Variance asymptotics for the area of planar cylinder processes generated by Brillinger-mixing point processes

We introduce cylinder processes in the plane defined as union sets of dilated straight lines (appearing as mutually overlapping infinitely long strips) generated by a stationary independently marked point process on the real line, where the marks describe the width and orientation of the individual cylinders. We study the behavior of the total area of the union of strips contained in a space-filling window ϱK as ϱ → ∞. In the case the unmarked point process is Brillinger mixing, we prove themean-square convergence of the area fraction of the cylinder process in ϱK. Under stronger versions of Brillinger mixing, we obtain the exact variance asymptotics of the area of the cylinder process in ϱK as ϱ → ∞. Due to the long-range dependence of the cylinder process, this variance increases asymptotically proportionally to ϱ3.


Introduction and preliminaries
Cylinder processes (CPs) in R d defined as countable unions of dilated affine subspaces R k , k = 1, . . . , d − 1, are basic random set models in stochastic geometry; see, for example, [15,19,21] and [16]. Meanwhile CPs also receive much attention due to numerous applications (for d = 2, 3) in modern technologies as models for materials consisting of long fibers or to model dynamic telecommunication networks. Until now, so far as we know, asymptotic properties of CPs in expanding domains were exclusively studied under Poisson assumptions; see [13,14,20] and [1]. The Poisson property of the generating stationary point process (PP) provides, among others, the stationarity of the associated CP. In this paper, we focus on planar CPs generated by a more general class of stationary independently marked PPs on R 1 . We assume that the corresponding unmarked (ground) PP is Brillinger mixing, that is, its reduced cumulant measures of any order exist and have finite total variation.
Throughout this paper, all random elements are defined on a common probability space [Ω, F, P], and by E and Var we denote the expectation and variance with respect to P. Next, we describe a CP in R 2 in terms of its generating stationary independently marked PP on R 1 . Let (Φ 0 , R 0 ) be the generic random vector taking values in the mark space [0, π) × [0, ∞) that describes the orientation Φ 0 and cross-section (or base) Ξ 0 := [−R 0 , R 0 ] of the typical cylinder. In addition, we assume that R 0 ∼ F and Φ 0 ∼ G are independent, that is, P(R 0 r, Φ 0 ϕ) = F (r)G(ϕ). Now we introduce a stationary independently marked PP as a locally finite simple counting measure Ψ P F,G := i∈Z δ [Pi,(Φi,Ri)] defined on the Borel sets of R 1 × [0, π) × [0, ∞) whose finite-dimensional distributions are shift-invariant in the first component; see, for example, [2,6] or [19]. The stationary ground PP Ψ = i∈Z δ Pi ∼ P with finite and positive intensity λ := E Ψ ([0, 1]) > 0 is assumed to be independent of the i.i. where |·| k denotes the Lebesgue measure on R k . Each triplet is the circle in R 2 with radius r 0 and center in the origin o, ⊕ stands for pointwise addition (Minkowski sum) of sets in R 2 , and g(p, ϕ) := {(x, y) ∈ R 2 : x cos ϕ + y sin ϕ = p} denotes the unique line with signed distance p ∈ R 1 from o and an angle ϕ ∈ [0, π) measured anticlockwise between the x-axis and normal vector v(ϕ) = (cos ϕ, sin ϕ) on the line with direction in the half-plane not containing o. Now we are in a position to define the main subject of this paper. DEFINITION 1. A CP Ξ = Ξ P F,G in the Euclidean plane R 2 derived from the stationary independently marked PP Ψ P F,G is defined by the random union set Our first aim is to prove the mean-square convergence (and thus the convergence in probability) of the ratio |Ξ ∩ K| 2 /| K| 2 to the deterministic limit 1 − exp{−λE|Ξ 0 | 1 } as → ∞ for a compact star-shaped set K ⊂ R 2 with respect to the origin o, an inner point of K; see Theorem 1. The Brillinger-mixing condition put on the ground PP Ψ ∼ P is essential to obtain this result. Our second main result, Theorem 2, shows the existence and explicit shape of the asymptotic variance Theorems 1 and 2 generalize some of the results obtained in [13] and [14] (in particular, Theorem 2 in [14]) for Poisson CPs in R d to planar CPs generated by Brillinger-mixing PPes.
The limit σ 2 P (K, F, G) is positive and finite (if E|Ξ 0 | 2 1 = 4ER 2 0 < ∞) and depends on the shape of K, the intensity λ, the first and second moments of F , and the distribution function G, which is assumed to be continuous (not necessarily absolutely continuous). A purely discrete distribution function G yields different expressions for σ 2 P (K, F, G) even if Ψ ∼ P = Π λ is a stationary Poisson PP with intensity λ > 0; see [13,14]. A distribution function G without jumps implies that P(Φ 0 = Φ 1 ) = 0 if the angles Φ 0 , Φ 1 ∼ G are independent. Note that the order 3 of the growth of Var |Ξ ∩ K| 2 is much faster than the growth of the area | K| 2 = 2 |K| 2 , which reveals a typical feature of long-range dependence within the random set (1.1) and in general need not be closed or stationary.

Basic assumptions and main results
Recall that the kth-order factorial cumulant measure γ (k) (·) of a PP Ψ ∼ P for k ∈ N is defined by where α (k) denotes the kth-order factorial moment measure of Ψ ∼ P defined by for bounded Borel sets B 1 , . . . , B k ⊂ R 1 , where the sum = runs over k-tuples of pairwise distinct integers. Formula (2.1) is based on the general relationship between mixed moments and mixed cumulants; see [2] or [18]. Note that γ (k) is a locally finite signed measure on Due to the stationarity of Ψ ∼ P , we may define the kth-order reduced cumulant measure γ red ) − are given by the Jordan decomposition of the signed measure γ red possesses a Lebesgue density c (k) red on R k−1 (called the kth-order reduced cumulant density), then we need the usual L q -norm c (k) red (x, p)| q dx) 1/q dp for k 3, and c (2) red * q := c (2) red q , where 1 q < ∞. Formally, we may put γ (1) red T V := 1 and c (1) red q = c (1) red * q := 1. Note that the existence and integrability of c (k) red imply that c (2) red 1 = γ (2) red T V and c (k)

Remark 1.
In general, the Brillinger-mixing condition is formulated for stationary PPs on R d , d 1. This condition expresses some kind of mutual asymptotic uncorrelatedness of the numbers of points in bounded sets with unboundedly increasing distance from each other. This type of weak dependence does not necessarily imply ergodicity (see [9]), but allows us to prove central limit theorems for various stochastic models related to point processes, for example, in stochastic geometry, statistical physics for d 1, or in queueing theory for d = 1; see, for example, [12]. In [10,11] the relations between (strong) Brillinger-mixing and classical mixing conditions are studied. Strong Brillinger mixing requires exponential moments of the number of points in bounded sets. For any dimension d 1, examples of such point processes are determinantal point processes (see [8]), Poisson cluster processes if the number of daughter points has an exponential moment, certain Cox processes, and Gibbsian PPs under suitable restrictions; see, for example, [17]. For d = 1, renewal processes with an exponentially decaying interrenewal density (see [12]) and, among them, the Erlang and Macchi processes (see [2, p. 144]) are strongly Brillinger mixing.
Our first result can be regarded as a mean-square ergodic theorem for the random set (1.1). Theorem 1. Assume that a stationary PP Ψ ∼ P on R 1 is Brillinger mixing. Further suppose that ER 0 < ∞ and Φ 0 ∼ G has a continuous distribution function G. Then

2)
which immediately implies the convergence in probability of the ratio |Ξ ∩ K| 2 /| K| 2 and its L p (P)convergence for any p 1. The limit (2.2) is the same as that for the Poisson PP Ψ ∼ Π λ .
Our second result provides an exact asymptotic behavior of the variance of the area of the cylinder process (1.1) that is contained in a star-shaped set K growing unboundedly in all directions. For this purpose, in comparison with Theorem 1, we need a strengthening and quantification of the usual Brillinger-mixing condition.
Theorem 2. Assume that the stationary PP Ψ ∼ P on R 1 is either strongly Brillinger mixing with b < 1/2 or strongly L q -Brillinger mixing with (E|Ξ 0 | 1 ) 1−1/q b q < 1/2 or strongly L * q -Brillinger mixing with (E|Ξ 0 | 1 ) 1−1/q b * q < 1/2 for some q > 1. Further suppose that ER 2 0 < ∞ and Φ 0 ∼ G has a continuous distribution function G. Then the limit (1.2) is positive and finite with Remark 2. In the particular case K = b(o, 1), we can show that C G,K 1 = 16/3 and C G,K 2 = 8/3 are independent of the distribution function G. If Φ 0 is uniformly distributed on [0, π], then we get The latter double integral is known as the second-order chord power integral of K; see, for example, [14, p. 327] and [19,Chap. 7] for integral geometric background.
The proofs of our results are based on series expansions of expectation and variance of the area |Ξ ∩ K| 2 in terms of the factorial moment and cumulant measures of the PP Ψ ∼ P ; see [2,Chap. 5.5]. These expansions and their convergence are studied in Section 3 using the above-defined Brillinger-mixing conditions. The asymptotics of these expansions (as → ∞) are studied in Sections 4 and 5. Note that the study of non-Poisson CPs is much more complex and difficult than that of Poisson CPs. To emphasize the basic ideas on the one hand and to keep our mathematical tools as simple as possible on the other hand, we study only planar CPs. In the proofs of our results, we have shortened some longer calculations at certain points. For more detailed derivations, we refer the interested readers to [5].
3 Factorial moment expansions of E|Ξ ∩ K| 2 and Var |Ξ ∩ K| 2 The distribution of a random closed set Ξ is determined by its Choquet functional (see, e.g., [15] or [16]) where K 2 denotes the family of nonempty compact sets in R 2 . The following lemma shows the connection between the Choquet functional and the probability generating functional where N denotes the set of locally finite simple counting measures on the Borel σ-algebra B(R 1 ).

Lemma 1.
For any X ∈ K 2 , we have the representation To simplify the notation, for k 2 (not necessarily distinct) points x 1 , . . . , x k ∈ R 2 , we define Proof of Lemma 1. To prove formula (3.2), we use the orthogonal matrix which represents an anticlockwise rotation by the angle ϕ ∈ [0, π), so that we Using the PGF (3.1) and the independence assumption in the definition of (1.1), we obtain To verify (3.5), we use the fact that Obviously, (3.5) coincides with (3.2). Hence the proof of Lemma 1 is complete.
We recall the fact that the probability space [Ω, F, P] on which the marked point process Ψ P F,G is defined can be chosen in such a way that the mapping ( [7]. This enables us to apply Fubini's theorem to the family of indicator variables 3) in terms of T Ξ and the PGF (3.1).
Let us fix a star-shaped set K ∈ K 2 containing the origin o as an inner point. Further, let 1 be a scaling factor tending to infinity implying that K ↑ R 2 as → ∞. The second-order mixed moment functions (3.6) fulfill the relation By applying Fubini's theorem together with (3.2) and (3.4) we get that (3.7) By Corollary 1, together with the equality we obtain the following:

D. Flimmel and L. Heinrich
As a consequence of (3.7) and the definition of the factorial moment measures α (k) of Ψ (see [3,Chap. 9.5]), we get the following series expansion: provided that the series on the right-hand side is convergent, where p i,k := (p i , . . . , p k ) for k i and i = 1 or i = 2. By the Bonferroni inequalities (see [3,Prop. 9.5.VI]) it follows that for all m 1 and x ∈ R 2 . Consequently, the right-hand side of (3.9) is convergent if and only if To show this convergence, we express α (m) by the factorial cumulant measures Representation (3.11) follows by inverting formula (2.1). This gives us a tool to prove the following: Lemma 3. If a stationary PP Ψ ∼ P is strongly Brillinger mixing with b < 1/2 and ER 0 < ∞, then which, together with (3.10), implies (3.9). If Ψ ∼ P is strongly L q -(L * q )-Brillinger mixing for some q > 1 such that Proof of Lemma 3. Using representation (3.11), we obtain red (dp 2,k ) dp 1 for k = 1, . . . , m and fixed 1 and Hence, together with some elementary combinatorics, we arrive at Combining (3.13) with the latter bound for b < 1/2 immediately leads to estimate (3.12). Under the strong L q -Brillinger-mixing condition, we may express f (k) for k 2 as follows: red (p 2,k ) dp 2,k dp 1 , The latter estimate with a * q and b * q instead of a q and b q can be shown under the strong L * q -Brillinger-mixing condition. The details are omitted. Finally, we have to repeat the foregoing steps with the latter bound for f (k), which completes the proof of Lemma 3.

Some auxiliary lemmas
The following Lemmas 4-6 are essential for the calculation of the terms on the right-hand side of (2.3). It is interesting that the assumptions in these lemmas are rather mild in comparison with the Brillinger-mixing-type conditions in Theorems 1 and 2.

This inequality is also valid if
) exists, and combining this with (3.7) reveals that which immediately yields Corollary 2.
Lemma 5. Let Ψ be a second-order stationary PP on R 1 satisfying γ (2) red T V < ∞. Further, suppose that ER 0 < ∞ and Φ 0 ∼ G with (not necessarily continuous) distribution function G. Then Proof of Lemma 4. We apply (3.11) for k = m to α (m) (dp 1,m ), which allows us to express the left-hand side of (4.1) as follows: w xj (p j ) dp 1,m .
Applying Fubini's theorem shows that the last summand coincides with the right-hand side of (4.1). Hence the proof of Lemma 4 is complete if the remaining summands in the foregoing line disappear as → ∞, and this follows by showing that, for k = 2, . . . , m, red (dp 2,k ) dp 1 converges to 0 as → ∞. Since 0 w xi (p i + p 1 ) 1 for i = 3, . . . , k, it suffices to prove that Since the total-variation measure |γ red | is finite on R k−1 and the inner integral over R 1 is bounded by E|Ξ 0 | 1 , we have only to verify that the inner integral vanishes as → ∞. For this purpose, we rewrite this integral as the expectation over indicator functions and Φ 0 , respectively, and R 1 , R 2 , Φ 1 , Φ 2 are mutually independent random variables. The right-hand expectation in the last line converges to 0 as → ∞. This can be verified as follows: For i = 1, 2, fix two points x i = x i (cos(α i ), sin(α i )) ∈ R 2 and two points v(ϕ i ) = (cos(ϕ i ), sin(ϕ i )) on the unit circle line. We easily see that the equality v(ϕ 1 ), x 1 = v(ϕ 2 ), x 2 or, in other words, x 1 cos(ϕ 1 − α 1 ) = x 2 cos(ϕ 2 − α 2 ) holds for finitely many pairs ϕ 1 , ϕ 2 ∈ [0, π]. Hence, for two independent random angles Φ 1 , Φ 2 with continuous distribution function G, we have P( v(Φ 1 ), x 1 = v(Φ 2 ), x 2 ) = 1 for any two points x 1 , x 2 ∈ R 2 with x 1 + x 2 > 0.
Proof of Lemma 6. We rewrite the integral J (K) as follows: with r K (ψ) as defined in Theorem 2. Here we have used the substitution y = r v(ψ) and the independence of Φ 0 and R 0 . Further, since v(ψ + π) = −v(ψ), by the shift-invariance of |·| 1 , the motion-invariance of |·| 2 , and the fact that by the definition of r K (ψ), Variance asymptotics for planar cylinder processes 69 by substituting ψ = arccos(z/r) for z ∈ [−r, r] and changing the order of integration. After changing once more the integration order over z and r and substituting z = u/ , we can proceed with the abbreviation where 0 h(r, z, ϕ) = r r K := max{r K (ψ): 0 ψ π} diam(K), which shows that In the last step, we could apply Lebesgue's dominated convergence theorem since the inner integral in (4.5) over r is bounded by |K| 2 diam(K) and the mapping z → h(r, z, ϕ) is continuous in z = 0 with h(r, 0, ϕ) = r v(ϕ + π/2) and arccos(0) = π/2. Finally, we use the relation R 1 |Ξ 0 ∩ (Ξ 0 + u)| 1 du = |Ξ 0 | 2 1 = 4R 2 0 and the independence of Φ 0 and Ξ 0 , which provide the statement of Lemma 6.

Proofs of Theorems 1 and 2
Proof of Theorem 1. According to the definition of L 2 (P)-convergence, the limit (2.2) is proved if vanishes as → ∞. In view of Corollary 2, it remains to prove that −4 Var |Ξ ∩ K| 2 → 0 as → ∞. For this purpose, we use the variance formula (3.8) of Lemma 2 and show that for any distinct points x, y ∈ K \ {o}.
For this, we make use of the finite expansion (3.10) of the PGF G P [1 − w x (·)] with remainder term, where w x can be replaced by any Borel-measurable function w : Hence (3.10) reads as |G P [1 − w(·)] − S m (w)| T m (w), which leads us to the following estimate: where we have additionally used that G P [1 − w(·)] 1 and S m (w) G P [1 − w(·)] + T m (w). We are now in a position to apply the limit (4.1) of Lemma 4, which yields that for any x ∈ K \ {o} and m ∈ N, for all m ∈ N and x 0. Next, we have to find the limit of T m (w ∪ x, y ) as → ∞. Using the relation w ∪ x,y (p) = w x (p) + w y (p) − w ∩ x,y (p) and taking into account that the factorial moment measure α (m) is invariant under permutations of its m components, we may write There is at least one term ) in each summand of the last line that will be integrated over R 1 with respect to dp i , so that after expressing α (m) by cumulant measures (see (3.11)), the expectation E|Ξ 0 ∩ (Ξ 0 + v(Φ 0 ), y − x )| 1 emerges and vanishes as → ∞ if x = y. Thus the last line completely vanishes as → ∞, whereas the line (5.3) converges to the limit (2λE|Ξ 0 | 1 ) m /m! as → ∞ by applying the limit (4.1) once more. Therefore for any m ∈ N and x = y, we for some θ 2 ∈ [−1, 1]. The last limit, combined with the above limits of S m (w x ) and S m (w y ), leads to For a given ε ∈ (0, 1], we find a large enough m(ε) such that (2λE|Ξ 0 | 1 ) m /m! ε for all m m(ε). Thus the right-hand side of the last inequality does not exceed 2 ε + ε 2 for sufficiently large m. The same bound can be obtained for the limit (as → ∞) of the four terms in line (5.2). Finally, after summarizing all ε-bounds of the above limiting terms, we arrive at This, together with (5.3), implies −4 Var |Ξ ∩ K| 2 → 0 as → ∞, completing the proof of Theorem 1.
Proof of Theorem 2. Lemma 2 yields the equality (x, y). (5.5) Variance asymptotics for planar cylinder processes 71 We first rewrite the integrand on the right-hand side of (5.5) as follows: To study the behavior of L( x, y) (as → ∞), we use an expansion of log G P [1 − w(·)] in terms of the factorial cumulant measures γ (k) of Ψ ∼ P (see (2.1)): provided the sum in (5.7) is convergent. In view of (5.5) and the inequality |e x − 1 − x| x 2 e max(x,0) /2, we have to find a uniform bound of L( x, y) and to calculate the limits We start by noting that relations (3.10), (4.1), and (5.4) under the assumptions of Theorem 1 imply that L( x, y) d(x, y), (5.11) where the existence of the limit (5.11) has yet to be shown. A rigorous proof that the second limit in (5.8) vanishes as → ∞ together with a uniform bound of L( x, y) will be given after calculation of the first limit in (5.8). By combining (5.5) and (5.6) with limits (5.8)-(5.10) and (5.11) we see that the limit of −3 Var |Ξ ∩ K| 2 as → ∞ (if it exists) coincides with By using expansion (5.7) we are able to express the double integral of (5.8) as follows: where the integrands T n ( x, y) for n ∈ N are defined by w y (p j ) γ (n) (dp 1,n ). (5.12) Since γ (1) (dp) = λ dp and w ∪ x, y (p) − w x (p) − w y (p) = −w ∩ x, y (p), we get −T ( ) where the right-hand limit is just the statement of Lemma 6. The proof of Lemma 6 reveals that |T ( ) We easily see that |T 1 ( x, y)| λE|Ξ 0 | 1 and T 1 ( x, y) In the next step, we derive a uniform bound of T ( ) 2 (K) and determine its limit as → ∞.
Since the integrand in (5.12) for n = 2 is symmetric in x, y and in p 1 , p 2 , it follows that where, in view of |γ (2) red |(R 1 ) < ∞ and the dominated convergence theorem, In the last line, we have used the same arguments as in the proof of Lemma 6, among them, the uniform estimate J (K) 2E|Ξ 0 | 2 1 |K| 2 diam(K). Finally, Lemma 5 and (5.13) show that In addition, we can derive a uniform bound of T ( ) 2 (K). From (5.14) and the above bound of T ( ) Hence we see from (4.4) and (5.12) that for two independent pairs (Ξ i , Φ i ), i = 1, 2, with the same distribution as (Ξ 0 , Φ 0 ), we have the following estimate: red defined (in differential notation) by γ (n) (dp 1,n ) = λ γ (n) red ((dp i − p j : i = j)) dp j for j = 1, . . . , n and the boundedness of the total-variation measure |γ (n) red | on R n−1 , after some Variance asymptotics for planar cylinder processes 73 elementary calculations, we obtain the following estimates: red (dp 2,n ) dp (5.16) and T n,2 (x, y) red (dp 2,n ) dp.
For a Brillinger-mixing PP Ψ ∼ P , it is easy to show that T n,1 ( x, y) λE|Ξ 0 | 1 γ red T V n2 n for n 1. If Ψ ∼ P is strongly Brillinger mixing, that is, γ (n) red T V ab n n! for n 1 (see Definition 2), then we get that |T n ( x, y)| T n,1 ( x, y)+T n,2 ( x, y) λE|Ξ 0 | 1 × (n + 1)a(2b) n n!. Thus we obtain a uniform estimate of L( x, y): Similar uniform bounds of L( x, y) can be shown if Ψ ∼ P is strongly L q -Brillinger mixing (resp., strongly L * q -Brillinger mixing). The derivation of these bounds is completely analogous to the proof of estimate (5.22) (resp., (5.24)) below. The details are left to the reader.
Remark 3. Note that in Theorem 1 (resp., Theorem 2) the interval Ξ 0 := [−R 0 , R 0 ] with ER k 0 < ∞ can be replaced by a finite union of random closed intervals Ξ 0 ⊂ R 1 satisfying E|Ξ 0 | k 1 < ∞ for k = 1 (resp., k = 2). This extension is based on the definition of a process of cylinders with nonconvex bases; see, for example, [20]. Furthermore, the exponential shape of the PGF of a Poisson cluster PP Ψ ∼ P (see [2,3]) allows us to simplify the function L(x, y) in (5.6) and to reduce the Brillinger-mixing-type conditions in Theorems 1 and 2.

Conclusion
Strong Brillinger mixing with b < 1/2 is a rather restrictive condition for the one-dimensional PP Ψ ∼ P . Equivalently formulated, the power series ∞ n=2 (z n /n!)|γ (n) red |(R n−1 ) is analytic in the interior of the disk b(o, 2) in the complex plane. Such a strong condition has been used for statistical analysis of point processes in [4]. The Gauss-Poisson process, Poisson cluster processes with a finite number of nonvanishing cumulant measures, and, among them, certain Neyman-Scott processes (see, e.g., [2]) satisfy this condition. On the other hand, if a PP Ψ ∼ P is strongly L q -Brillinger mixing (resp., strongly L * q -Brillinger-mixing) for some q > 1 with b q > 0 (resp., b * q > 0), then we can choose E|Ξ 0 | 1 sufficiently small to fulfill the assumptions of Theorem 2, which greatly expands its applicability.
Variance asymptotics for planar cylinder processes 79 Another question concerns the asymptotic normality of the scaled and centered total area Z ( ) (K) := −3/2 (|Ξ ∩ K| 2 − E|Ξ ∩ K| 2 ) of the union set (1.1) in K as → ∞. To achieve this goal, we need to find conditions (as mild as possible, but certainly stronger than in Theorem 2) implying that all cumulants Cum k (Z ( ) (K)) = −3k/2 Cum k (|Ξ ∩ K| 2 ) of order k 3 vanish as → ∞. With the notation and by using and extending some results in Section 3 (in particular, Lemma 2) we easily see that Cum k (Z ( ) (K)) → 0 as → ∞ for any fixed k 3 if and only if A profound modification of the recursive technique applied in Section 2 of [13] to prove (6.1) for Poisson CPs seems to be promising. The details will be subject of a separate paper.
Acknowledgment. The authors would like to thank the anonymous reviewer for his careful reading of the original manuscript and for pointing out a gap in the proof of Theorem 2. His critical comments resulted in a substantially improved paper.
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