The distribution of the number of nodes in the relative interior of the typical I-segment in homogeneous planar anisotropic STIT Tessellations

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1 Comment.Math.Univ.Caroin. 51,3(21) The distribution of the number of nodes in the reative interior of the typica I-segment in homogeneous panar anisotropic STIT Tesseations Christoph Thäe Abstract. A resut about the distribution of the number of nodes in the reative interior of the typica I-segment in homogeneous and isotropic random tesseations stabe under iteration (STIT tesseations) is extended to the anisotropic case using recent findings from Schreiber/Thäe, Typica geometry, second-order properties and centra imit theory for iteration stabe tesseations, arxiv:11.99 [math.pr] (21). Moreover a new expression for the vaues of this probabiity distribution is presented in terms of the Gauss hypergeometric function 2 F 1. Keywords: hypergeometric function, iteration/nesting, random tesseation, segments, stochastic geometry, stochastic stabiity Cassification: Primary 6D5; Secondary 6G55, 52A22 1. Introduction and resut Since their introduction by Nage and Weiss in [4], homogeneous (i.e. spatia stationary) random tesseations that are stabe under the operation of iteration have attracted considerabe interest in stochastic geometry. One of their main features is that they admit an expicit oca and goba construction that aows an interpretation as a random process of ce division. It can roughy be expained as foows: et us fix a ocay finite measure Λ on the space of ines in the pane R 2 that can be written as Λ R, where is the ebesgue measure on the rea axis and R a probabiity measure on the space of directions, i.e. the space of ines through the origin (the non-oriented Grassmannian in the sense of [5]). We use here the standard parameterization as identifying a ine with its direction and its signed distance from the origin. We assume from now on that R is not concentrated on a singe direction (ine), which ensures that our tesseations wi have bounded ces with probabiity one. Beside Λ, we fix some compact convex poygon W R 2 in which our construction is carried out. Now, assign to W an exponentiay distributed random ife time with parameter Λ([W]), where by [W] we mean the coection of a (parameterized) ines in the pane that hit W R 2. Upon expiry of this random ifetime, the ce W dies and spits into two poygona sub-ces W + and W separated by a ine in [W] that is chosen This work was supported by the Swiss Nationa Science Foundation, Grant PP /1.

2 54 C. Thäe Figure 1. States of the random ce division process for different time instants t (, 6]; the respective new segments are dashed according to the aw Λ([W]) 1 Λ( [W]). The resuting two new ces W + and W are again provided independenty of each other with exponentia ifetimes with respective parameters Λ([W + ]) and Λ([W ]) (this ensures that smaer ces ive stochasticay onger) and the entire construction continues independenty and recursivey, unti some deterministic time threshod t > is reached. The ce-separating 1-dimensiona faces arising in subsequent ce spits are usuay referred to as I-segments, assuming shapes simiar to the etter I. This ce division procedure is iustrated in Figure 1 for different time instants t. The resuting random tesseation Y (t, Λ, W) is a homogeneous random tesseation inside W, but it can be shown that its aw is consistent in W, which impies the existence of random tesseation Y (tλ) in the whoe pane with the property that for any W R d we have Y (tλ) W D Y (t, Λ, W), where D stands for equaity in distribution, cf. [4]. Reaizations of Y (tλ) inside a quadratic window W for fixed t and different choices of Λ are shown in Figure 2. We ca the random tesseation Y (tλ) a random STIT tesseation with ine measure tλ, where the construction time t may be interpreted as the edge ength intensity of Y (tλ), i.e. the mean ength of edges per unit area. The abbreviation STIT refers to the characteristic property

3 The distribution of the number of nodes 55 Figure 2. Reaizations of Y (tλ) for fixed t and different directiona distributions R; the uniform distribution for R (eft) and R concentrated with equa weight on the coordinate directions ony (right) of the tesseations of being stabe under iteration, which formay means that Y (tλ) Y (sλ) D Y ((s + t)λ), s, t >, where denotes the operation of iteration, see [4]. The ine measure tλ is sometimes referred to as the generating or the driving measure of the homogeneous random STIT tesseation Y (tλ). We are concerned in this note with the typica I-segment of a homogeneous random STIT tesseation Y (tλ) with driving measure tλ, that is the amost surey uniquey determined I-segment of the tesseation that contains the origin when Y (tλ) is regarded under the Pam distribution with respect to the embedded point process of I-segment midpoints (see [5] for the technica background). Moreover, we can mark each of the midpoints by the direction of the I-segment through the particuar point, which opens the door to the theory of Pam distributions for marked point processes, see e.g. [5, Chapter 3.5]. This technique aows us to make mathematicay precise the notion of the typica I-segment of the tesseation with a particuar direction in the support of R. It is our main objective here to extend a recent resut from [3] concerning the probabiity distribution of the number of nodes in the reative interior of the typica I-segment obtained for homogeneous and isotropic (stochasticay rotation invariant) random STIT tesseations to the anisotropic case, i.e. to genera driving measures tλ satisfying our above formuated assumption. Our resut reads as foows:

4 56 C. Thäe Theorem. The probabiity p(k) that the typica I-segment with direction of a homogeneous panar random STIT tesseation Y (tλ) has k N {, 1, 2,...} nodes in its reative interior does not depend on and t and its precise vaue is given by p(k) 2 k+2 (k + 1)(k + 2)(k + 3) 2 F 1 (k + 1, k + 1; k + 4; 1), where 2 F 1 is the Gauss hypergeometric function (see [1, Chapter 15] or (11) beow). Some particuar vaues of p(k) for sma k are provided in the tabe. k k 1 k 2 exact 8 n n n2 72 numerica k 3 k 4 k 5 exact 34 n n n numerica From the formua for p(k) it can be cacuated that the mean number of points in the reative interior of the typica I-segment and of the typica I-segment with direction equas 2. This equaity was aready observed in [7] and was a first indication for the distributiona equaity provided by the Theorem. It can further be shown that no higher moments exist. The proof we give here wi be based on recent findings from [6] obtained in the framework of martingae theory, which fis the gap from [3] of a missing Sivnyak-type resut for I-segments. In fact, the martingae techniques from [6] yied a Sivnyak theorem for I-segments that hods in construction time t of the tesseation. This resut pays an important rôe in the background of the proof given beow. As a byproduct, we are aso abe to provide an aternative formua to that from [3] for the probabiities p(k) in terms of the Gauss hypergeometric function 2 F 1. Our methods provide information on anisotropic random STIT tesseations, namey that the probabiity distribution of the number of nodes in the reative interior of the typica I-segment in direction is the same for a directions. This might seem impausibe at first sight if one thinks for exampe of a directiona distribution R that is concentrated on two different directions 1 and 2, 1 with weight ǫ > and 2 with weight 1 ǫ. However, intuitivey one coud argue as foows: I-segments with direction 1 are rare and short. But they are hit by a arge number of I-segments with direction 2. On the other hand, I-segments

5 The distribution of the number of nodes 57 with direction 2 appear more often and they are comparativey ong. But they are sedom hit by I-segments with direction 1. Surprisingy, these two effects are baancing and ead to the independence of the crucia distribution from the direction of the segment. Another way to make the resut pausibe is to say that the distribution of the number of nodes in the reative interior of the typica I-segment in direction cannot depend on its ength, because in contrast to the ength, the number of nodes does not change under diations of the tesseation. But the ength depends on the direction, so the number of nodes and the direction coud be independent, too. 2. Proof of the Theorem et Y (tλ) be a homogeneous random STIT with driving measure tλ, t >, where Λ has the product structure Λ R with R as above. Denote the random segment process of I-segments of Y (tλ) by X I and for each I-segment s X I, by β(s) (, t] its birth time that it gets in the spatio-tempora construction of the tesseation. We consider now the birth time augmented process ˆX I : {(s, β(s)) : s X I } and denote by ˆQ Y (tλ) the distribution of the birth-time marked typica I-segment in the sense of [5, Chapter 4.1]. From the genera theory presented in [5, Chapter 3] Y (tλ) we deduce that there exists a reguar famiy of distributions ˆQ,, such that (1) ˆQY (tλ) ˆQ Y (tλ) R(d). ˆQ Y (tλ) We may interpret a random segment with distribution as the typica birthtime marked I-segment of Y (tλ) with direction. We regard now a homogeneous Poisson ine tesseation PT(sΛ) in the pane with ine measure sλ and < s < t. Then we can consider simiary to what we did with the I-segments above the distribution Q PT(sΛ) of the typica edge of the Poisson ine tesseation and the famiy (Q PT(sΛ) ) of distributions of random segments that may be interpreted as typica edges of PT(sΛ) with direction. Having these concepts in mind, we reca Equation (33) from [6], which states that the distribution ˆQ Y (tλ) of the typica birth-time marked I-segment of Y (tλ) is a mixture of distributions ˆQ PT(sΛ), < s < t, and the mixing distribution is a beta-distribution on [, t] with parameters 2 and 1. More formay, we have ˆQ Y (tλ) t t 2 [QPT(sΛ) δ s ] ds, where δ s stands for the unit mass Dirac measure concentrated at s (, t]. Using now (1) and its equivaent for the Poisson ine tesseations, we arrive at

6 58 C. Thäe (2) f(z)ˆq Y (tλ) (dz)r(d) t t 2 f(z)(q PT(sΛ) δ s )(dz)dsr(d), for any non-negative measurabe function f on the space of birth-time marked ine segments. Above, we have constructed from the distribution Q PT(sΛ) a famiy of distributions Q PT(sΛ) theory that for each Q PT(sΛ). Simiary to that, we can infer once again from the genera there exists a reguar famiy Q PT(sΛ),x, such that Q PT(sΛ) (, ) Q PT(sΛ),x λ,s (dx), where by λ,s we mean the ength distribution of a random ine segment with distribution Q PT(sΛ). Consequenty, (2) can be written as f(z)ˆq Y (tλ) (dz)r(d) (3) t t 2 f(z)(q PT(sΛ),x δ s )(dz)λ,s (dx)dsr(d). (, ) However, it is we known that λ,s has a density p,s (x) with respect to the ebesgue measure on the positive rea haf-axis (see the remarks in [5] or equation (6) beow). To simpify the notations, we denote by I a random birth-time marked segment with distribution segment with distribution Q PT(sΛ),x (4) Ef(I )R(d) t t 2 ˆQ Y (tλ) and by Z,s,x the random marked ine δ s. Then (3) can be rewritten as p,s (x) Ef(Z,s,x )dxdsr(d). We define now the foowing specia function F on the space of birth-time marked ine segments by F(Z) : 1 [in the reative interior of Z there appear after its birth time and unti time t exacty k nodes] with k N {, 1, 2,...}. Furthermore, we introduce the abbreviations p(k ) : EF(I ) and p(z,s,x, k) : EF(Z,s,x ).

7 The distribution of the number of nodes 59 Then (4) with f repaced by F h() with an arbitrary non-negative measurabe function h() on reads whence h()p(k )R(d) (5) p(k ) t t h() t 2 t 2 p s, (x)p(z,s,x, k)dxdsr(d), p s, (x)p(z,s,x, k)dxds foows. In order to cacuate p(k ), it remains to determine the ength density p,s (x) with respect to the ebesgue measure on the positive rea haf-axis and the counting density p(z,s,x, k). To this end, notice at first that it is we known from the theory of Poisson ine tesseations that the ength distribution of Z,s is an exponentia distribution with parameter sλ([e()]), where e() denotes a segment of unit ength on (see for exampe the remarks on Gamma-type resuts for Poisson based random tesseation in [5, Chapter 1] or [6] and the references given therein). Thus, we have (6) p,s (x) sλ([e()])e sλ([e()])x. It remains to determine the counting density p(z,s,x, k). To this end, we make the foowing considerations: Note, that at time s, the random segment Z,s has by definition the same distribution as the typica marked I-segment with direction that has birth-time s. Furthermore, at the time of its birth s (, t], such an I-segment does not contain any node in its reative interior due to the construction, whence the nodes can ony appear during the remaining time interva (s, t]. Reca now that homogeneous random STIT tesseations have the foowing section property (cf. [2] and the references cited therein): The intersection of Y (tλ) with any ine g induces a homogeneous Poisson point process on g with intensity tλ([e()]), where is the unique ine parae to g. Moreover, observe that from the iteration stabiity of Y (tλ) it foows (7) Y (tλ) D Y (sλ) Y ((t s)λ). The ast equation says that both resuts are the same, either to continue the random ce division process from time s unti t or to stop the process at time s and to perform an iteration of Y (sλ) with tesseations distributed as Y ((t s)λ). We consider the second mentioned possibiity and regard the effect of the iteration (7) on each of two the sides of the I-segment Z,s. On each side of Z,s, an iteration is performed and it induces on the ine containing the segment a homogeneous Poisson point process with intensity (t s)λ([e()]), because of the section property for STIT tesseations mentioned afore. This is because iteration

8 51 C. Thäe with Y ((t s)λ) is for the considered ine the same as intersection with the STIT tesseation Y ((t s)λ). The same does independenty aso hod for the other side of the segment. As a resut, there appears on the ine containing the I-segment Z,s the superposition of two independent and homogeneous Poisson point processes, which is again a homogeneous Poisson point process but with intensity 2(t s)λ([e()]). Thus, in the reative interior of the random segment Z,s,x, which is born without an inner structure, there appears a random number of nodes that is Poisson distributed with parameter 2(t s)λ([e()])x, i.e. (8) p(z,s,x, k) (2(t s)λ([e()])x)k e 2(t s)λ([e()])x, k N, k! because we have conditioned additionay on its ength x (, ). Thus, we obtain from (5) in view of (6) and (8) the foowing identity for the probabiity p(k ): p(k ) t t 2 sλ([e()])e sλ([e()])x (2(t s)λ([e()])x)k e 2(t s)λ([e()])x dxds. k! The ast expression can further be evauated and we obtain (9) p(k ) t t t t (2(t s)λ([e()])) k t 2 sλ([e()]) k! (2(t s)λ([e()])) k t 2 sλ([e()]) k! 2 (2(t s)) k Λ([e()])Λ([e()]) k t 2 (2t s) k+1 Λ([e()]) k+1 ds 2 (2(t s)) k t 2 ds (2t s) k+1 x k e Λ([e()])(2t s)x dxds k! ds (Λ([e()])(2t s)) k+1 which is independent of the direction. This impies that the probabiity p(k ) for the typica I-segment with a particuar direction and p(k )R(d) for the typica I-segment itsef coincide. For this reason we wi write from now on p(k) instead of p(k ). To proceed, we appy the substitution a 1 s/t, which eads to (1) (9) 1 which is independent of t. 2((1 a)t) 2 t k+1 (1 a) 2 a k (2(t (1 a)t)) k t da (2t (1 a)t) k+1 da p(k), (1 + a) k+1

9 The distribution of the number of nodes 511 We can derive from the ast formua an aternative expression for the vaues p(k), i.e. for the probabiity that the number of nodes in the reative interior of the typica I-segment equas k N. Reca for that purpose that the Gauss hypergeometric function 2 F 1 (u, v; w; z), u, v, w >, z R is defined by (11) 2F 1 (u, v; w; z) (u) n (v) n (w) n n Γ(w) Γ(u)Γ(v) n zn n! Γ(u + n)γ(v + n) Γ(w + n) zn n!, where (x) n, n N, is the Pochhammer symbo (x) n Γ(x + n). Γ(x) From [1, Equation (15.3.1)] we know that this function has the foowing integra representation (the so-caed Euer integra): 2F 1 (u, v; w; z) Γ(w) 1 da Γ(v)Γ(w v) a 1 v (1 a) 1 w+v (1 za) u. Using the ast formua with the parameters u v k + 1 >, w k + 4 >, k N, and z 1 yieds 2F 1 (k + 1, k + 1; k + 4; 1) Γ(k + 4) Γ(k + 1)Γ(3) 1 (k + 1)(k + 2)(k + 3) 2 da a k (1 a) 2 (1 + a) k+1 1 (1 a) 2 a k da. (1 + a) k+1 A comparison of this expression and the expression for the probabiities p(k) from (1) impies now 1 p(k) 2 k+1 (1 a) 2 a k (1 + a) 1+k da 2 k+2 (k + 1)(k + 2)(k + 3) 2 F 1 (k + 1, k + 1; k + 4; 1) and competes the proof. Acknowedgement. The author highy vaues the comments and remarks by Werner Nage (Jena) who consideraby heped to improve the presentation.

10 512 C. Thäe References [1] Abramowitz M., Stegun I.A., Handbook of Mathematica Functions with Formuas, Graphs, and Mathematica Tabes, Dover, New York, 1965, onine version under [2] Mecke J., Nage W., Weiss V., ength distributions of edges in panar stationary and isotropic STIT tesseations, J. Contemp. Math. Ana. 42 (27), [3] Mecke J., Nage W., Weiss V., Some distributions for I-segments of panar random homogeneous STIT tesseations, Math. Nachr. (21), to appear. [4] Nage W., Weiss V., Crack STIT tesseations: characterization of stationary random tesseations stabe with respect to iteration, Adv. in App. Probab. 37 (25), [5] Schneider R., Wei W., Stochastic and Integra Geometry, Springer, Berin, 28. [6] Schreiber T., Thäe C., Typica geometry, second-order properties and centra imit theory for iteration stabe tesseations, arxiv:11.99 [math.pr] (21). [7] Thäe C., Moments of the ength of ine segments in homogeneous panar STIT tesseations, Image Ana. Stereo. 28 (29), University of Fribourg, Mathématiques, Chemin de Musée 23, CH 17 Fribourg, Switzerand E-mai: (Received Apri 1, 29, revised February 24, 21)