LARGE TYPICAL CELLS IN POISSON DELAUNAY MOSAICS

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1 LARGE TYPICAL CELLS IN POISSON DELAUNAY MOSAICS DANIEL HUG an ROLF SCHNEIDER Deicate to Tuor Zamfirescu on the occasion of his sixtieth birthay It is prove that the shape of the typical cell of a Poisson Delaunay tessellation of R tens to the shape of a regular simplex, given that the surface area, or the inraius, or the minimal with, of the typical cell tens to infinity. Typical cells of large iameter ten to belong to a special class of simplices, istinct from the regular ones. In the plane, these are the rightangle triangles. AMS 000 Subject Classification: 60D05, 5A. Keywors an phrases: Ranom mosaic, Poisson Delaunay tessellation, typical cell, ranom polytope, D.G. Kenall s problem, shape, regular simplex. 1. INTRODUCTION In Stochastic Geometry, ranom tessellations of R, also known as ranom mosaics, are a frequently stuie topic. This is ue to their various practical applications in two an three imensions, but also to their great geometric appeal. Introuctions to ranom tessellations are foun in [13, ch. 5], [15, ch. 10], [14, ch. 6], an important source is [1]. Particularly tractable are the ranom tessellations that are erive from stationary Poisson processes, either in the space of hyperplanes, leaing to Poisson hyperplane tessellations, or in R, where the corresponing Voronoï tessellations an their uals, the Delaunay tessellations, yiel interesting classes of ranom mosaics. For such mosaics, the asymptotic shape of large cells has recently become an issue of investigation. The starting point was a conjecture of D.G. Kenall (see [15], forewor to the first eition), accoring to which This work was supporte in part by the European Network PHD, FP6 Marie Curie Actions, RTN, Contract MCRN

2 the shape of the zero cell of a stationary, isotropic Poisson line tessellation in the plane, given that the area of the cell tens to infinity, shoul ten to circular shape. Contributions to the planar case in [, 7, 8, 9, 11], incluing an affirmative answer by Kovalenko, were followe by higher imensional versions an extensions in various irections, see [4, 6, 5]. Alreay in [11] an [5], the size of the zero cell was not only measure by the volume, but also by other functions, like intrinsic volumes or the inraius. In work in progress it has turne out that for stationary Poisson hyperplane tessellations (not necessarily isotropic) an for Poisson Voronoï tessellations, asymptotic shapes of large zero cells can be etermine for quite general interpretations of large, but that these shapes may epen on the function by which the size is measure. For example, large cells in the sense of iameter have shapes egenerating to segments. In the case of a Poisson Delaunay mosaic, all cells are simplices, with probability one. It was prove in [6] that the asymptotic shapes of typical cells of large volume are regular simplices. In the present paper, we exhibit the same phenomenon for further functions, the surface area, the inraius, an the minimal with. If the size is measure by the iameter, the asymptotic shapes of large typical cells are no longer those of regular simplices, but of simplices for which one ege contains the centre of the circumscribe sphere. In the plane, this class consists of the right-angle triangles. Both types of results are special cases of a general theorem, where large refers to an abstract size function. The paper conclues with a result on the asymptotic istribution of the size of the typical cell.. RESULTS We refer to [14] for etails about Poisson Delaunay tessellations, but we recall here the basic efinitions. Let X be a stationary Poisson point process in R with positive intensity. With probability one, any + 1 points x 0,..., x of X lie on the bounary of a unique ball. If the interior of this ball contains no other point of X, then the simplex convx 0,..., x } is calle a cell. The collection Y of all cells obtaine in this way is a tessellation of R, calle the Poisson Delaunay tessellation inuce by X. For a -imensional simplex S R, we enote by z(s) the circumcentre, that is, the centre of the sphere through the vertices of S, an by r(s) the raius of that sphere. Let 0 be the space of all -simplices in R with z(s) = 0, equippe with the Hausorff metric δ.

3 Let Y be the Poisson Delaunay tessellation inuce by X, consiere as a stationary particle process. As such, it has a grain istribution Q 0. This is a probability measure on 0 (that is, on the Borel σ-algebra of 0 which is inuce by the topology of the Hausorff metric). Explicitly, choosing any convex boy W with 0 in the interior, Q 0 (A) = E cars Y : z(s) W, S z(s) A} E cars Y : z(s) W } cars Y : z(s) rw, S z(s) A} = lim, r cars Y : z(s) rw } for Borel sets A 0, where E enotes mathematical expectation an the secon equality hols with probability one, ue to the fact that the Poisson Delaunay mosaic Y is ergoic. The typical cell of Y is, by efinition, the ranom simplex with istribution Q 0 ; thus, the typical cell is only etermine up to stochastic equivalence. The intuitive heuristic iea behin this is that one takes a large region in a realization of Y, picks out at ranom one of the cells within this region (with equal chances), translates it so that its circumcentre becomes the origin, an thus obtains a realization of the typical cell. We want to stuy asymptotic shapes of large typical cells. In orer to be able to interpret large in ifferent ways, we introuce a class of abstract functions for measuring the size. By a size function we unerstan a positive function Σ : 0 R which has the following properties (V enotes volume): Σ is continuous. Σ is homogeneous of some egree k > 0. On the set of -simplices inscribe to the unit sphere, Σ attains a maximum, an V /Σ 1/k is boune. The maximum that Σ attains on the set of -simplices inscribe to the unit sphere will be enote by τ. By homogeneity, (1) Σ(S) r(s) k τ for all S 0. Every S 0 yieling equality in (1) will be calle an extremal simplex (for the given Σ). As we want to estimate the probability for large eviations of the shape of large typical cells from the shapes of extremal simplices, we nee a measure for that eviation. By a eviation function for Σ we unerstan a nonnegative function ϑ : 0 R with the following properties: 3

4 ϑ is continuous. ϑ is homogeneous of egree zero. ϑ(s) = 0 if an only if S is an extremal simplex. By a stability function for Σ an ϑ we unerstan a continuous function f : [0, 1) [0, 1] with the properties f(0) = 0, f(ɛ) > 0 for ɛ > 0 an () Σ(S) (1 f(ɛ))r(s) k τ for all S 0 with ϑ(s) ɛ. Now we can formulate our main result. By P we enote the unerlying probability, an P( ) is a conitional probability. THEOREM 1. Let Z be the typical cell of the Poisson Delaunay tessellation erive from a stationary Poisson process with intensity λ > 0 in R. Suppose that functions Σ, ϑ, f with the properties liste above are given. There is a constant c 0 epening only on these functions an the imension such that the following is true. If ɛ (0, 1) an I = [a, b) is an interval (b = allowe) with a /k λ σ 0 > 0 for some constant σ 0, then } P(ϑ(Z) ɛ Σ(Z) I) c exp c 0 f(ɛ)a /k λ, where c is a constant epening only on, ɛ, Σ, ϑ, f, σ 0. It follows from Theorem 1 that lim P(ϑ(Z) < ɛ Σ(Z) a) = 1 a for every ɛ (0, 1). Thus, the shapes of typical cells of large size have small eviation from the shape of extremal simplices, with high probability. Here, it oes not matter how the eviation is measure. In this sense, the shapes of extremal simplices are the asymptotic shapes of typical cells of large size. In concrete cases, where the eviation function is explicit an has an intuitive geometric meaning, the estimate of Theorem 1 provies much stronger information, an even more so when a stability function is explicitly known. We consier some special cases of this type. In [6] the case of the volume, Σ = V, was treate. The extremal simplices in that case are precisely the regular simplices. A measure of eviation of the shape of a simplex from the shape of a regular simplex can be efine as follows. Let v 0,..., v S 1 be points such that T = 4

5 convv 0,..., v } is a regular simplex. Let S be a simplex in R. We efine ϑ 1 (S) as the smallest number α with the following property. There are points x 0,..., x S 1 such that convx 0,..., x } is similar to S an x i v i α for i = 0,...,. Clearly, ϑ 1 has the properties that we require of a eviation function for V. The following stability estimate was prove in [6]. There is a positive constant c 1 () such that, for every ɛ [0, 1] an for every simplex S R, (3) V (S) (1 c 1 ()ɛ )r(s) τ for all S 0 with ϑ 1 (S) ɛ (recall that τ is always the maximum of Σ on the simplices inscribe to the unit sphere). With this, our present Theorem 1 reuces to Theorem 1 of [6]. The inclusion of any other concrete size functions requires the etermination of the simplices inscribe to the unit sphere for which the size function attains its maximum. This is a purely geometric task, which may be ifficult. For example, it seems to be unknown, for 3, whether the regular simplices yiel the maximum for the mean with (see the iscussion in Gritzmann an Klee [3, Section 9.10.]). For the surface area, it follows from a more general result of Tanner [16] that the extremal simplices are the regular ones. For 3, we o not have an explicit stability estimate in this case. Nevertheless, the existence of a stability function (say, for the eviation function ϑ 1 ) can be shown, an then Theorem 1 is sufficient to ensure that typical cells of large surface area are asymptotically close to regular shape, with high probability. In the planar case, where the surface area reuces to the perimeter L (an also to π times the mean with), the following stability estimate will be prove in Section 4. Let S R be a triangle inscribe to the unit circle, an let ɛ [0, 1]. Then (4) L(S) (1 ɛ /36)τ if ϑ 1 (S) ɛ. Tanner s [16] general result yiels some other size functions for which the regular simplices are extremal, for example, the sum of the ege lengths. Further functions which satisfy the requirements for a size function (as shown in Section 4) an for which the extremal simplices can be etermine, are the inraius, the minimal with (or thickness), an the iameter. The inraius ρ(s) of a simplex is the largest raius of a ball containe in S. The maximal simplices for the inraius are the regular ones, an the stability estimate (5) ρ(s) (1 c ()ɛ )r(s)τ for all S 0 with ϑ 1 (S) ɛ 5

6 (with c () = c 1 ()/) hols (see Section 4). Hence, in this case the estimate of Theorem 1 takes again an explicit form, with ϑ = ϑ 1 an f(ɛ) proportional to ɛ. The minimal with w(s) of a simplex is the minimal istance between any two parallel supporting hyperplanes of S. The maximal simplices are again the regular simplices, as shown by Alexaner [1]. Also in this case, we must be satisfie with the mere existence of a stability function. The situation changes if the size is measure by the iameter D. For a simplex S inscribe to the unit sphere we have D(S), an equality hols if an only if S has an ege of length. Generally, we say that a simplex S is iametral if its iameter is equal to the iameter of the circumscribe sphere. Thus, the maximal simplices for the size function Σ = D are precisely the iametral simplices in 0. We efine a measure of eviation of the shape of a simplex from the shape of a iametral simplex. Let w 0, w 1 be two antipoal points of the unit sphere S 1. For a simplex in S R, let ϑ (S) be the smallest number α with the following property. There are points x 0,..., x S 1 such that convx 0,..., x } is similar to S, x 0 = w 0 an x 1 w 1 α. It is a trivial task to prove the stability estimate (6) D(S) (1 ɛ /8)r(S)τ for all S 0 with ϑ (S) ɛ. With this, Theorem 1 provies an explicit estimate for the eviation of the shape of typical cells of large iameter from the shape of iametral simplices. 3. PROOF OF THEOREM 1 For the proof of Theorem 1, we assume that a stationary Poisson process X in R with intensity λ > 0 an three functions Σ, ϑ, an f with the properties liste in Section are given. We recall that the number k is the egree of homogeneity of Σ an the constant τ > 0 is the maximal value of Σ(S) for S 0 with r(s) = 1, accoring to (1). By σ we enote the spherical Lebesgue measure on the unit sphere S 1, an κ is the volume of the unit ball. The proof to follow extens the one given in [6], for the case of the volume an for special explicit eviation an stability functions. We make use of the integral representation for the istribution Q 0 ue to Miles (see, 6

7 e.g., [14, Satz 6..10]). For a Borel set A 0, Q 0 (A) = α()λ 1 A (convru 0,..., ru }) 0 S 1 S 1 with e λκ r r 1 V (convu 0,..., u }) σ(u 0 ) σ(u ) r α() := +1 π 1 Γ ( ) Γ ( +1 ) [ ( Γ +1 ) ] Γ ( + 1). For functions F efine on -simplices, we use the abbreviation F (convx 0,..., x }) =: F (x 0,..., x ). In the following, c 1, c,... enote positive constants which may epen on, the chosen functions Σ, ϑ, f, an the given number ɛ > 0. LEMMA 1. Let ɛ > 0 an efine the number h 0 = h 0 (ɛ) by (1+h 0 (ɛ)) /k = 1 + (ɛ/(5 + 4ɛ)). Then there is a constant c 1 such that, for 0 < h h 0 an a > 0, P(Σ(Z) a[1, 1 + h]) c 1 h(a /k λ) exp κ } τ /k (1 + ɛ)a/k λ. Proof. Let ɛ > 0, h (0, h 0 ] an a > 0 be given. Using the formula of Miles, we obtain P(Σ(Z) a[1, 1 + h]) = α()λ 0 S 1 S 1 1 } r k Σ(u 0,..., u ) a [1, (1 + h)] e λκ r r 1 V (u 0,..., u ) σ(u 0 ) σ(u ) r. Substituting s = λκ r, we get P (Σ(Z) a [1, 1 + h]) = c S 1 S 1 0 } 1 s λκ (a/σ(u 0,..., u )) /k [1, (1 + h) /k ] e s s 1 s V (u 0,..., u ) σ(u 0 ) σ(u ). 7

8 For fixe u 0,..., u S 1 in general position, we apply to the inner integral the mean value theorem for integrals. This gives the existence of a number (7) ξ(u 0,..., u ) λκ (a/σ(u 0,..., u )) /k [1, (1 + h) /k ] such that the inner integral is equal to ( ) λκ (a/σ(u 0,..., u )) /k (1 + h) /k 1 exp ξ(u 0,..., u )}ξ(u 0,..., u ) 1. Using (1 + h) /k 1 = (/k)(1 + θh) (/k) 1 h with 0 θ 1 an h h 0 (in the case /k < 1), we obtain P (Σ(Z) a [1, 1 + h]) ) = c 3 ((1 + h) /k 1 a /k λ exp ξ(u 0,..., u )} S 1 S 1 ξ(u 0,..., u ) 1 V (u 0,..., u ) Σ(u 0,..., u ) /k σ(u 0) σ(u ) c 4 ha /k λ exp ξ(u 0,..., u )}ξ(u 0,..., u ) 1 } } R(,ɛ) V (u 0,..., u ) Σ(u 0,..., u ) /k σ(u 0) σ(u ), where we choose (up to a set of σ +1 -measure zero) R(, ɛ) := (u 0,..., u ) (S 1 ) +1 : Σ(u 0,..., u ) /k g(ɛ)τ /k, with g(ɛ) := 1 4ɛ 5 + 4ɛ V (u 0,..., u ) c 5 } an a positive constant c 5 etermine in the following way. The set of all (u 0,..., u ) (S 1 ) +1 with Σ(u 0,..., u ) /k > g(ɛ)τ /k is nonempty an open, hence we can choose c 5 in such a way that σ +1 (R(, ɛ)) =: c 6 is positive. 8

9 For (u 0,..., u ) (S 1 ) +1 in general position we have Σ(u 0,..., u ) τ an hence, by (7), ξ(u 0,..., u ) κ τ /k a/k λ. For (u 0,..., u ) R(, ɛ) in general position we can estimate an ξ(u 0,..., u ) κ τ /k g(ɛ) 1 (1 + h 0 ) /k a /k λ V (u 0,..., u ) Σ(u 0,..., u ) /k c 5 τ /k. Since σ +1 (R(, ɛ)) = c 6, we finally obtain where we use that P (Σ(Z) a [1, 1 + h]) c 7 h(a /k λ) exp κ } τ /k g(ɛ) 1 (1 + h 0 ) /k a /k λ = c 7 h(a /k λ) exp κ } τ /k (1 + ɛ)a/k λ, (1 + h 0 (ɛ)) /k = 1 + ɛ 5 + 4ɛ = This completes the proof. ( 1 4ɛ ) (1 + ɛ) ɛ LEMMA. For each ɛ (0, 1), there is a constant c 8 such that, for h (0, 1] an a > 0, P(Σ(Z) a[1, 1 + h], ϑ(z) ɛ) c 8 h(a /k λ) 1/ exp κ } τ /k (1 + f(ɛ)/k)a/k λ. Proof. Similarly as in the proof of Lemma 1, we obtain P(Σ(Z) a[1, 1 + h], ϑ(z) ɛ) = α()λ S 1 S 1 0 [ 1 r (a/σ(u 0,..., u )) 1/k 1, (1 + h) 1/k]} e λκ r r 1 r 1ϑ(u 0,..., u ) ɛ} V (u 0,..., u ) σ(u 0 ) σ(u ). 9

10 For fixe u 0,..., u S 1 in general position, there is a number (8) ξ(u 0,..., u ) (a/σ(u 0,..., u )) 1/k [1, (1 + h) 1/k ] such that the inner integral is equal to ( ) (a/σ(u 0,..., u )) 1/k (1 + h) 1/k 1 exp λκ ξ(u 0,..., u ) }ξ(u 0,..., u ) 1. Estimating (1 + h) 1/k 1 c 9 h, we get P (Σ(Z) a [1, 1 + h], ϑ(z) ɛ) c 10 ha 1/k λ S exp λκ ξ(u 0,..., u ) } ξ(u 0,..., u ) 1 1 S 1 1ϑ(u 0,..., u ) ɛ} V (u 0,..., u ) Σ(u 0,..., u ) 1/k σ(u 0) σ(u ). By our assumptions on Σ, (9) V (u 0,..., u ) Σ(u 0,..., u ) 1/k c 11. If (10) 1ϑ(u 0,..., u ) ɛ} 0, then, using (8) an (), ξ(u 0,..., u ) We etermine g(ɛ) by a 1/k Σ(u 0,..., u ) 1/k a 1/k a1/k (1 f(ɛ)) 1/k τ 1/k τ 1/k (1 + f(ɛ))1/k. (1 g(ɛ))(1 + f(ɛ)) /k = 1 + f(ɛ)/k. Since (1 + f(ɛ)) /k > 1 + f(ɛ)/k (using f(ɛ) 1 if k > ), we have 0 < g(ɛ) < 1. There exists a constant c 1 such that, for ξ 0, exp λκ ξ } ξ 1 c 1 λ ( 1)/ exp (1 g(ɛ))λκ ξ }. 10

11 This shows that if (10) is satisfie, then Finally, this yiels exp λκ ξ(u 0,..., u ) }ξ(u 0,..., u ) 1 c 1 λ ( 1)/ exp κ } τ /k (1 g(ɛ))(1 + f(ɛ))/k a /k λ. P(Σ(Z) a[1, 1 + h], ϑ(z) ɛ) c 13 h(a /k λ) 1/ exp κ } τ /k (1 + f(ɛ)/k)a/k λ. Let ɛ (0, 1). In Lemma 1, we can replace ɛ by f(ɛ)/4k. Then it follows that there exists a constant c 14 (now epening also on f) such that, for 0 < h h 0 := h 0 (f(ɛ)/4k) an a > 0, P(Σ(Z) a[1, 1 + h]) c 14 h(a /k λ) exp κ } τ /k (1 + f(ɛ)/4k)a/k λ. From this result an Lemma, Theorem 1 is now euce in precisely the same way as in [6] Theorem 1 was euce from Lemmas an SPECIAL CASES For the special cases of size functionals Σ consiere in Section, we have to show that they satisfy the requirements. Continuity an homogeneity are trivial, an also the existence of a maximum on the set of -simplices inscribe to the unit sphere is clear in each case. Hence, it remains to show that V /Σ 1/k is boune on the -simplices S inscribe to S 1. For the case of the surface area A, an estimate V (S)/A(S) 1/( 1) c() follows from the isoperimetric inequality if A(S) 1 an is trivial if A(S) 1. Similarly one can argue for the iameter, using the isoiametric inequality. From this case, the corresponing assertion for the sum of the ege lengths is obtaine. To treat the inraius, let F 0,..., F be the facets of the simplex S inscribe to the unit sphere, an let F i be the ( 1)-volume of F i. Then V (S)/ρ(S) = ( F F ) / is boune from above by a constant epening only on. For the minimal with w, we have V (S)/w(S) κ 1. The stability estimate (3) was prove in [6]. From this, (5) is obtaine as follows. Let S be a -simplex inscribe to the unit sphere. Among all simplices of given inraius, the regular ones have the smallest volume (a 11

12 proof is inicate, e.g., in [10, p. 318]). From this fact an from (3) we conclue for ϑ(s) ɛ that ( ) ρ(s) ρ(t V (T ) V (S) (1 c 1 ()ɛ )V (T ), ) where T is a regular simplex inscribe to the unit sphere. This yiels (5). As mentione, (6) is easily obtaine. The remaining stability estimate (4) is prove in Lemma 3 below. In the case where Σ is the surface area or the minimal with, we o not have an explicit stability estimate, but the existence of a stability function, say for the eviation function ϑ 1, is easy to see. For this, let ɛ (0, 1) be given, an let A ɛ be the closure of the set S 0 : r(s) = 1, ϑ 1 (S) ɛ}. This set is compact, hence the continuous function Σ attains a maximum M ɛ on this set. Since the maximum is not attaine at a regular simplex, we have M ɛ < τ an can put f(ɛ) := 1 M ɛ /τ. The function f efine in this way, together with f(0) := 0, has the require properties. LEMMA 3. Let S be a triangle inscribe to S 1, an let ɛ [0, 1]. Then L(S) (1 ɛ /36)L(T ) if ϑ 1 (S) ɛ. Proof. Let S be a triangle inscribe to S 1 an satisfying the inequality L(S) > (1 ɛ /36)L(T ), where T is a regular triangle inscribe to S 1, hence L(T ) = 3 3. Then 0 int S. Let α, β, γ be the angles at 0 spanne by the eges of S. Then L(S) = (sin α + sin β + sin γ) an α + β + γ = π. We can choose the notation in such a way that the angles ϕ := α π/3 an ψ := β π/3 are either both non-negative or both non-positive. We get hence L(S) L(T ) [ ( π ) ( π ) ( )] π = sin 3 + ϕ + sin 3 + ψ + sin 3 + ϕ + ψ 3 3 = 3[cos ϕ + cos ψ + cos(ϕ + ψ)] + sin ϕ + sin ψ sin(ϕ + ψ) 3 3, (11) L(S) L(T ) = 3 (cos ϕ + cos ψ ) + 1

13 with := 3 cos(ϕ + ψ) 3 + sin ϕ + sin ψ sin(ϕ + ψ). We show that 0. For this, we rewrite as follows: = ( 3 cos ϕ + ψ ) + sin ϕ + sin ψ sin(ϕ + ψ) = 3 sin ϕ + ψ = 3 sin ϕ + ψ = 3 sin ϕ + ψ = 4 sin ϕ + ψ + sin ϕ + ψ + sin ϕ + ψ cos ϕ ψ sin ϕ + ψ ( cos ϕ ψ cos ϕ + ψ + sin ϕ + ψ sin ϕ sin ψ ( sin ϕ sin ψ 1 ) ϕ + ψ 3 sin. cos ϕ + ψ ) Here we have sin ϕ sin ψ 1 3 sin ϕ + ψ = sin ϕ ( ψ sin π ) + sin ψ ( ϕ 3 sin π ) 3 = sin ϕ ( β sin π ) + sin ψ ( α sin π ) ( = sin ϕ cos β + sin ψ cos α ) an therefore = 4 sin ϕ + ψ ( sin ϕ cos β + sin ψ cos α ). Since either ϕ 0, ψ 0 or ϕ 0, ψ 0 (an ϕ + ψ < π) an α/, β/ [0, π/], we euce that 0, as asserte. Thus (11) yiels ɛ < L(S) L(T ) 3 (cos ϕ ) ϕ, where the latter inequality follows from ϕ/ < π/6 an the Taylor formula. This gives ϕ < ɛ/ an similarly ψ < ɛ/. From this, we euce that ϑ 1 (S) < ɛ. 13

14 5. ON THE DISTRIBUTION OF THE SIZE OF LARGE CELLS Lemmas 1 an also permit us to obtain a limit relation for the probability P(Σ(Z) a). The analogue for the area of the zero cell of a stationary, isotropic Poisson line process in the plane was first obtaine by Golman []. THEOREM. Let Z be the typical cell of the Poisson Delaunay tessellation erive from a stationary Poisson process in R with intensity λ > 0. Let Σ be a size function, of homogeneity k an with maximum τ on the simplices inscribe to the unit sphere. Then lim a a /k ln P(Σ(Z) a) = κ τ /k λ. Proof. For ɛ > 0 an a > 0, Lemma 1 with h = h 0 (ɛ) gives P(Σ(Z) a) c 15 (a /k λ) exp κ } τ /k (1 + ɛ)a/k λ, hence lim inf a a /k ln P(Σ(Z) a) κ (1 + ɛ)λ. τ /k With ɛ 0, this yiels (1) lim inf a a /k ln P(Σ(Z) a) κ λ. τ /k From the first part of the proof of Lemma, where we choose h = 1, omit the conition ϑ(z) ɛ, an use the estimate (9), we obtain P(Σ(Z) a[1, ]) c 16 a 1/k λ S exp λκ ξ(u 0,..., u ) } ξ(u 0,..., u ) 1 1 S 1 σ(u 0 ) σ(u ). where ξ(u 0,..., u ) satisfies (8) (for h = 1). For fixe u 0,..., u S 1 in general position, it follows from (8) an Σ(u 0,..., u ) τ that ξ(u 0,..., u ) (a/τ) 1/k. 14

15 Let ɛ (0, 1/3). There exists a constant c 17 such that, for ξ 0, exp λκ ξ } ξ 1 c 17 λ ( 1)/ exp (1 ɛ)λκ ξ }. These estimates together imply that, for ξ = ξ(u 0,..., u ), exp λκ ξ } ξ 1 c 17 λ ( 1)/ exp κ } τ /k (1 ɛ)a/k λ. Therefore, P(Σ(Z) [1, ]) c 18 (a /k λ) 1/ exp κ } τ /k (1 ɛ)a/k λ. Assuming a /k λ σ 0 > 0, we obtain P(Σ(Z) [1, ]) c 19 exp κ } τ /k (1 ɛ)a/k λ, with a constant c 19 epening also on σ 0. For ɛ (0, 1/3) an a /k λ σ 0, we conclue that P(Σ(Z) a) ( = P Σ(Z) ) a i [1, ] i=0 P ( Σ(Z) a i [1, ] ) i=0 i=0 c 19 exp κ } τ /k (1 ɛ)(ai ) /k λ c 19 exp κ τ /k (1 3ɛ)a/k λ } c 0 exp κ } τ /k (1 3ɛ)a/k λ. From this result, we can conclue that lim sup a for any ɛ (0, 1/3), an therefore lim sup a i=0 exp κ } τ /k ɛ(ai ) /k λ a /k ln P(Σ(Z) a) κ (1 3ɛ)λ τ /k a /k ln P(Σ(Z) a) κ λ. τ /k Together with (1), this proves Theorem. 15

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