Limit laws for area coverage in non Boolean models
Tamal Banerjee∗
January 28, 2009
Abstract. Sensor nodes being deployed randomly, one typically models its location by
a point process in an appropriate space. The sensing region of each sensor is described by
a sequence of independent and identically distributed random sets. Hence sensor network
coverage is generally analyzed by an equivalent coverage process. Properties of both area
coverage and path coverage are well known in the literature for homogeneous sensor deployments. We study two models where the sensor nodes are deployed according to a stationary
Cox point process and a non homogeneous Poisson point process respectively. We derive
asymptotic properties of vacancy in both the models.
Key words. Cox point process, non homogeneous Poisson point process, Boolean
model, area coverage, sensor networks.
AMS subject classifications (2000). Primary 60D05; 60G70
Secondary 60K37; 60G55
Department
of
Mathematics,
India, Email: banerjee@math.iisc.ernet.in
∗
Indian
Institute
1
of
Science,
Bangalore-560012,
1
1.1
Introduction
Motivation
Coverage by a sensor network has always been a challenging problem from both theoretical
and application viewpoint. A sensor is a device that measures a physical quantity over a
region and converts it into a signal which can be read by an instrument or an observer. The
union of all such sensing regions in the sensor field is the coverage provided by the sensor
network. Coverage of a sensor network provides a measure of the quality of surveillance
that the network can provide. Some of the common applications of sensor network includes
environmental monitoring, emergency rescue, ambient control and surveillance networks.
Adopting homogeneous scenarios in modeling the sensor locations is often too simplistic.
For example, in applications like battlefield surveillance, an exact area of deployment is
not known, at the same time a finite number of sensors are to be deployed over a large
area. Even after deploying sensors, its location may change over time due to environmental
factors like wind, river stream, rain, etc. Sometimes, a priori knowledge of the sensor field
can be used to determine the concentration of sensors in the sensor field. This may result in
higher concentration of sensors in some parts and lower concentration in others. Non uniform
operational characteristics like interference, frequency of data collection and communication,
etc., also results in non uniform degradation of the network ([9]). An appropriate way to
model such deployments is to assume a non homogeneous distribution for the location of
sensors. For instance, the stochastic environmental heterogeneity for the distribution of the
sensor nodes may be modeled by a Cox point process. This motivates us to consider two
such models for sensor deployment and study their coverage properties. In the first model we
assume that the sensor locations are distributed according to a stationary Cox point process
and in the later model we consider a non homogeneous Poisson distribution for the sensor
locations.
2
1.2
Model Description
Let P ≡ {ξi, i > 1} be a stochastic point process in Rd , d ≥ 1 and {S1 , S2 , ...} be i.i.d random
sets in Rd , independent of P. Then C ≡ {ξi + Si , i > 1} is called a coverage process. If in
addition, P is a stationary Poisson point process then C is a Boolean model. The points of P
may be interpreted as the location of sensors in a random sensor network and the shapes Si
may be thought of as the sensing area about the ith sensor. Instead of working with random
sets Si , we assume the sets Si0 s to be a fixed (non random) non empty, Borel measurable
subset (say S) of Rd with finite content c (i.e, 0 < c = kSk < ∞). We consider the following
two modes of deployment of sensors:(i) Model I
P ≡ {ξi , i ≥ 1} be a Cox point process in Rd . We have the following definition of Cox
point process from [5] (pp. 186-187). Let ∧(x), x ∈ Rd , be a non negative random field
( i.e, a non negative stochastic process indexed in Rd ) defined over some probability
space. Conditional on ∧(x) = λ(x), for x ∈ Rd , let P(λ) be a non homogeneous
Poisson process with intensity function λ. Then P ≡ P(∧) is a Cox point process. We
assume that ∧ is stationary, i.e, E[∧(x)] = λ, a constant not depending on x.
(ii) Model II
P ≡ {ξi , i ≥ 1} be a non homogeneous Poisson process in Rd with intensity function
λ(x). We assume the following bounds on the intensity function λ(x), for every fixed
R ⊆ Rd , with finite content
0 ≤ λl (R ± S) ≤ λ(x) ≤ λu (R ± S) < ∞ ∀ x ∈ R ± S,
(1.1)
the addition and substraction being understood in the Minkowski sense. (1.1) may be
interpreted as the constraints on the number of sensors to be deployed in an operational
3
area R, which we assume to be known for each fixed R. However in Section 3 we study
the limit laws keeping R fixed. Hence we drop the argument and denote λl (R ± S)
(respectively λu (R ± S)) as λl (respectively λu ) for the rest of the paper.
1.3
Previous Work
The coverage problem is one of the oldest in geometric probability. Coverage processes arising
naturally in stochastic geometry have been studied in [12] along with their applications.
However the Boolean model seems to be the most famous random set model in stochastic
geometry. The area coverage properties for the Boolean model have been extensively studied
in the literature, most notably in [5]. One may also see [8] for some recent results in the case of
one coverage. The coverage of a line by a two dimensional Boolean model was investigated in
[13]. In [10] the authors considers a coverage process where the shapes Si have distributions
that depend on the locations of their centers. They analyze the probability of the event
that the whole of Rd will be covered by such a coverage process. The growth of tumor cells
have been modeled using coverage process in [2]. The statistical properties of the coverage
of a one dimensional path induced by a two dimensional non homogeneous random sensor
network have been studied in [9]. In [7] coverage by a finite number of heterogeneous sensors
is analyzed using integral geometry. The Cox process have been actively applied in Finance
and Risk Theory [1, 6] and in forestry [11]. To the best of our knowledge, the problem of
area coverage in the two models, described above has not been addressed before.
1.4
Organization of the Paper and Summary of Results
Our paper is in the same sprit as those that study area coverage (as in [5]). In Section
2 we consider a coverage process arising out of a Cox point process (Model I). We derive
the expectation and variance of vacancy. Finally we study the asymptotic vacancy under
suitable scaling and obtain a central limit theorem for the vacancy.
4
We carry out similar type of analysis in Section 3 when P is a non homogeneous Poisson
process (Model II). We study the asymptotic vacancy under suitable conditions on the intensity function of the non homogeneous Poisson process and derive a central limit theorem
for the vacancy. The techniques of the proofs in both cases are in general similar to those in
Chapter 3 of [5]. We provide a detailed proof for the Cox process and indicate a sketch in
the other case.
2
2.1
Coverage in a Cox point process (Model I)
Expectation and Variance of Vacancy
Consider a coverage process C ≡ {ξi + Si , i > 1} when P is as in (i) of Section 1.2. Let R be
a Borel subset of Rd , 0 < kRk < ∞ and Si ≡ S ∀ i ≥ 1, S being a fixed (non random) non
empty, Borel measurable subset of Rd with finite content c. Define the following indicator
function for a point x ∈ Rd


 1 if x ∈
/ ξi + S, ∀ i > 1,
χ(x) =

 0 otherwise.
We define the vacancy VC within R, to be the d-dimensional volume of the part not covered
by C, i.e.,
VC = VC (R) =
Z
R
5
χ(x) dx.
(2.1)
By Fubini’s theorem and stationarity of the Cox point process we obtain from [5]
Z
χ(x)dx
E(VC ) = E
Z R
=
P r[ x ∈
/ ξi + S, ∀ i ≥ 1] dx
ZR
=
P r[ ξi ∈
/ x − S, ∀ i ≥ 1] dx
ZR
=
P r[ ξi ∈
/ −S, ∀ i ≥ 1] dx
R
h R
= kRkE e− −S ∧(x)
dx
i
.
Similarly, we have
i
h R
−
∧(x) dx
E[χ(x1 ) χ(x2 )] = E e (x1 −x2 −S)∪(−S)
R
R
−
∧(x) dx+ −S ∧(x)
= E e x1 −x2 −S
h R
= E e−2 −S ∧(x)
dx
R
.e
dx−
(x1 −x2 −S)∩(−S)
R
(x1 −x2
∧(x) dx
i
∧(x) dx
−S)∩(−S)
.
(2.2)
The last line following from the stationarity of the Cox point process.
Cov[χ(x1 ) χ(x2 )] = E[χ(x1 ) χ(x2 )] − E[χ(x1 )]E[χ(x2 )]
h R
i h R
R
∧(x) dx
= E e−2 −S ∧(x) dx .e (x1 −x2 −S)∩(−S)
− E e− −S ∧(x)
dx
i2
.
(2.3)
Hence
V AR(VC ) =
Z Z
Cov[χ(x1 ) χ(x2 )] dx1 dx2
Z Z h R
R
∧(x)
E e−2 −S ∧(x) dx .e (x1 −x2 −S)∩(−S)
=
R
R
R
R
6
dx
i
h R
− E e− −S ∧(x)
dx
i2 dx1 dx2 .
(2.4)
2.2
Limit Laws for Model I
Consider a coverage process C(δ) with fixed shapes S and P same as in (i) of Section 1.2.
We scale the shapes S by δ (δ < 1) in the C(δ) model. Let VC be the vacancy within the
region R (0 < kRk < ∞) arising from the C(δ) model. An excellent discussion for studying
limit laws in scaled models can be found in Section 3.4 of [5]. Recall that λ = E[∧(x)].
R
∧(δz) dz
→ ∞ a.s.
Theorem 2.1. Consider the scaled coverage process C(δ). Let δ → 0 as B kBk
R
R
dR
∧(δz) dz
∧(δz) dz
such that δ d B kBk
→ ρ a.s. (0 < ρ < ∞), B kBk
V AR e−δ −S ∧(δz) dz →
R
0 a.s. ∀ B ⊆ Rd , 0 < kBk < ∞, B ∧(x) > 0 a.s. and δ d λ → l, l being any positive
constant. Then,
(i)
E(VC ) →k R k e−ρkSk .
(2.5)
V AR(VC ) → 0.
(2.6)
(ii)
(iii)
E|VC − E(VC )|p → 0,
7
1 ≤ p < ∞.
(2.7)
(iv)
R
B
∧(δz) dz
kBk
a.s.
V AR(VC ) −−→ σ 2
−2ρkSk
≡ ρkRke
Z
Rd
eρk(y−S)∩(−S)k − 1 dy.
R
(2.8)
∧(δz) dz
→ ∞ a.s.
Theorem 2.2. Consider the scaled coverage process C(δ). Let δ → 0 as B kBk
R
R
dR
−δ −S ∧(δz) dz
d
B ∧(δz) dz
B ∧(δz) dz
such that δ
→ ρ a.s. (0 < ρ < ∞),
V AR e
→
kBk
kBk
R
0 a.s. ∀ B ⊆ Rd , 0 < kBk < ∞,
∧(x) > 0 a.s. and δ d λ → l. Then,
B
R
B
where
∧(δz) dz
kBk
1/2
d
(2.9)
∧(δz) dz
kBk
→ ∞ a.s. ⇒ λ → ∞. Hence the
{VC − E(VC )} −
→ N(0, σ 2 ) a.s.
σ 2 is same as in (2.8).
Remark 2.1. From the scaling laws we have
R
B
condition δ d λ → l is needed for the above results to hold.
dR
−δ −S ∧(δz) dz
Remark 2.2. If ∧(x) ≡ γ, a constant then V AR e
= 0. Therefore the condition
dR
R
B ∧(δz) dz
V AR e−δ −S ∧(δz) dz → 0 a.s. is not required in [5] to prove the central limit
kBk
theorem for vacancy in a Boolean model.
Remark 2.3. The limit laws of vacancy for the Boolean model can be derived from the above
theorems by choosing ∧(x) ≡ γ, a constant.
8
3
Coverage in a Non Homogeneous Poisson Process
(Model II)
3.1
Expectation and Variance of Vacancy
Consider a coverage process C ≡ {ξi + Si , i > 1} when P is as in (ii) of Section 1.2. Let R
be a non empty Borel subset of Rd with finite content and Si ≡ S ∀ i ≥ 1, S being a fixed
(non random) non empty, Borel measurable subset of Rd with finite content c. Define the
following indicator function for a point x ∈ Rd


 1 if x ∈
/ ξi + S, ∀ i > 1,
χ(x) =

 0 otherwise.
We define the vacancy VN within R, to be the d-dimensional volume of the part not covered
by C, i.e.,
VN = VN (R) =
Z
χ(x) dx.
(3.1)
R
By similar calculations as in Section 2.1 we obtain
Z
χ(x)dx
E(VN ) = E
Z R
=
P r[ x ∈
/ ξi + S, ∀ i ≥ 1] dx
R
Z
=
P r[ ξi ∈
/ x − S, ∀ i ≥ 1] dx
R
Z R
=
e− x−S λ(t) dt dx.
R
9
(3.2)
Similarly, we have
E[χ(x1 )χ(x2 )] = P r[ x1 ∈
/ ξi + S, ∀ i ≥ 1 and x2 ∈
/ ξi + S, ∀ i ≥ 1 ]
= P r[ ξi ∈
/ x1 − S, ∀ i ≥ 1 and ξi ∈
/ x2 − S, ∀ i ≥ 1 ]
= P r[ ξi ∈
/ (x1 − S) ∪ (x2 − S), ∀ i ≥ 1 ]
−
R
−
R
=e
=e
(x1 −S)∪(x2 −S)
x1 −S
λ(t) dt
λ(t) dt+
R
λ(t) dt−
x2 −S
R
(x1 −S)∩(x2 −S)
λ(t) dt
.
(3.3)
Cov[χ(x1 ) χ(x2 )] = E[χ(x1 ) χ(x2 )] − E[χ(x1 )]E[χ(x2 )]
−
=e
R
x1 −S
λ(t) dt+
R
x2 −S
λ(t) dt−
R
(x1 −S)∩(x2 −S)
λ(t) dt
R
R
−
λ(t) dt
−
λ(t) dt
− e x1 −S
e x2 −S
.
(3.4)
Hence
V AR(VN ) =
Z Z
R
−
e
R
R
x1 −S
λ(t) dt+
R
x2 −S
λ(t) dt
h R
λ(t)
e (x1 −S)∩(x2−S)
dt
i
− 1 dx1 dx2 .
(3.5)
We have the following bounds for V AR(VN ) from (1.1)
−2λu kSk
e
×
Z Z
R
R
≤ V AR(VN )
−2λl kSk
≤e
×
eλl k(x1 −x2 +S)∩Sk − 1 dx1 dx2
Z Z
R
3.2
R
eλu k(x1 −x2 +S)∩Sk − 1 dx1 dx2 .
(3.6)
Limit Laws for Model II
Consider a coverage process C(δ) with fixed shapes S, P same as in (ii) of Section 1.2 and the
intensity function λ(x) of the non homogeneous Poisson point process satisfying the bound
10
in (1.1). We scale the shapes S by δ (δ < 1) in this C(δ) model. Let VN be the vacancy
within the region R (0 < kRk < ∞) arising from the C(δ) model. To obtain non trivial
coverage we assume λl > 0 and λu > 0. We obtain the following results by same techniques
as in [5].
Lemma 3.1. If δ → 0, as λl → ∞ and λu → ∞, such that δ d λl → ρl and δ d λu → ρu , where
0 < ρl ≤ ρu < ∞, then for the scaled process C(δ),
lim sup E(VN ) ≤ kRke−ρl kSk
(3.7)
λl ,λu →∞
and
lim inf E(VN ) ≥ kRke−ρu kSk .
(3.8)
E(VN ) → kRke−ρkSk .
(3.9)
λl ,λu →∞
If ρl = ρ = ρu , then
Lemma 3.2. If δ → 0, as λl → ∞ and λu → ∞, such that δ d λl → ρl and δ d λu → ρu , where
0 < ρl ≤ ρu < ∞, then for the scaled process,
(i)
V AR(VN ) → 0,
(even if ρl 6= ρu) .
(3.10)
(ii)
E|VN − E(VN )|p → 0,
1 ≤ p < ∞.
(3.11)
(iii)
−2ρl kSk
lim sup λl V AR(VN ) ≤ ρl kRke
λl ,λu →∞
Z
Rd
and
−2ρl kSk
lim sup λu V AR(VN ) ≤ ρu kRke
λl ,λu →∞
Z
Rd
11
eρu k(y+S)∩Sk − 1 dy
eρu k(y+S)∩Sk − 1 dy.
(3.12)
(3.13)
(iv)
−2ρu kSk
lim inf λl V AR(VN ) ≥ ρl kRke
λl ,λu →∞
Z
Rd
and
−2ρu kSk
lim inf λu V AR(VN ) ≥ ρu kRke
λl ,λu →∞
Z
Rd
eρl k(y+S)∩Sk − 1 dy
eρl k(y+S)∩Sk − 1 dy.
(3.14)
(3.15)
Lemma 3.3. If ρl = ρ = ρu then under the same scaling law as in Lemma 3.1 we have
for the scaled process,
λi V AR(VN ) → σ 2
−2ρkSk
≡ ρkRke
Z
Rd
eρk(y+S)∩Sk − 1 dy
f or i ∈ {l, u}.
(3.16)
Lemma 3.4. Consider the scaled coverage process C(δ). If δ → 0, as λl → ∞, and λu → ∞,
such that δ d λl → ρ and δ d λu → ρ, 0 < ρ < ∞, then for the scaled process,
1/2
d
→ N(0, σ 2 ) f or i ∈ {l, u}
λi {VN − E(VN )} −
where
(3.17)
σ 2 is same as in (3.16).
Remark 3.1. As mentioned in Section 1.2 both ρu and ρl will depend on R and S. However,
since R and S are kept fixed in the entire analysis, we do not indicate their dependence in
the notation.
Remark 3.2. As λl → ∞ and λu → ∞, the intensity function λ(x) → ∞, pointwise for all x
in the area of interest. Hence the scaling law implies δ d λ(x) → ρ, pointwise for all x in that
region.
12
4
Proof of Main results
4.1
Proof of Results in Model I
Proof of Theorem 2.1. (2.5) follows trivially by dominated convergence theorem. By (2.4)
and the inequality ex − 1 ≤ xex , x ≥ 0, we have,
V AR(VC )
Z Z h R
i2 i h R
R
∧(x) dx
− −δS ∧(x) dx
−2 −δS ∧(x) dx
(x1 −x2 −δS)∩(−δS)
dx1 dx2
− E e
=
E e
.e
R R
R
R
Z Z
Z
−2 −δS ∧(x) dx
(x1 −x2 −δS)∩(−δS) ∧(x) dx
∧(x) dx
e
dx1 dx2
≤
E e
R
(x1 −x2 −δS)∩(−δS)
R
R
+ kRk V AR e− −δS ∧(x)
Z
Z Z
R
∧(x) dx dx1 dx2 + kRk2 V AR e− −δS ∧(x) dx
≤
E
(x1 −x2 −δS)∩(−δS)
R R
R
x1 − x2
− −δS ∧(x) dx
2
d
≤ kRk δ λk(
− S) ∩ (−S)k + V AR e
.
δ
2
dx
(4.1)
The last line following from Fubini’s theorem and the fact that E[∧(x)] = λ. Both the terms
in (4.1) converges to zero under the given scaling law and the fact that S has a finite content.
Hence (2.6) follows. By Chebychev’s inequality,
P [|VC − E(VC )| > ] ≤
V AR(VC )
→ 0.
2
Hence VC − E(VC ) → 0 in probability. Since 0 ≤ VC ≤ kRk, the dominated convergence
theorem gives us the Lp convergence in (2.7). Applying the change of variables x1 − x2 = y
and x2 = x, we obtain from (2.4)
V AR(VC ) = δ
d
Z
R
dx
Z
(x−R)
δ
h
R
d
E e−2δ −S ∧(δz)
dz
δd
.e
R
(y−S)∩(−S)
∧(δz) dz
i
h dR
− E e−δ −S ∧(δz)
dz
i2 (4.2)
13
dy.
It is easy to see under the given scaling law
h
R
R
d
d
fδ (x) =
E e−2δ −S ∧(δz) dz .eδ (y−S)∩(−S) ∧(δz)
δ−1 (x−R)
Z
−2ρkSk
eρk(y−S)∩(−S)k − 1 dy.
−→ m ≡ e
Z
dz
i
h dR
− E e−δ −S ∧(δz)
dz
i2 dy
(4.3)
Rd
By dominated convergence theorem,
R
B
as
R
B
∧(δz) dz
kBk
∧(δz) dz
kBk
V AR(VC ) → ρ
Z
m dx ≡ ρmkRk,
R
→ ∞ a.s. Hence we have (2.8).
Proof of Theorem 2.2. Let r be a large positive constant. We divide all of Rd into a regular
lattice of d dimensional cubes of side length (τ rδ),where τ = 2c = 2kSk, so that each cube is
separated from its adjacent cube by a spacing strip of width (2τ δ). Let A1 denote the union
of all the cubes which are wholly contained within R. A2 be the union of all the spacing
strips that are wholly contained within R. Finally we denote A3 as the intersection with R
of all the spacing strips and the cubes that are contained only partially within R. The above
(i)
configuration is illustrated in Figure 1 for dimension d=2. Let VC be the vacancy within
the region Ai . The vacancy within R can be expressed as,
(1)
(2)
(3)
V = VC + VC + VC .
As δ → 0, under the assumption of the theorem the cubes gets finer and we have
kA3 k → 0 a.s.
14
(4.4)
R
Figure 1: The region shaded in dark represents A1 whereas the lightly shaded region represents A2 . The unshaded part within the region R represents A3 .
We also have,
l
kA2 k ≤ ,
r
where l is a constant independent of r.
(4.5)
(i)
The vacancy VC within the region Ai satisfies
(i)
V AR(VC )
Z Z h R
i h R
i2 R
−2 −δS ∧(x) dx
− −δS ∧(x) dx
(x1 −x2 −δS)∩(−δS) ∧(x) dx
=
E e
− E e
dx1 dx2
.e
Ai Ai
R
R
Z Z
Z
∧(x)
dx
−2 −δS ∧(x) dx
≤
E e
∧(x) dx e (x1 −x2 −δS)∩(−δS)
dx1 dx2
Ai
≤
Ai
Z Z
Ai
Rd
(x1 −x2 −δS)∩(−δS)
E
Z
(x1 −x2 −δS)∩(−δS)
R
+ kAi k2 V AR e− −δS ∧(x)
R
∧(x) dx dx1 dx2 + kAi k2 V AR e− −δS ∧(x) dx
R
≤ kAi kδ 2d λ(kSk)2 + kAi k2 V AR e− −δS ∧(x)
dx
.
dx
(4.6)
The last line following from Fubini’s theorem and stationarity of the Cox process. From
(4.4), (4.5) and Remark 2.1, we have,
R
lim
B ∧(δz) dz→∞
R
B
∧(δz) dz
kBk
15
(3)
V AR(VC ) = 0 a.s.
(4.7)
and
lim
lim sup
r→∞ R ∧(δz) dz→∞
B
R
∧(δz) dz
B
kBk
(2)
V AR(VC ) = 0 a.s.
(4.8)
(1)
Hence we observe that the only significant term involves VC and to prove the central limit
theorem it is enough to show that,
(1)
(1)
(1)
d
{VC − E(VC )}/(V AR(VC ))1/2 −
→ N(0, 1) a.s.
(4.9)
and
lim
lim sup
r→∞ R ∧(δz) dz→∞
B
R
B ∧(δz) dz
1
V AR(VC ) − V AR(VC ) = 0 a.s.
kBk
(4.10)
Let n denote the number of small cubes of length τ rδ which make up the region A1 , and let
Di denote the ith of these cubes, for 1 ≤ i ≤ n. We further denote by Ui the contribution
P
(1)
(1)
to VC from Di . Then we have VC = ni=1 Ui . Since the shape δS is contained within a
sphere of radius τ δ, and the cubes Di are least 2τ δ distance apart, the shape δS can not
intersect more than one cube. Due to stationarity of the Cox point process the variables Ui
are identically distributed. Hence Ui0 s are i.i.d random variables and we have
V
(1)
AR(VC )
=
n
X
V AR(Ui ) = nV AR(Ui ).
i
Let D be a d dimensional cube of side length τ r having the same orientation as D1 . For any
two real sequences an and bn an ∼ bn implies an /bn → 1 as n → ∞. From (2.4) we obtain
(1)
V AR(VC )
Z Z h R
i h R
R
−2 −δS ∧(x) dx
(x1 −x2 −δS)∩(−δS) ∧(x) dx
=n
E e
.e
− E e− δS ∧(x)
Di Di
Z Z
2d −2ρkSk
∼ nδ e
eρk(x1 −x2 −S)∩(−S)k − 1 dx1 dx2 .
D
D
16
dx
i2 dx1 dx2
Since n = O
R
∧(δz) dz
B
a.s. we have
P
3
E|Ui − E(Ui )|2 E|Ui − E(Ui )|
i E|Ui − E(Ui )|
P
P
=
( i V ar(Ui ))3/2
( i V ar(Ui ))3/2
2 k Di k V ar(Ui )
≤ P
( i V ar(Ui ))3/2
Z
−1/2 !
2(τ rδ)d
∧(δz) dz
= P
=O
→0
( i V ar(Ui ))1/2
B
as
R
B
(4.11)
∧(δz) dz → ∞ a.s. Hence (4.9) follows by Lyapunov’s central limit theorem. By
Cauchy-Schwarz inequality we have
(1) V AR(VC ) − V AR(VC )
i2 h
(1)
2
= E [VC − E (VC )] − E VC − E VC
h
i
(2)
(3)
(2)
(3)
(2)
(3)
(2)
(3)
×
2V
−
V
−
V
−
E
2V
−
V
−
V
= E VC + VC − E VC + VC
C
C
C
C
C
C
h
i1/2 h
i1/2
(2)
(3)
(2)
(3)
≤ 4 V AR VC + V AR VC
V AR (VC ) + V AR VC + V AR VC
.
(4.12)
(4.10) follows from (2.8), (4.7), (4.8) and (4.12). Hence as
Theorem 2.2.
Remark 4.1. The condition
R
showing (4.3), (4.7) and (4.8).
4.2
B
∧(δz) dz
kBk
−δd
V AR e
R
−S
R
B
∧(δz) dz → ∞ a.s. we have
∧(δz) dz
→ 0 a.s. is used only in
Proof of Results in Model II
By using asymptotic properties of vacancy for the Boolean model (as in [5], pp. 141-158),
one can derive all the results for asymptotic vacancy in Section 3.2. While investigating the
asymptotic properties of vacancy for the Cox point process in Section 2.2, we have modified
the proofs given in [5] and illustrated the detailed techniques. Hence to avoid repetitions we
17
indicate only a sketch of the proofs in the non homogeneous model. In all the proof that
follows the operational area R is kept fixed.
Proof of Lemma 3.1. We have from (3.2)
kRke−λu kSk ≤ E(VN ) ≤ kRke−λl kSk .
(4.13)
Thus the expected vacancy is bounded from above (respectively from below) by the expectation of vacancy in a Boolean model driven by a Poisson point process with intensity λl
(respectively λu ) and generated by the same shapes S. Now scaling the shapes S by δS we
have,
kRke−λu δ
d kSk
≤ E(VN ) ≤ kRke−λl δ
d kSk
.
Hence (3.7) and (3.8) follows under the scaling law in Lemma 3.1. In the case when ρu =
ρ = ρl , (3.9) follows trivially.
Proof of Lemma 3.2. From equation (3.6) for the scaled process we have
−2λl δd kSk
V AR(VN ) ≤ e
Z Z h
i
x −x
λu δd k( 1 δ 2 +S)∩Sk)
(
×
e
− 1 dx1 dx2 .
R
(4.14)
R
(3.10) now follows by the same argument as in (2.6) of Theorem 2.1. By Chebychev’s
inequality,
P [|VN − E(VN )| > ] ≤
V AR(VN )
→ 0.
2
Hence VN − E(VN ) → 0 in probability. Since 0 ≤ VN ≤ kRk, the dominated convergence
theorem gives us the Lp convergence in (3.11). By making the change of variable, x1 −x2 = y
and x2 = x, as in the proof of (2.8) in Theorem 2.1 one can prove (3.12), (3.13), (3.14) and
18
(3.15).
Lemma 3.3 follows trivially from Lemma 3.2 when ρl = ρ = ρu .
Proof of Lemma 3.4. We proceed exactly as in the proof of Theorem 2.2. Unless otherwise
stated all the notations that are used in the proof of Theorem 2.2 will have an analogous
meaning for Model II. We obtain the following equations for the scaled process from (3.6)
and [5].
V
(i)
AR(VN )
−2λl kδSk
≤e
×
Z Z
Ai
≤ λu δ 2d eλu
δd kSk
Ai
(λu k(x1 −x2 +δS)∩δSk)
e
− 1 dx1 dx2
kAi kkSk2.
(4.15)
(3)
λu V AR VN = 0
(4.16)
Hence
lim
λl ,λu →∞
and
lim lim sup λu V AR
r→∞ λ ,λu →∞
l
(2)
VN
= 0.
(4.17)
Hence to prove the central limit theorem it is sufficient to prove that
(1)
(1)
(1)
d
{VN − E(VN )}/(V AR(VN )1/2 −
→ N(0, 1),
(4.18)
(1)
lim lim sup λu V AR VN − V AR(VN ) = 0.
(4.19)
and
r→∞ λ ,λu →∞
l
19
Pn
3
(τ rδ)d
i=1 E|Ui − E(Ui )|
P
P
≤
n
n
( i=1 V AR(Ui ))3/2
( i=1 V ar(Ui ))1/2
≤
(ne−2λu kδSk ×
R
Di
R
(τ rδ)d
Di
[eλl k(x1 −x2 +δS)∩δSk − 1] dx1 dx2 )1/2
= O(λu−1/2 ) −→ 0 under the given scaling law.
(4.20)
Therefore (4.18) follows by Lyapunov’s central limit theorem. The rest of the proof follows as
in ([5] pp. 157-158). The other statement in Lemma 3.4 can be proved in a similar way.
5
Related Problems
The results of [5] for asymptotic vacancy in a Boolean model have been generalized in [4],
using the notion of “associated random measures”. It could be interesting to obtain similar
generalization results for the coverage processes studied in our paper. It is not clear if one
can obtain a central limit theorem for the vacancy in non homogeneous deployment of sensors
without imposing condition (1.1) on the intensity function. One can carry out a similar type
of analysis of path coverage as in [9, 13] for the Cox point process.
Acknowledgements
The present author would like to thank Debleena Thacker for drawing the figure and providing important suggestions in preparing this report.
20
References
[1] S. BASU AND A. DASSIOS, A Cox process with log-Normal intensity, Insurance: Mathematics and Economics, Volume 31, pp. 297-302, 2002.
[2] N. CRESSIE AND F. L. HUNTING, A spatial statistical analysis of Tumor growth,
Journal of the American Statistical Association, Volume 87, No. 418, 1992.
[3] N. EISENBAUM, A Cox process involved in the Bose-Einstein condensation, Annales
Henri Poincare, Birkhauser Basel, Volume 9, No. 6, Oct. 2008.
[4] STEVEN N. EVANS, Rescaling the Vacancy of a Boolean Coverage Process, Seminar
on Stochastic Processes, Birkhauser, pp. 23-34, 1989.
[5] P .HALL, Introduction to the Theory of Coverage Process , John Wiley and Sons, 1988.
[6] DAVID LANDO, On Cox processes and credit risky securities, Review of Derivatives
Research, Springer Netherlands, Volume 2, No. 2-3, pp. 99-120, Dec. 1998.
[7] L. LAZOS AND R. POOVENDRAN, Stochastic coverage in heterogeneous sensor networks, ACM Transactions on Sensor Networks 2, 3 (August), pp. 325 - 358, 2006.
[8] B. LIU AND D. TOWSLEY, A study on the coverage of large scale networks, Proceedings of ACM MobiHoc, 2004.
[9] P. MANOHAR, S. SUNDHAR RAM AND D. MANJUNATH , On the path coverage
properties by a non homogeneous sensor field, Proceedings of IEEE Globecom, San
Francisco CA, USA, Nov 30-Dec 02, 2006.
[10] I. MOLCHANOV AND V. SCHERBAKOV, Coverage of the Whole Space, Advances in
Applied Probabilty (SGSA) 35, pp. 898-912, 2003.
21
[11] J. MOLLER, A. R. SYVERSVEEN AND R. P. WAAGEPETERSEN, Log Gaussian Cox
processes: A statistical model for analyzing stand structural heterogeneity in forestry,
Proceedings of First European Conference for Information Technology in Agriculture,
Copenhagen, 1997 (to appear).
[12] D. STOYAN, W. S. KENDALL AND J. MECKE, Stochastic geometry and its Applications, John Wiley and Sons, 2nd edition, 1995.
[13] S. SUNDHAR RAM, D. MANJUNATH, S. K. IYER, AND D. YOGESHWARAN, On
the path sensing properties of random sensor networks, IEEE Transactions on Mobile
Computing, pp. 446 - 458, May 2007.
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