Random Marked Sets

0.35 MB·

by  Felix Ballani

9 Random Marked Sets
0
0
2 Felix Ballani, Zakhar Kablu
hko, Martin S
hlather

r
a
M Institute for Mathemati
al Sto
hasti
s, Georgia Augusta University,
Golds
hmidtstr. 7, D-37077 Göttingen, Germany,

3
1 ballanimath.uni-goettingen.de, kablu
hmath.uni-goettingen.de,

s
hlathermath.uni-goettingen.de
]
R
P Mar
h 13, 2009
.
h
t
a
m Abstra
t. We introdu
eRad new
lass of sto
hasti
pro
esses whi
h are
[ de(cid:28)ned on a random set in . These pro
esses
an be seen as a link be-

tween random (cid:28)elds and marked point pro
esses. Unlike for random (cid:28)elds,
1
v the mark
ovarian
e fun
tion need in general not be positive de(cid:28)nite. This
8
implies that in many situations the use of simple geostatisti
al methods ap-
8
pears to be questionable. Surprisingly, for a spe
ial
lass of pro
esses based
3
2 on Gaussian random (cid:28)elds, we do have positive de(cid:28)niteness for the
orre-
.
3 sponding mark
ovarian
e fun
tion and mark
orrelation fun
tion.
0
9
Classi(cid:28)
ation. Primary: 60G60, 60G55; se
ondary: 60G15, 60D05
0
:
v Keywords. random (cid:28)eld, random set, marked point pro
ess, mark
or-
i
X relation fun
tion, mark
ovarian
e fun
tion
r
a
1 Introdu
tion
Rd
Quantities measured in are mostly modelled as so-
alled regionalized
variables, i.e., one usuallyRdassumes that these quantities
an, inprin
iple, be
measured everywhere in and that the
hoi
e of sampling points does not
depend on the values of these quantities. Based on this assumption, several
geostatisti
al methods like variogram analysis or kriging
an be applied, see
[6℄. However, there are two types of situations where this assumption does
not hold [23℄ and hen
e, un
riti
al use of geostatisti
al methods might
ause
in
orre
t or meaningless results.
1
The (cid:28)rst type of problems is
aused by the investigators themselves by
some kind of preferential sampling [9℄. For instan
e, this happens when data
are sampled only at pla
es where high values of the variable of interest are
expe
ted. The se
ond type of problems is intrinsi
to the investigated obje
t
itself. An obvious situation is the investigation of individuals, e.g. trees in a
forest, where intera
tions among individuals are present. In this parti
ular
situation the theory of marked point pro
esses provides a formal framework
for data analysis [10, 12℄.
Here, we like to draw the attention to some further, de
eptive situations
whereimpli
it
onditioning hasbeenmostlyignoredinliterature[13,16,29℄.
For instan
e, the investigation of pesti
ides in soil is restri
ted to
ropland
andtheheightofforestlitterisrestri
tedtosilvi
ulturalareas. Inboth
ases,
a presele
tion
annot be ex
luded sin
e environmental
onditions dire
tly
in(cid:29)uen
e the kind of land use. A further, simple example has motivated this
work and appears when the altitude is predi
ted by geostatisti
al methods
based on measurements that are taken above sealevel only.
Su
h kind of
onditioning might be
onsidered as minor, but
an
ause
major e(cid:27)e
ts, nonetheless. We advise
aution be
ause of the following fa
ts:
1. Any
hara
teristi
, su
h as the
ovarian
e fun
tion or the variogram,
has to be understood as a
onditional quantity given measurements
an be
taken at
ertain lo
ations.
2. In general, neither the
ovarian
e fun
tion is positive de(cid:28)nite nor the
variogram is
onditionally negative de(cid:28)nite.
Sin
e Gaussian random (cid:28)elds are rather popular, a bigger part of this
paper deals with the following model: the sealevel is at 0 and the altitude is
Z t
given by some (smooth) stationary Gaussian random (cid:28)eld with mean −
1
and varian
e . Then we fa
e the following oddities when inferen
e is based
on measurements above sealevel only:
1. The theoreti
al variogram is not
onditionally negative de(cid:28)nite, in
general.
C(x,y)
2. A naive de(cid:28)nition of the
ovarian
e fun
tion by
C(x,y) = E[Z(x)Z(y) Z(x) 0,Z(y) 0] m2
| ≥ ≥ −
m R
leads in general to a fun
tion whi
h is not positive de(cid:28)nite, for any ∈ .
3. A more suitable de(cid:28)nition of the
ovarian
e fun
tion for the altitude
Z(x) 0 Z(y)
above sealevel asthe
onditional
ovarian
e given that ≥ and ≥
0
leads to a fun
tion whi
h is never di(cid:27)erentiable.
t = 0
4. If , the
onditional
ovarian
e fun
tion is positive de(cid:28)nite.
Though, no random (cid:28)eld exists that is independent of the sampling lo
a-
tions and that
an model the altiditude above sealevel.
2
Before dis
ussing the above set up in detail, we will introdu
e a theo-
reti
al framework so that both a meaningful de(cid:28)nition of se
ond-order
har-
a
teristi
s is possible and usual random (cid:28)elds as well as marked point pro-

esses are in
luded as parti
ular
ases. For this reason we extend the no-
tRio=n o[f a r,and]om uRppder semi-
ontinuous (u.s.
.) fun
tion (taking vaRludes in
−∞ ∞ ) on su
h that the domain is a random subset of . To
this end we make use of Matheron's [17℄ idea and
onsider the hypograph
A = (x,t) X R : t f(x) , X Rd
f
{ ∈ × ≤ } ⊆
f : X R A f
f
of a fun
tion → . In fa
t, is
losed if and only if is u.s.
. on
X f A
f

losed , and the mapping 7→ is a bije
tion.
The paper is organized as follows. In Se
tion 2 we formally introdu
e
the notion of a random marked
losed set and dis
uss some examples. In
Se
tion 3 we generalise the de(cid:28)nition of several
hara
teristi
s for random
(cid:28)elds to random marked sets. We show that, in general, they do not share
the same de(cid:28)niteness properties as their random (cid:28)eld analogues. In Se
tion
Z(x) Z(x)
4 we studyt GaRussian random (cid:28)elds given that ex
eeds a
ertain
threshold ∈ . In Se
tion 5 some resultson the di(cid:27)erentiablity of the mark

ovarian
e fun
tion ofrandommarkedsetsaregiven. InSe
tion6,we
olle
t
the proofs of the statements of the pre
eding se
tions.
2 Random marked
losed sets
R = R ,+
Denote by ∪{−∞ ∞} the extended real line. Let
Φ = (X,f) : X Rd , f : X R .
usc
{ ⊆ is
losed → is u.s.
.}
Φ A Rd R
usc cl
is isomorphi
to the system U of all
losed sets ⊆ × whi
h
satisfy
x Rd t R: (x,t) A x [ ,t] A
∀ ∈ ∀ ∈ ∈ ⇒ { }× −∞ ⊆ (1)
by the bije
tion
τ : Φ
usc cl
→ U
(X,f) (x,t) X R : t f(x) , (X,f) Φ .
usc
7→ { ∈ × ≤ } ∈
Th(RedsubsReq)uent propositionfollowRsdimmRediately from thefa
t thatthespa
e
cl
F × (oRfd
losRe)d subsets of × is
ompa
t [17, 19℄ and that U is

losed in F × .
3
Φ
usc cl
Proposition 1. is
ompa
t in the topology indu
ed by U .
(Ω, ,P) (Ξ,Z) :
De(cid:28)nition 1. Let be A a
omplete probability spa
e and let
Ω Φ
usc
→ be a mapping with
ω Ω : τ(Ξ,Z) B = ∅
{ ∈ ∩ 6 } ∈ A
B Rd R (Ξ,Z)
for every
ompa
t set in × . Then is
alled a random marked

losed set.
The distribution law of a random
losed set is
hara
terized by the prob-
abilities of hitting
ompa
t sets [18, 19℄ whereas, by [18, Prop. 2.3.1℄, it
su(cid:30)
es to restri
t to a suitable base. When
hoosing the same base of all
B [t , ]
i i
(cid:28)nite unions of halfR
ydlinders × ∞ as in [24, Thm. XII-6℄ for random
u.s.
. fun
tions on , we obtain the following
hara
terization of random
marked
losed sets.
(Ξ,Z)
Theorem 1. The distribution of a random marked
losed set (as a
Φ
usc
probability measure on ) is
ompletely determined by the joint probabili-
ties
P sup Z(x)< t , B Ξ = ∅, i I; B Ξ = ∅, j 1,...,n I ,
i i j
∩ 6 ∈ ∩ ∈ { }\
x∈Bi∩Ξ
(cid:0) (cid:1)
B ,...,B Rd t ,...,t R I
1 n 1 n
wher1e,...,n n aNre
ompa
t subsets of , ∈ , and is a subset
of { }, ∈ .
(Ξ,Z)
De(cid:28)nition 2. A random marked
losed set is
alled stationary if
P(τ(Ξ,Z)+(x,0) ) = P(τ(Ξ,Z) )
∈ · ∈·
x Rd
for all ∈ , and it is
alled isotropi
if
P(θτ(Ξ,Z) ) = P(τ(Ξ,Z) )
∈ · ∈ ·
θ SO θ(Rd 0 ) = Rd 0
d+1
for all rotations ∈ with ×{ } ×{ }.
Example 1. A parti
ular model of a random marked
losed set that de-
Z
s
ribes an unbiased sampliRngd of a random (cid:28)eld [27℄ is given when is a
random u.s.
. fun
tion on that is independent of the random
losed set
Ξ (Ξ,Z)
. We
all a random-(cid:28)eld model.
If the data are
onsistent with a random-(cid:28)eld model, any analysis sim-
pli(cid:28)es
onsiderably sin
e the domain and the marks
an be investigated sep-
arately (see also Remark 4) by using standard te
hniques for random sets
[26℄ and for geostatisti
al data [6, 10℄. For the parti
ular
ase of marked
point pro
esses, several tests for the random-(cid:28)eld model hypothesis have
been developed [11, 23℄.
4
Ξ Z(x) = d(∂Ξ,x)
Example 2. Let x beRad random
losed set aΞnd Z the Euk-
lidean distan
e of ∈ to the boundary of . Then is even
ontinuous
Ξ Z
on . Sin
e lo
al maxima of are only attained at lo
ations in the interior
Ξ (Ξ,Z)
of , the random marked set is a random-(cid:28)eld model if and only if
Ξ = ∂Ξ Z
almost surely, in whi
h
ase is trivial.
Example 3. Cressie et al. [7℄
onsider the spatial predi
tion on a river
Ξ
network. Here, is the (cid:29)ow of the river (as a one-dimensional line or a
Z
two-dimensional stripe) and models the dissolved oxygen.
Ξ
Example 4. LCet2 be a random
losed setRrdepresented as a lo
ally (cid:28)nite
union of
losed -smooth hypersurfa
es in su
h that any two hypersur-
(d 1)
fa
es interse
t at most in a set of measure zero with respe
t to the − -
x Ξ Z(x)
dimensional Hausdor(cid:27) measure. For any ∈ , the mark is the maxi-
x
mum of the mean
urvatures of the hypersurfa
es at . The mean
urvature
has its importan
e for example in the analysis of foams [15℄.
3 Chara
teristi
s for random marked
losed sets
For the des
ription of random (cid:28)elds a set of se
ond-order
hara
teristi
s
like the variogram, the
ovarian
e fun
tion and the
orrelation fun
tion are
used [6℄. In analogy to these summary fun
tions, several se
ond-order
har-
a
teristi
s for marked point pro
esses have been introdu
ed as
onditional
quantitiesgiventheexisten
eofpointsoftherespe
tiveunmarkedpointpro-

ess [22, 26℄. Sin
e point pro
esses
an be des
ribed as random (
ounting)
measures, these quantities have been derived as Radon-Nikodym derivatives
of
ertain se
ond-order moment measures [3, Se
tion 2.7℄. Nevertheless, ran-
dom measures are not always appropriate for the de(cid:28)nition of se
ond-order

hara
teristi
s as the following example illustrates.
Ξ R1
Example 5. Let the stationary random
losed set in be given by
Ξ = ξ+ [2z p,2z+p] 2z+1 ,
− ∪{ }
z∈Z
[
p (0, 1) ξ [0,1]
where ∈ 3 and isuniformlydistributedin . Obviously,interpoint
r (0,2p]
distan
es ∈ are only possible if both points belong to the same
ξ + [2z p,2z + p] r (1 p,1 + p]
segment − , and interpoint distan
es ∈ −
ξ +[2z p,2z +p]
are only possible if one point belongs to a segment − and
ξ +2z 1 ξ +2z +1
tPh(eo,ortherΞ)is=fr0om one rof th(1e sipn,g1le+topn]s, { − } or { }. Sin
e
∈ forall ∈ − , theapproa
h ofde(cid:28)ningse
ond-order
5
hara
teristiR
s1usingarandom measure, whi
h isherebasedontheLebesgue
measure on ,
annot a

ount forsegment-singleton point pairs, andhen
e,
r (1 p,1+p]
these
hara
teristi
s are unde(cid:28)ned for ∈ − . Nonetheless, it does
make sense also to
onsider the
orrelation of two marks given that the
r r (1 p,1+p]

orresponding points are a distan
e , ∈ − , apart.
B (x) Rd
ε
x InRwd hat follows,ε 0 denotes the Eu
lidean ball in with
entre
∈ and radius ≥ , ⊕ denotes Minkowski addition, and we write
Ξ Ξ B (o)
⊕ε ε
shortly (Ξ,Zf)or ⊕ . Rd
RLet εbe a0stationary random marked
losed setZin with marks
ε
in . For any ≥ de(cid:28)ne the (stationary) random (cid:28)eld by
max Z(y), x Ξ , e
⊕ε
Zε(x) = y∈Ξ∩Bε(x) ∈

0, .
 otherwise
e
f :R2 R  h Rd
Let → be a right-
ontinuous fun
tion. For all ∈ de(cid:28)ne
κ (h) = lim E f Z (o),Z (h) o,h Ξ
f ε ε ⊕ε
ε→0+ | ∈ (2)
h (cid:16) (cid:17) i
κ (h) < P(o,h eΞ )e> 0 ε > 0 κ (h)
|f| ⊕ε f
whenever ∞ and ∈ for all , otherwise
is unde(cid:28)ned.
f
In parti
ular, for the following
hoi
es of ,
e(m ,m )= m , c(m ,m )= m m , v(m ,m ) = m2,
1 2 1 1 2 1 2 1 2 1
(3)
de(cid:28)ne
E(h) = κ (h)
e
(4)
1
γ(h) = (κ (h)+κ ( h)) κ (h)
v v c
2 − − (5)
cov(h) = κ (h) κ (h)κ ( h)
c e e
− − (6)
κ (h) κ (h)κ ( h)
c e e
cor(h) = − −
(κ (h) κ (h)2)1/2(κ ( h) κ ( h)2)1/2 (7)
v e v e
− − − −
k (h) = (m)−2κ (h), (m = 0),
mm c
6 (8)
where
m = E[Z(o) o Ξ]
| ∈
is the mean mark.
6
γ cov cor
We
all the mark variogram, the mark
ovarian
e fun
tion, the
k k (Ξ,Z)
mm mm
mark
orrelatiΞon funR
dtion and Stoyan's -fun
tion of [22℄.
Note that, if ≡ , these de(cid:28)nitions are
ompatible with the
lassi
al
de(cid:28)nitions for random (cid:28)elds (see Remark 4).
(Ξ,Z)
Whenever isassumedtobebothstationaryandisotropi
the
har-
a
teristi
sgiveEn(bry) (r4)(cid:21)(8[0),are)rotationinvaEri(ahn)t. hBysRligdhtabuseofnotation
we will write , ∈ ∞ , instead of , ∈ . The same applies
for the fun
tions de(cid:28)ned in Eq. (5)(cid:21)(8).
Ψ = ν ( Ξ )
ε d ⊕ε
Remark 1. Let ·∩ bethe randomvolume measure asso
iated
Ξ ν d
⊕ε d
with the random
losed set . Here, is the -dimensional Lebesgue
(2)
µ Ψ
ε ε
Bme,aBsure. If(Rd) denotes the se
ond-order moment measure of then, for
1 2
∈ B , we have
E f Z (x),Z (y) 1 (x)1 (y) dxdy
ε ε Ξ⊕ε Ξ⊕ε)
ZB2ZB1 h (cid:16) (cid:17) i
= E ef Ze(x),Z (y) Ψ (dx)Ψ (dy)
ε ε ε ε
(cid:20)ZB2ZB1 (cid:16) (cid:17) (cid:21)
= f(me ,m )eQ (d(m ,m ))µ(2)(d(x,y))
1 2 ε;x,y 1 2 ε
ZB1×B2ZR2
= f(m ,m )Q (d(m ,m ))P(x,y Ξ )dxdy,
1 2 ε;x,y 1 2 ⊕ε
ZB2ZB1ZR2 ∈
Q
ε;x,y
where isthetwo-point markdistribution oftheweighted randommea-
(Ψ ,Z ) (x,y) P(x,y Ξ ) > 0
ε ε ⊕ε
sure [3℄. Hen
e, for almost all with ∈ , we
have
e
E f Z (x),Z (y) x,y Ξ = f(m ,m )Q (d(m ,m )).
ε ε ⊕ε 1 2 ε;x,y 1 2
| ∈ R2
h (cid:16) (cid:17) i Z
e e P(o,h Ξ) > 0 h Rd
Remark 2. In
ase ∈ , ∈ , the above de(cid:28)nition takes the
simpler form
κ (h) = E[f(Z(o),Z(h)) o,h Ξ]
f
| ∈
if we impose the integrability
onditions
E[Z(o)1 (o)1 (h)] < , κ (h) < ,
Ξ Ξ |e|
| | ∞ ∞
f = e
for , and,
E[Z(o)21 (o)1 (h)] < , κ (h) < ,
Ξ Ξ |v|
| | ∞ ∞
7
f = c f = v a a
+ −
for or . This
an be seaen aRs follows. Denoting by and the
positive and the negative part of ∈ , respe
tively, we always have
Z (o) 1 (o)1 (h) Z (o) 1 (o)1 (h) Z (o)1 (o)1 (h)
ε + Ξ⊕ε Ξ⊕ε ≤ ε + Ξ⊕ε Ξ⊕ε ≤ | ε | Ξ⊕ε Ξ⊕ε
ε ε κ (h) <
e e e |e|
for all ≤ , where the right-hand side is integrable due to ∞.
Similarly,
Z (o) 1 (o)1 (h) Z(o) 1 (o)1 (h) Z(o)1 (o)1 (h).
ε − Ξ⊕ε Ξ⊕ε ≤ − Ξ Ξ ≤ | | Ξ Ξ
e
In the same way we obtain
Z (o)21 (o)1 (h)+ Z(o)21 (o)1 (h)
| ε | Ξ⊕ε Ξ⊕ε | | Ξ Ξ
as an integrable uepper bound of |Zε(o)|21Ξ⊕ε(o)1Ξ⊕ε(h)| and
|Z(o)|2 +|Z(h)|2 1Ξ(o)1Ξ(h)+e |Zε(o)|2 +|Zε(h)|2 1Ξ⊕ε(o)1Ξ⊕ε(h)
(cid:16) (cid:17)
(cid:0) (cid:1)
Z (eo)Z (h)1 e (o)1 (h) Z
as an integrable upper bound of | ε ε | Ξ⊕ε Ξ⊕ε . Sin
e is
Ξ ε > 0 x Ξ δ > 0
u.s.
. on , a value exists for every ∈ and for every su
h
Z(y) Z(x)+δ y Be(x) eΞ Z (x) Z(x)
ε ε
that ≤ for all ∈ ∩ . Hen
e, we have →
ε 0+ x / Ξ x / Ξ
⊕ε
from above as → . Further, ∈ implies ∈ for all su(cid:30)
iently
ε e
small . We then have
f(Z (o),Z (h))1 (o)1 (h) f(Z(o),Z(h))1 (o)1 (h)
ε ε Ξ⊕ε Ξ⊕ε → Ξ Ξ a.s.
ε 0+
e e
as → . Hen
e, by the dominated
onvergen
e theorem, we have
E f Z (o),Z (h) 1 (o)1 (h)
ε ε Ξ⊕ε Ξ⊕ε
κ (h) = lim
f ε→0+ h (cid:16) P(o,h (cid:17)Ξ⊕ε) i
e e ∈
E[f (Z(o),Z(h))1 (o)1 (h)]
Ξ Ξ
= .
P(o,h Ξ)
∈
Remark3. Thereexistsanalternative
on
eptofrandommarkedsetswhi
h
is inspired by the notion of random (cid:28)elds and where se
ond-order
hara
ter-
isti
s inRth∅e=senRse ofζt∅he pre
eding remark
anRbe de(cid:28)nedζ.∅
(RL∅e)t ∪{ } beσthe extension of by some . WeBd1enoBt2e by
BB1 (tRhe)respeB
t2ive Borel, -(cid:28),eζl∅d whi
h is generated by all sets ∪ for
∈ B and ⊂ {−∞ ∞ Z}.(,x) :Ω R∅ x Rd
A fam(ilΩy,of,rPa)ndom variables · → , ∈ , on thΞe probabil-
ity spa
e A is
alled a random (cid:28)eld with random domain , if
Ξ = x Rd : Z(,x) = ζ∅ .
{ ∈ · 6 }
8
Z ζ∅ , ,ζ∅
Clearly, when takes only values di(cid:27)erent frRom Ror {−∞ ∞ } this
notRiodn of a random marked set in
ludes usual - or -valued random (cid:28)elds
on .
Ξ
Note that is a ran1do(mx)s=et 1in(aZv(exr)y) geneZral sense [18℄, entirely deter-
Ξ R
mZined( by its(Rindd),i
a(tRor∅)) . If is jointly meaΞsurable, i.e.,
is A⊗B B -measurable, then the realizations of are almost
surely Borel measurable. Ifwehave evenalmostsurely
losed(open) realiza-
Ξ Z
tions of then is
alled a random (cid:28)eld with random
losed (open) domain,
see alsPo([o19℄. Ξ) > 0 Z
If ∈ holds for a stationary random (cid:28)eld with random
Ξ
domain we
an de(cid:28)ne se
ond-order
hara
teristi
s without any further
Z
assuZm(xp)tio=nZon(xp)ath rxegulΞarity. LZet(x)b=e t0he (stationary) rafnd:oRm2 (cid:28)eldRgiven
by for ∈ , ahnd Rde otherwise. Let → be a
measurable fun
tion. For all ∈ de(cid:28)ne
e e
κ (h) = E[f(Z(o),Z(h)) o,h Ξ]
f
| ∈
P(o,h Ξ)> 0 E[fe(Z(o)e,Z(h))1Ξ(o)1Ξ(h)] <
whenever ∈ and | | ∞.
(Ξ,Z)
Remark 4. Let be a stationaery reeal-valued random-(cid:28)eld model and
max Z(y), x Ξ ,
⊕ε
Zε(x) = y∈Ξ∩Bε(x) ∈

Z(x), .
 otherwise
Z Ξ  Z (x) Z(x) x Ξ
ε
Sin
e is u.s.
. on we hZave x→ Rd frεom a0b+ove for ∈ , and
ε
hen
e, by the de(cid:28)nition of , for all ∈ as → . Then, using the
Z Ξ
independen
e of and , we obtain
κ (h) = lim E[f(Z (o),Z (h)) o,h Ξ ]
f ε ε ⊕ε
ε→0+ | ∈
= lim E[f(Z (o),Z (h))]
ε ε
ε→0+
= E[f (Z(o),Z(h))]
h Rd P(o,h Ξ ) > 0 ε > 0
⊕ε
for all ∈ whi
h satisfy ∈ for all and, depending
f
on the
hoi
e oΞf a

ording Rtod(3), one of the integrability
onditions in
Remark 2 with repla
ed by .
κ
f
Remark5. Thede(cid:28)nitionof a

ordingto(2)is,inimportantsitutations,

onsistent with the
lassi
al de(cid:28)nition of the se
ond-order
hara
teristi
s of
Φ
stationary markedRpdointRpro
esses [Ξ22℄: Let be a stationary simple marked
point pro
ess on × . Then, is the support of the unmarked point
e
9
Φ = Φ( R)
pro
ess ·× . We assume that the se
ond-order moment measure
µ(2) Φ h > 0
of is lo
ally (cid:28)nite. For k k we have
e
E f Z (o),Z (h) 1 (o)1 (h)
ε ε Ξ⊕Bε(o) Ξ⊕Bε(o)
h (cid:16) (cid:17) i
= E f Z (o),Z (h) 1 1
e eε ε {Φ(Bε(o))=1} {Φ(Bε(h))=1}
h (cid:16) (cid:17) i
+E f Z (o),Z (h) 1 1
e ε e ε {Φ(Bε(o))>1} {Φ(Bε(h))≥1}
h (cid:16) (cid:17) i
+E f Z (o),Z (h) 1 1 .
eε eε {Φ(Bε(o))=1} {Φ(Bε(h))>1}
h (cid:16) (cid:17) i
0 < ε< h /2 e e
For any k k the (cid:28)rst summand equals
E f(m ,m )1 (x )1 (x ) =:µ(2)(B (o) B (h)).
 1 2 Bε(o) 1 Bε(h) 2  f ε × ε
e
(x1,m1)X,(x2,m2)∈Φ
 
We
an extend the argumentation in [8, Prop. 9.3.XV℄ in order to
on
lude
that
P(o,h Ξ ) P(Φ(B (o)) 1,Φ(B (h)) 1)
⊕ε ε ε
∈ = ≥ ≥ 1
µ(2)(B (o) B (h)) µ(2)(B (o) B (h)) →
ε ε ε ε
× ×
ε 0+ ε > 0
as → . If we additionally impose the
ondition that for some ,
E f Zε(o),Zε(h) 1{Φ(Bε(o))>1}1{Φ(Bε(h))≥1}1{|f(Zeε(o),Zeε(h))|>M}
sup 0
h(cid:12) (cid:16) P(cid:17)((cid:12)Φ(B (o)) 1,Φ(B (h)) 1) i →
ε∈(0,ε) (cid:12)(cid:12) e e (cid:12)(cid:12) ε ≥ ε ≥
M
as → ∞, we obtain
(2)
µ (B (o) B (h))
κ (h) = lim f ε × ε
f ε→0+ µ(2)(Bε(o) Bε(h))
×
µ(2)
whi
h equals -a.e. the Radon-Nikodym derivative
(2)
dµ (x,x+h)
f
.
dµ(2)(x,x+h)
α
E f Z (o),Z (h)
ε ε
For instan
e, the above
ondition is satis(cid:28)ed if is uni-
(0,ε) α > 1 (cid:12) (cid:16) (cid:17)(cid:12)
formly bounded on for some . (cid:12) e e (cid:12)
(cid:12) (cid:12)
10