Statistical aspects of birth--and--growth stochastic processes

1 Statistical asp ects of birth–and–gro wth sto c hastic pro cesses Giacomo Aletti, Enea G. Bongiorno, Vincenzo Capasso Department o f Mathematics, Universit y of Milan, via Saldini 50, 10133 Milan Italy giacomo.al etti@mat.u nimi.it bongio@mat .unimi.it vincenzo.c apasso@mat .unimi.it Summary . The pap er considers a particular family of set–v alued sto c hastic pro- cesses mo deling birth–and–grow th pro ces ses. The prop osed setting allow s us to in- vestig ate the nucleation and t he gro wth pro cesses . A d eco mp osi tion theorem is es- tablished to c haracterize the nucleatio n and the gro wth . As a consequence, differen t consisten t set–v alued estimators are studied for growth pro cess. Moreov er, th e nu- cleation process is studied via the hitting function, and a consistent estimator of the nucleati on hitting function is derived. In tr oduction Nucleation and gro wth pro cess es arise in several natural and technolog ical applica- tions (cf. [5, 6] and th e references therein) su c h as, for example, solidification and phase–transition of materials, semiconductor crystal grow th, biomineralization, and DNA replication (cf., e.g., [17]). During the years, several authors studied sto c hastic spatial pro cesses (cf. [10, 23, 31] and references therein) nevertheless they essen tially consider static app ro ac hes mo deling real phenomenon s. F or what concerns the dy- namical p oin t of view, a parametric birth–and–gr owth pr o c ess was studied in [25, 26]. A birth–and–gro wth pro cess is a RaCS family given by Θ t = S n : T n ≤ t Θ t T n ( X n ), for t ∈ R + , where Θ t T n ( X n ) is the RaCS obtained as the evol u tion up to time t > T n of the germ b orn at (random) t ime T n in ( ra ndom) location X n , according t o some gro wth model. A n analytical approac h is often used to mo del birth–and–gro wth pro- cess, in particular it is assumed that the gro wth of a sph eric al nucleus of infinitesimal radius is driven according to a non–negativ e normal velocit y , i.e. for ev ery instan t t , a b order point of the crystal x ∈ ∂ Θ t “gro ws” along the out w ards normal unit (e.g. [3, 4, 8, 16]). In view of the c h o sen framew ork, different parametric and n o n– parametric estimations are prop osed o ver the y ears (cf. [2, 5, 7, 9, 12, 24, 27] and references therein). Note that the existence of the outw ards normal vector imp oses a regulari ty condition on ∂ Θ t (and also on the nucleatio n process: it cannot be a p oi nt pro cess). On th e other hand, it is wel l k no wn th at random sets are particular cases of fuzzy 2 Statistic al asp e cts of set–val ue d c ontinuous t i me st o chastic pr o c esses sets. No w, in the class of all convex fuzzy sets ha ving compact supp o rt, Doob –typ e decomp os ition for sub- and sup er–martingales w as studied (e.g. [13–15, 32]). Nev- ertheless, a more general case (than the conv ex one) has not yet b een considered; surely , in order to do this, the first easiest step is to consider decomp ositio n for random set–v alued p rocesses. After which, the step forw ard, to b e considered in a follo wing pap er, can b e to generalize results of this pap er to birth –and – gro wth fuzzy set–v alue pro cesses. This paper is an attempt t o offer an original approac h based on a p u rely geometric stochastic p oin t of view in order to a voi d regularit y assumptions describing birth– and–gro wth pro cess es. The pioneer w ork [21] studies a growth mod el for a single conv ex crystal based on Minko wski sum, whilst in [1], the authors derive a compu- tationally tractable math ematical mo del of such p rocesses that emp hasi zes th e geo- metric growth of ob jects without regularit y assumptions on the boun dary of cry sta ls. Here, in v ie w of this approac h , we introd uce different set–v alued p ara metric estima- tors of the rate of growth of the pro cess . They arise naturally from a decomp ositio n via Minko wski sum and they are consistent as th e observ ation window exp an d s t o the whole space. On the other h an d , keeping in mind that distribut ions of random closed sets are determined by Choqu et capacity functionals and that the nucleation process cannot be observed directly , the pap er p ro vides an estimation pro cedures of the hitting function of the nuclea tion pro ces s. The article is organized as follow s. Section 1.1 contains p reliminary properties. Sec- tion 1.2 introduces a birth–and–gro wth model for random closed sets as the com bi- nation of t wo set–v alued pro cesses (nucleation and gro wth resp ectiv ely) . F urth er, a decomp os ition theorem is established to chara cterize the nuclea tion and the gro wth. Section 1.3 stu dies different estimators of the gro wth p rocess and correspondent con- sisten t prop erties are prove d. I n Section 1.4, t h e nucleation pro cess is studied via the hitting function, and a consistent estimator of the nucleation hitting function is derived. 1.1 Pr eli minary results Let N , Z , R , R + b e the sets of all non–negative integer, integ er, real and non–negative real n u m b ers resp ectiv ely , and let X = R d . Let F be the family of all closed subsets of X and F ′ = F \ {∅} . The suffixes b , k and c denote bound edness, compactness and conv exity properties resp ectiv ely (e.g. F kc denotes t he family of all compact con vex subsets of X ). F or all A, B ⊆ X and α ∈ R + , let us define A + B = { a + b : a ∈ A, b ∈ B } = S b ∈ B b + A, (Minko wski Sum) , α · A = αA = { αa : a ∈ A } , (Scalar Prod uct) , A ⊖ B = ` A C + B ´ C = T b ∈ B b + A, (Minko wski Subtraction) , ˇ A = {− a : a ∈ A } , (Symmetric Set) , where A C = { x ∈ X : x 6∈ A } is the complementary set of A , x + A means { x } + A (i.e. A translate by vector x ) , and, by defi nitio n, ∅ + A = ∅ = α ∅ . I t is well kn o wn that + is a comm u tativ e and associative op eration with a neutral element but, in general, A ⊆ X does not admit opposite ( cf . [18, 29]) and ⊖ is n ot the inve rse op eration of +. The follo wing relations are useful in the sequel (see [30]): for every A, B , C ⊆ X Nov ember 3, 2018 Giacomo Aletti, Enea G. Bongiorno, Vincenzo Capasso 3 ( A ∪ B ) + C = ( A + C ) ∪ ( B + C ) , if B ⊆ C, A + B ⊆ A + C , ( A ⊖ B ) + ˇ B ⊆ A and ( A + B ) ⊖ ˇ B ⊇ A, ( A ∪ B ) ⊖ C ⊇ ( A ⊖ C ) ∪ ( B ⊖ C ) . In the follo wing, w e shall work with closed sets. In general, if A, B ∈ F then A + B does not belong to F (e.g. , in X = R let A = { n + 1 /n : n > 1 } and B = Z , then { 1 /n = ( n + 1 /n ) + ( − n ) } ⊂ A + B and 1 /n ↓ 0, but 0 6∈ A + B ) . I n view of this fact, we define A ⊕ B = A + B where ( · ) den o tes th e closure in X . It can b e prov ed that, if A ∈ F and B ∈ F k then A + B ∈ F (see [30]). F or any A, B ∈ F ′ the Hausdorff distanc e (or metric ) is defin ed by δ H ( A, B ) = max  sup a ∈ A inf b ∈ B k a − b k X , sup b ∈ B inf a ∈ A k a − b k X ff . A random closed set (RaCS) is a map X d efined on a probabilit y space ( Ω , F , P ) with v alues in F such th at { ω ∈ Ω : X ( ω ) ∩ K 6 = ∅} is measurable for eac h compact set K in X . It can b e pro ved (see [19]) that, if X, X 1 , X 2 are RaCS and if ξ is a measurable real–v alued function, then X 1 ⊕ X 2 , X 1 ⊖ X 2 , ξ X and (Int X ) C are RaCS. Moreo ver, if { X n } n ∈ N is a sequence of RaCS then X = S n ∈ N X n is so. Let X b e a RaCS, then T X ( K ) = P ( X ∩ K 6 = ∅ ), for all K ∈ F k , is its hitting function (or Cho quet c ap acity functional ). The well k no wn Matheron Theorem states that, the probability law P X of any RaCS X is uniquely determined by its hitting function (see [20]) and hence by Q X ( K ) = 1 − T X ( K ). Remark 1.1 ( Se e [ 22].) If b oth X and Y a re R aC S , then, for every K ∈ F k , T X ⊕ Y ( K ) = E ˆ E ˆ T X ` K ⊕ ˇ Y ´ ˛ ˛ Y ˜˜ . Moreo ver, if X, Y are indep endent, then, for every K ∈ F k , T X ∪ Y ( K ) = T X ( K ) + T Y ( K ) − T X ( K ) T Y ( K ) . A RaCS X is stationary if the probability law s of X and X + v coincide for every v ∈ X . Thus, t h e hitting function of a stationary RaCS clearly is in vari ant up to translation T X ( K ) = T X ( K + v ) for each K ∈ F k and an y v ∈ X . A stationary RaCS X is er go dic , if, for all K 1 , K 2 ∈ F , 1 | W n | Z W n Q X (( K 1 + v ) ∪ K 2 ) dv → Q X ( K 1 ) Q X ( K 2 ) , as n → ∞ ; where { W n } n ∈ N is a c onvex aver aging se quenc e of sets in X (see [11]), i.e. eac h { W n } is con vex and compact, W n ⊂ W n +1 for all n ∈ N and sup { r ≥ 0 : B ( x, r ) ⊂ W n for some x ∈ W n } ↑ ∞ , as n → ∞ . Proposition 1.2 Let X , Y be R a CS with Y ∈ F ′ k a.s. and X stationary , t hen X + Y is a stationary R aC S . Moreo ver, if X is ergod i c, then X + Y is so. Nov ember 3, 2018 4 Statistic al asp e cts of set–val ue d c ontinuous t i me st o chastic pr o c esses Proof. Let Z = X + Y , it is a RaCS. Note that T Z ( K ) = E ˆ E ˆ T X ` K + ˇ Y ´ ˛ ˛ Y ˜˜ = E ˆ E ˆ T X ` K + ˇ Y + v ´ ˛ ˛ Y ˜˜ = T Z ( K + v ) , for every K ∈ F k and v ∈ X , then Z = X + Y is stationary . F urther, let us supp ose that X is ergodic, then, by T onelli’s Theorem and by dominated conv ergence theorem, w e obtain Z W n Q Z (( K 1 + v ) ∪ K 2 ) | W n | dv = E » E » 1 | W n | Z W n Q X ((( K 1 + v ) ∪ K 2 ) + ˇ Y ) dv ˛ ˛ ˛ ˛ Y –– → E ˆ E ˆ Q X ( K 1 + ˇ Y ) Q X ( K 2 + ˇ Y ) ˛ ˛ Y ˜˜ = Q Z ( K 1 ) Q Z ( K 2 ) , for every K 1 , K 2 ∈ F k . Hence X + Y is ergo dic.  1.2 A Birt h–a nd–Gro wth pro cess Let ( Ω , F , { F n } n ∈ N , P ) b e a filtered probability sp ace with th e usual prop erties. Let { B n : n ≥ 0 } and { G n : n ≥ 1 } b e tw o families of R aC S such that B n is F n – measurable and G n is F n − 1 –measurable. These pro ces ses rep resent th e birth ( o r nucle ation ) pr o c ess and the gr owth pr o c ess respectively . Th us, let us define recur- sive ly a birth–and–gro wth pro ces s Θ = { Θ n : n ≥ 0 } by Θ n =  ( Θ n − 1 ⊕ G n ) ∪ B n , n ≥ 1 , B 0 , n = 0 . (1.1) Roughly sp eaking, Equ atio n (1.1) means that Θ n is the enlarge ment of Θ n − 1 due to a Minko wski gr owth G n while nucle ation B n occurs. Without loss of general ity let us consider the foll o wing assumption. (A-1) F or every n ≥ 1, 0 ∈ G n . Note that, Assumption (A-1) is equiv alent to Θ n − 1 ⊆ Θ n . In [1], the authors d erive (1.1) from a contin uous time birth–and–gro wth pro cess ; here, in order to make in ference, the discrete t ime case is su fficient. Indeed, a sample of a b ir th–and–gro wth pro ces s is usually a time sequence of pictures that represent process Θ at d ifferent temp oral step; namely Θ n − 1 , Θ n . Thus, in view of (1.1), it is interes ting to inv estigate { G n } and { B n } ; in particular, we sh a ll estimate the maximal grow th G n and the capacity functional of B n . F or the sak e of simplicity , Y , X , G and B will d enote RaCS Θ n , Θ n − 1 , G n and B n respectively (then X ⊆ Y ). Thus, let us consider the follo wing general definition. Definition 1.3 Let Y , X be RaCS with X ⊆ Y . A X –de c omp osition of Y is a couple of RaCS ( G, B ) for whic h Y = ( X ⊕ G ) ∪ B . (1.2) Note that, since we can consider ( G, B ) = ( { 0 } , Y ), there alw ays exists a X – decomp os ition of Y . It can happ en that G and B in (1.2) are not uniqu e. As ex ample, let Y = [0 , 1] and X = { 0 } , then b oth ( G 1 , B 1 ) = ( Y , Y ) and ( G 2 , B 2 ) = ( X, Y ) Nov ember 3, 2018 Giacomo Aletti, Enea G. Bongiorno, Vincenzo Capasso 5 satisfy (1.2). As a consequence, since we can not d istinguish b et w een tw o different decomp os itions, w e shall choose a maximal one according t o the follo wing prop osi- tion. Proposition 1.4 (Se e [30]) Let Y , X b e RaCS with X ⊆ Y . Then G = Y ⊖ ˇ X = { g ∈ X : g + X ⊆ Y } . (1.3) is the greatest R aC S, with respect to set inclusion, such that ( X ⊕ G ) ⊆ Y . Corollary 1.5 The couple ( G = Y ⊖ ˇ X , B = Y ∩ ( X ⊕ G ) C ) is th e m ax -min X – decomp os ition of Y . As a consequ en ce, ( G, B ) is a X –decomp ositio n of Y and for any other X –decomposition of Y , say ( G ′ , B ′ ), then G ′ ⊆ G and B ′ ⊇ B . In other words, if X , G ′ , B ′ are RaCS and Y = ( X ⊕ G ′ ) ∪ B ′ , th en G = Y ⊖ ˇ X ⊇ G ′ and Y = ( X ⊕ G ) ∪ B ′ . Let Θ be as in (1.1). F rom now on, G n denotes Θ n ⊖ ˇ Θ n − 1 that, as a conseq u ence of Assumption (A-1), con tains the origin. Moreo ver, w e shall supp os e (A-2) There exists K ∈ F ′ b such that G n = Θ n ⊖ ˇ Θ n − 1 ⊆ K for eve ry n ∈ N . (A-3) F or every n ≥ 1, ` B n ⊖ ˇ Θ n − 1 ´ = ∅ almost surely . Roughly sp eaking, Assumption (A -2) means t h at pro cess Θ does not grow too “fast” , whilst Assumption (A-3) m eans th at it cannot b orn something that, up to a trans- lation, is larger (or equal) than what there already exists. Let us remark that A ssumption (A-2) implies { G n } ⊂ F ′ k and X ⊕ G n = X + G n , for any RaCS X . 1.3 Est im ators of G On th e one h an d Prop o sition 1.4 giv es a theoretical formula for G , but, on the other hand, in practical cases , data are b ounded by some observ ation window and edge effects may cause problems. Hence, as the standard statistical scheme for spatial processes (e.g. [23]) suggests, we w onder if there exists a consisten t estimator of G as the observ ation window expands to the whole space X . Proposition 1.6 If { W i } i ∈ N ⊂ F ′ ck is a con vex a veraging sequence of sets, then , for any K ∈ F ′ k , X = S i ∈ N W i ⊖ ˇ K . In this case, we say that { W i } i ∈ N expands t o X and w e shall write W i ↑ X . Proof. At first note that X = S i ∈ N Int W i and for an y i ∈ N , W i ⊆ W i +1 . Let x ∈ X and K ∈ F ′ k . Note t hat, x + K ∈ F ′ k is a compact set. Then there exists a finite family of ind i ces I ⊂ N suc h th at, if N = max I , then x + K ⊆ [ j ∈ I Int W j = Int W N . Hence, w e hav e that x ∈ Int W N ⊖ ˇ K ⊆ W N ⊖ ˇ K , i.e., for any x ∈ X , there exists n 0 ∈ N such that x ∈ W n 0 ⊖ ˇ K .  Nov ember 3, 2018 6 Statistic al asp e cts of set–val ue d c ontinuous t i me st o chastic pr o c esses Let W ∈ { W i } i ∈ N b e an observ ation window and let us denote b y Y W and X W , th e (random) observ ation of Y an d X through W , i.e. Y ∩ W and X ∩ W respectively . Let u s consider the estimator of G given by th e maximal X W –decomposition of Y W : b G W = ` Y W ⊖ ˇ X W ´ (1.4) so that X W ⊕ b G W ⊆ Y W ⊆ W . Notice th a t, whenever Y and X are b ounded, then there exists W j ∈ { W i } i ∈ N such that Y ⊆ W j and ˇ X ⊆ W j , hence b G W j = Y ⊖ ˇ X = G . In other words, on th e set { ω ∈ Ω : X ( ω ) , Y ( ω ) boun ded } , th e estimator (1.4) is consisten t b G W i ( Y , X | Y , X b ounded) → G, as W i ↑ X ; otherwise, as we already said, if Y and X are unbounded, edge effects ma y cause problems and the estimator (1.4) is, in general, not consisten t as w e discussed in the follo wing example. Example 1.7 Let X = R 2 , let us consider X = ( { x = 0 } ∪ { y = 0 } ) and Y = X + B (0 , 1) where B (0 , 1) is the closed unit ball centered in the origin. S urely X ⊂ Y , and they are unbound ed. N ote that Y = ( X + G ) for any G such that ( { 0 } × [ − 1 , 1] ∪ [ − 1 , 1] × { 0 } ) ⊆ G ⊆ B (0 , 1). On the other hand , by Prop osi tion 1.4, there ex i sts a un iq ue G that is the greatest set, with resp ect to set inclusion; in this case G = [ − 1 , 1] × [ − 1 , 1]. Let us su p pose 0 ∈ W 0 and let W ∈ { W i } i ∈ N , then, by Eq u atio n (1.4), the estimator of G is b G W = { 0 } 6 = G . This is an edge effect due to th e fact t hat, for every G ′ with { 0 } ⊂ G ′ ⊆ G , it holds ( X W + G ′ ) ∩ W C 6 = ∅ and then X W + G ′ 6⊆ Y W that do es not agree with Proposition 1.4. Edge effects can b e reduced by considering the foll o wing estimators of G b G 1 W = ` Y W ⊖ ˇ X W ⊖ ˇ K ´ ∩ K , (1.5) b G 2 W = “h Y W ∪ “ ∂ + K W X W ”i ⊖ ˇ X W ” ∩ K ; (1.6) where K is given in A ss umption (A-2) and where ` ∂ + K W X W ´ = ( X W + K ) \ W . The role of K will be clarified in Prop ositio n 1.8 where it guarantees the monotonicit y of b G 1 W . Note th at, estimators (1.5) (1.6) are b ounded (i.e. compact) RaCS, moreo ver, if Y and X are b ounded, then b G 1 W j , b G 2 W j even tually coincide with the estimator (1.4); i.e. there exists n 0 such that for all j ≥ n 0 , b G W j = b G 1 W j = b G 2 W j = G . Let us explain ho w b G 1 W and b G 2 W w ork . Estimator b G 1 W is obtained by reducing th e information given by X to th e smaller window W ⊖ ˇ K , whilst Y is observed in W . Then b G 1 W is the grea test sub se t of K , with respect to set inclusion, suc h that X W ⊖ ˇ K + b G 1 W ⊆ Y W (see Prop os ition 1.4). Estimator b G 2 W is obtained by observing X in W (and not W ⊖ ˇ K ), whilst Y is increased (at least) by ` ∂ + K W X W ´ , that is the greatest p ossible set of gro wth for X outside of th e observed window W . Then b G 2 W is th e greatest subset of K , with respect to set inclusion, such that ( X W + b G 2 W ) ∩ W ⊆ Y W , or, alternativel y , X W + b G 2 W ⊆ Y W ′ , where Y W ′ = Y W ∪ ` ∂ + K W X W ´ (see Proposition 1.4 ). Note that by definition of Minko wski Subtraction Nov ember 3, 2018 Giacomo Aletti, Enea G. Bongiorno, Vincenzo Capasso 7 b G 1 W = T x ∈ X W ⊖ ˇ K x + (( − x + K ) ∩ Y W ) , b G 2 W = T x ∈ X W x + (( − x + K ) ∩ Y W ′ ) ; i.e. every x ∈ X W ⊖ ˇ K (resp. x ∈ X W ) “gro ws” at most as ( − x + K ) ∩ Y W (resp. ( − x + K ) ∩ Y W ′ ). Now , we are ready to show the consistency p roperty of b G 1 W i and b G 2 W . In particular, Proposition 1.8 prov es that b G 1 W i decreases, with resp ect to set inclusion, to the theoretical G , whenever W i expands to the whole sp a ce ( W i ↑ X ). Prop osition 1.9 prov es that, for every W ∈ F ′ , b G 2 W is a better estimator than b G 1 W and hence it is a consisten t estimator of G . Proposition 1.8 Let Y , X b e RaCS, let 0 ∈ G = Y ⊖ ˇ X ⊆ K . The follow ing statements hold for b G 1 W : (1) G ⊆ b G 1 W for every W ; (2) b G 1 W 2 ⊆ b G 1 W 1 if W 2 ⊇ W 1 ; (3) If W i ↑ X , then T i ∈ N b G 1 W i = G . Moreo ver, lim i →∞ δ H ( b G 1 W i , G ) = 0 . (1.7) Proof. (1) Since 0 ∈ K , T k ∈ K − k + W = W ⊖ ˇ K ⊆ W and then X W ⊖ ˇ K ⊆ W . Let g ∈ G , t hen g + X ⊆ Y . Since g ∈ K , last inclusion still holds when X and Y are substituted b y X W ⊖ ˇ K and Y W respectively: g + X W ⊖ ˇ K ⊆ Y W . Thus g ∈ b G 1 W follo ws by Definition (1.5) and Prop os ition 1.4. (2) In order to obtain b G 1 W 2 ⊆ b G 1 W 1 , it is su ffici ent to prove that X W 1 ⊖ ˇ K + b G 1 W 2 ⊆ Y W 1 (1.8) since b G 1 W 1 is th e greatest set, with resp ect to set in clu sion, for which the inclusion (1.8) holds. In fact, W 1 ⊖ ˇ K ⊆ ` W 1 ⊖ ˇ K ´ + K ⊆ W 1 ⊆ W 2 , th en X W 1 ⊖ ˇ K ⊆ X W 2 . Let x ∈ X W 1 ⊖ ˇ K = X ∩ ` W 1 ⊖ ˇ K ´ , then x ∈ X W 2 . By definition of b G 1 W 2 , w e have x + b G 1 W 2 ⊆ Y W 2 ⊆ Y . On the other hand, since x ∈ W 1 ⊖ ˇ K and b G 1 W 2 ⊆ K , we ha ve x + b G 1 W 2 ⊆ ` W 1 ⊖ ˇ K ´ + K ⊆ W 1 ; i.e. x + b G 1 W 2 is included both in Y and in W 1 . (3) Since G ⊆ T i ∈ N b G 1 W i , it remains t o prov e that \ i ∈ N b G 1 W i ⊆ G ; i.e. if g ∈ b G 1 W i for eac h i ∈ N , then g ∈ G . T ak e g ∈ T i ∈ N b G 1 W i . By definition of b G 1 W 1 , w e have g + x ∈ Y for all x ∈ X W i ⊖ ˇ K and ∀ i ∈ N . (1.9) Nov ember 3, 2018 8 Statistic al asp e cts of set–val ue d c ontinuous t i me st o chastic pr o c esses By contradiction, assume g 6∈ G . Then g + X 6⊆ Y , i.e. there exists x ∈ X such that ( g + x ) 6∈ Y . On the one hand , Proposition 1.6 implies that there ex ists j ∈ N such that x ∈ W j ⊖ ˇ K . On the other hand, Equation (1.9) implies g + x ∈ Y which is a contra diction. Thus Theorem 1.1.18 in [19] implies (1.7).  Proposition 1.9 F or every W ∈ F ′ , G ⊆ b G 2 W ⊆ b G 1 W . Proof. Let us d ivide th e pro of in tw o parts; in t h e first one w e prove that b G 2 W ⊆ b G 1 W , in the second one that G ⊆ b G 2 W . Let g ∈ b G 2 W and x ∈ X W ⊖ ˇ K . S i nce b G 2 W ⊆ K , we hav e x + g ∈ ` W ⊖ ˇ K ´ + b G 2 W ⊆ ` W ⊖ ˇ K ´ + K ⊆ W ; (1.10) where w e u se prop erties of monotonicit y of th e Minkwoski S ubtraction and Sum. Moreo ver, by definition of b G 2 W , x + g ∈ Y W , or x + g ∈ “ ∂ + K W X W ” ⊆ W C . By ( 1 .10), x + g ∈ Y W . The arbitrary choice of x ∈ X W ⊖ ˇ K completes the first p a rt of the pro of. F or the second part, let g ∈ G and x ∈ X W . By definition of G , x + g ∈ Y . W e ha ve tw o cases: - x + g ∈ W , and therefore x + g ∈ Y W , - x + g 6∈ W . Since x ∈ X W , x + g ∈ ( X W + G ) \ W ⊆ ` ∂ + K W X W ´ .  Corollary 1.10 b G 2 W is consisten t (i.e. b G 2 W ↓ G whenever W ↑ X ). A Gener al Definition of b G 2 W . The follo wing p roposition shows that the esti- mator in (1.6) can be defi ned in an equiv alent w a y by b G 2 W ( Z ) = nh Y W ∪ “ ∂ + K W Z ”i ⊖ ˇ X W o K ; where ` ∂ + K W X ´ in ( 1 .6) is substituted by ` ∂ + K W Z ´ with X W \ ( W ⊖ ˇ K ) ⊆ Z ⊆ W. (1.11) In other w ords, we are sa y in g that, under condition (1.11), b G 2 W ( Z ) do es not d ep end on Z . F rom a computational point of view, this means that Z can b e c hosen in a w ay th at reduces th e computational costs. On the one hand, the best c hoice of Z seems to b e the smallest p ossible set, i.e. Z = X W \ ( W ⊖ ˇ K ) . On the other hand, in order to get X W \ ( W ⊖ ˇ K ) , we h a ve to compute ` W ⊖ ˇ K ´ that may be costly if at least on e b etw een W and K has a “bad shap e” (for instance it is not a rectangular one). Proposition 1.11 If Z 1 , Z 2 ∈ P ′ b oth satisfy condition (1.11), then b G 2 W ( Z 1 ) = b G 2 W ( Z 2 ). Nov ember 3, 2018 Giacomo Aletti, Enea G. Bongiorno, Vincenzo Capasso 9 Fig. 1. 1. W e consider t wo pictures of a simulated birth–and–growth p rocess, at tw o different time instan ts, that in our notations are X and Y . Emp hasi zing the differences, w e rep ort here the magnified pictures of t h e true gro wth used for the sim u la t io n , the compu t ed b G 2 W , b G 1 W and b G 1 W ⊖ ˇ K . Note t hat they agree with Prop o- sition 1.8 and Proposition 1.9 since b G 1 W ⊖ ˇ K ⊇ b G 1 W ⊇ b G 2 W . Proof. It is sufficien t to prov e: (1) Z 1 ⊆ Z 2 implies b G 2 W ( Z 1 ) ⊆ b G 2 W ( Z 2 ); (2) b G 2 W ( W ) ⊆ b G 2 W “ X W \ ( W ⊖ ˇ K ) ” . In fact, (1) and (2) imp ly that b G 2 W ( W ) = b G 2 W “ X W \ ( W ⊖ ˇ K ) ” . At the same time they imply b G 2 W ( Z ) = b G 2 W “ X W \ ( W ⊖ ˇ K ) ” holds for every Z that satisfies (1.11 ); that is the thesis. STEP (1) is a consequence of the follo wing implications Z 1 ⊆ Z 2 ⇒ Z 1 + K ⊆ Z 2 + K, ⇒ Y W ∪ [ ( Z 1 + K ) \ W ] ⊆ Y W ∪ [ ( Z 2 + K ) \ W ] , ⇒ b G 2 W ( Z 1 ) ⊆ b G 2 W ( Z 2 ); where the last one holds since X 1 ⊖ Y ⊆ X 2 ⊖ Y if X 1 ⊆ X 2 (see [30]). Before proving the second step, we show that b G 2 W ( Z ) = b G 2 W “ Z W \ ( W ⊖ ˇ K ) ” for all Z t h at satisfies (1.11). This statement is true if “ Z W \ ( W ⊖ ˇ K ) + K ” \ W and ( Z + K ) \ W are th e same set. Since Minko wski sum is distribu t i ve with resp ect to union, we get Nov ember 3, 2018 10 S tatistic al asp e cts of set–val ue d c ontinuous t i me st o chastic pr o c esses ( Z + K ) \ W = h“ Z W \ ( W ⊖ ˇ K ) ∪ Z W ⊖ ˇ K ” + K i \ W = h“ Z W \ ( W ⊖ ˇ K ) + K ” \ W i ∪ ˆ` Z W ⊖ ˇ K + K ´ \ W ˜ . Then w e hav e to pro ve that ˆ` Z W ⊖ ˇ K + K ´ \ W ˜ = ∅ : ` Z W ⊖ ˇ K + K ´ \ W = ˘ˆ Z ∩ ` W ⊖ ˇ K ´˜ + K ¯ \ W ⊆ ˘ ( Z + K ) ∩ ˆ` W ⊖ ˇ K ´ + K ˜¯ \ W ⊆ [( Z + K ) ∩ W ] \ W = ∅ . STEP (2) . Since b G 2 W ( X W ) = b G 2 W “ X W \ ( W ⊖ ˇ K ) ” , thesis b ecomes b G 2 W ( W ) ⊆ b G 2 W ( X W ). Let g ∈ b G 2 W ( W ) . W e m u st prov e g ∈ b G 2 W ( X W ), i.e. for every x ∈ X W g + x ∈ Y W , or g + x ∈ ( X W + K ) \ W. Since g ∈ b G 2 W ( W ) , for any x ∈ X W w e can h a ve tw o p ossi bilities (a) g + x ∈ Y W , (b) g + x ∈ ( W + K ) \ W . It remains to prove th at (b) implies g + x ∈ ( X W + K ) \ W . I n particular, (b ) implies g + x ∈ W C . At the same time g + x ∈ X W + K , i.e. g + x ∈ ( X W + K ) \ W .  1.4 Hit ting F u nction Asso ciated to B In many practical cases, an observer, through a window W and at tw o different instants, observes the nucleation and gro wth processes namely X and Y . According to Section 1.3 w e can estimate G v i a the consistent estimator b G 2 W or b G 1 W (in t h e follo wing we shall write b G W meaning one of them). F rom the birth–and–gro wth process p oint of view, it is also interesting to test whenever the nuclea tion process B = { B n } n ∈ N is a sp eci fic RaCS (for example a Bo ol ean model or a point p ro - cess). In general, we cannot directly observ e the n –th nucleation B n since it can b e o verlapped by other nuclei or by their evolutions. Nevertheless, we shall infer on the hitting function associated to the nucleation process T B n ( · ). Let us consider the decomp os ition given by (1.2) Y = ( X + G ) ∪ B then the follow ing prop osition is a consequence of Remark 1.1. Proposition 1.12 If ( G, B ) is a X –decomposition of Y suc h that B is indep enden t on X and on G , then, for each K ∈ F k , T Y ( K ) = T X + G ( K ) + T B ( K ) − T X + G ( K ) T B ( K ) , that, in terms of Q · ( K ) = (1 − T · ( K )), is equiv alen t t o Q Y ( K ) = Q B ( K ) Q X + G ( K ) . In other words, the probabilit y for the exploring set K to miss Y is the probability for K to miss B m ultiplied by the probabilit y for K to miss X + G . Nov ember 3, 2018 Giacomo Aletti, Enea G. Bongiorno, Vincenzo Capasso 11 Remark 1.13 W orking with d ata we shall consider tw o estimators of the h itting function (w e refer to [23, p. 57–63] and references th erei n). In particular, if X is a stationary ergo dic RaCS, then T X ( · ) can b e estimated by a single realization of X and t w o empirical estimators are giv en by b T X,W ( K ) = µ λ `` X + ˇ K ´ ∩ ( W ⊖ K 0 ) ´ µ λ ( W ⊖ K 0 ) , K ∈ F k ; where µ λ is the Lebesgue measure on X = R d and K 0 is a compact set suc h that K ⊂ K 0 for all K ∈ F k of in terest. A r e gular close d set in X is a closed set G ∈ F ′ for whic h G = Int G ; i.e. G is the closure (in X ) of its interio r. Proposition 1.14 Le t G ∈ F ′ k b e a regular closed subset in X . Then, for every X ∈ F ′ , X + G is a regular closed set. Proof. S ince X + G is a closed set, Int ( X + G ) ⊆ X + G . It remains to prov e t hat X + G ⊆ Int ( X + G ). Let y ∈ X + G , then there ex is ts x ∈ X and g ∈ G suc h that y = x + g . If g ∈ I n t G , then t h ere exists an op en neighborho od of g for whic h U ( g ) ⊆ Int G and x + U ( g ) is an open n eighborh o od of x + g included in X + G ; i.e. x + g ∈ Int ( X + G ). On the other hand, let g ∈ ∂ G = G \ Int G , then there exists { g n } n ∈ N ⊂ G su c h th at g n → g and g n ∈ Int G , for all n ∈ N . Th u s, for every n ∈ N , x + g n is an in terior point of X + G and x + g n → x + g ∈ Int ( X + G ).  Proposition 1.15 (Se e [23, The or em 4.5 p. 61] and r efer enc es ther ein) Let X b e an ergodic stationary random closed set. I f the random set X is almost surely regular closed sup K ∈ F k K ⊆ K 0 ˛ ˛ ˛ b T X,W ( K ) − T X ( K ) ˛ ˛ ˛ → 0 , a.s. (1.12) as W ↑ X an d for every K 0 ∈ F ′ . Remark 1.16 Proposition 1.14, together to Equation ( 1 .1) means that, if { G n } n ∈ N is a sequence of almost su rel y regular closed sets, t hen { Θ n } n ∈ N is so. The follow ing Theorem sh o ws that the hitting functional Q B of the hidden nucleatio n process can b e exstimated by the observ able quantit y e Q B,W , where for every K ∈ F k , e Q B,W ( K ) := b Q Y ,W ( K ) b Q X + b G W ,W ( K ) , (1.13) and b G W is giv en by (1.5) or (1.6). Nov ember 3, 2018 12 S tatistic al asp e cts of set–val ue d c ontinuous t i me st o chastic pr o c esses Theorem 1. 17 Let X , Y be tw o RaCS a.s. regular closed. Let ( G, B ) b e a X – decomp os ition of Y with B a stationary ergo dic RaCS indep endent on G and X . Assume that G is an a.s. regular closed set and e Q B,W defined in Equ atio n (1.13). Then, for any K ∈ F k , ˛ ˛ ˛ e Q B,W ( K ) − Q B ( K ) ˛ ˛ ˛ − → W ↑ X 0 , a.s. Proof. Let K ∈ F k b e fi xed. F or the sake of simplicit y , Q · , e Q · and b Q · denote Q · ( K ), e Q · ,W ( K ) and b Q · ,W ( K ) resp ectiv ely . Thus, ˛ ˛ ˛ e Q B − Q B ˛ ˛ ˛ = ˛ ˛ ˛ ˛ ˛ ˛ b Q Y b Q X + b G W − Q Y Q X + G ˛ ˛ ˛ ˛ ˛ ˛ = ˛ ˛ ˛ ˛ ˛ ˛ b Q Y Q X + G − Q Y b Q X + b G W b Q X + b G W Q X + G ˛ ˛ ˛ ˛ ˛ ˛ . Since Y ⊇ X + b G W , b Q X + b G W > b Q Y . Accordingly to (1.12), b Q Y conv erges to Q Y that is a p ositive quantit y . Th us, thesis is eq uiv alent to prov e th at ˛ ˛ ˛ b Q Y Q X + G − Q Y b Q X + b G W ˛ ˛ ˛ → 0 , a.s. as W ↑ X . The follo wing inequalities hold ˛ ˛ ˛ b Q Y Q X + G − Q Y b Q X + b G W ˛ ˛ ˛ ≤ Q X + G ˛ ˛ ˛ b Q Y − Q Y ˛ ˛ ˛ + Q Y ˛ ˛ ˛ Q X + G − b Q X + b G W ˛ ˛ ˛ ≤ Q X + G ˛ ˛ ˛ b Q Y − Q Y ˛ ˛ ˛ + Q Y ˛ ˛ ˛ Q X + G − Q X + b G W ˛ ˛ ˛ + Q Y ˛ ˛ ˛ Q X + b G W − b Q X + b G W ˛ ˛ ˛ . Proposition 1.2 and Proposition 1.14 gu arantee that X + G is a stationary ergo d ic RaCS and a.s. regular closed, then w e can apply (1.12) to the first an d the third addends. It remains to pro ve that ˛ ˛ ˛ Q X + G − Q X + b G W ˛ ˛ ˛ → 0 as W ↑ X . (1.14) Since Mink o wski sum is a contin uou s map from F × F k to F (see [30]), b G W ↓ G a.s. implies X + b G W ↓ X + G a.s. A s a consequence, we get that X + b G W ↓ X + G in distribution [28, p. 182], whic h is Equation (1.14 ).  References 1. G. Aletti, E. G. Bongiorno, and V. Capasso, A set–v alued framew ork for birth– and–gro wth processes. 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