ n ( ǫ ) . But, give n x ∈ R , there exits one an d only on e i , with 0 ≤ i ≤ k + 1, such that x ∈ [ x i , x i +1 ] then, by mon otony we ha v e that: f n ( x ) − f ( x ) ≤ f n ( x i +1 ) − f ( x ) = f n ( x i +1 ) − f ( x i +1 ) + f ( x i +1 ) − f ( x ) , so, by (20) and (21) w e ha v e: f n ( x ) − f ( x ) ≤ | f n ( x i +1 ) − f ( x i +1 ) | + | f ( x i +1 ) − f ( x ) | < ǫ 2 + ǫ 2 = ǫ. By an other h and, f ( x ) − f n ( x ) ≤ f ( x ) − f n ( x i ) = f ( x ) − f ( x i ) + f ( x i ) − f n ( x i ) , so, by (20) and (21) w e ha v e: f ( x ) − f n ( x ) ≤ | f ( x ) − f ( x i ) | + | f ( x i ) − f n ( x i ) | < ǫ 2 + ǫ 2 = ǫ. Then, w e ha v e that : f ( x ) − f n ( x ) ≤ ǫ and f n ( x ) − f ( x ) ≤ ǫ, for all n ≥ n ( ǫ ) , so | f n ( x ) − f ( x ) | < ǫ, for all n ≥ n ( ǫ ) and (22) follo ws.  Corollary 3.1 L et { Z n } b e a se quenc e of r andom variables with distribution functions F n . We assume Z n w → n →∞ Z and let F b e the distribution function of Z , c ontinuous, then sup x ∈ R | F n ( x ) − F ( x ) | → n →∞ 0 . Ac kno wledgemen ts The authors expr ess their gratitude to Dr. Go nzalo Perera for highly v alu- able s u ggestions. References [1] Billignsley , P . (1968). Conver genc e of Pr ob ability M e asur es . Wiley , New Y ork. [2] F alc´ on,C. & Pe rera, G. (2000). Fitting me an tr ansition kernels for evo- lution mo dels . (pre-pr in t). 19 [3] Guy on, X.(1995 ) R andom Field on a N etwork: Mo del ling, Statistics, and Applic ations. S pringer-V erlag. [4] P erera, G. (1997). Geometry in Z d and the C en tral Limit Theorem for w eakly dep enden t r amdom fi elds. J. The or et. Pr ob ab. 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