Local Times for Vector Functions:
Energy Integrals and LocaTGrowth: Rates
by
Donald Geman
Depavtment of Statistios
University of North CaPoUna at Chapel Hi."."
Institute of Statistics Mimeo Series No. 1088
September, 1976
This work was partially supported by National Science Foundation grant
and by the Office of Naval Research Contract N00014-7S-C-0809.
2
local Times for Vector Functions:
Energy Integra1sand Loc·t 1. ~f.'m'lt:1 ::~ tss
Donald Geman
Department of Statistics
University of North Carotina at ChapeZ Hitt
/
Abstract. Let F: E~Rm
n con~act) have a local time ~(x~dt)~
(EeR
xc;Rm ,and
denote the integral of ¢(s-t) against ~(x,ds)~(x,dt)dx; here ¢ is
a Yipotential kernel" on Rn , so that I(¢) is an (averaged) "energy integrallY
let I(¢)
for the distribution
¢
~(x,dt)
of mass on {tc;E: F(t)=x}.
We show that if
L1(dt)
and ¢ = sup ¢ for a sequence {¢n} of Fourier transfonns of
n n
positive Ll_functions~ then I (¢) is well approximated by certain ftmctionals
E
of the increments of F. We then draw the following conclusions about the local
growth and fluctuations of F:
if F is continuous,
I (¢)<co and ¢ is
radial, then (i) ap lim I IF(s)-F(t)i I/V(I Is-tl I) = co a.e.
VeE)
= (En¢(E))
-
11
(dt) on E, where
s~t
m, and (ii)
1
IE
_ 0
¢(r)rn-ldr]m = o(
sup
IIF(s)-F(t) II)
s, tc;E
II s-t II s;e:
as e:tO.
TI1is work was partially supported by National Science Foundation grant
and by the Office of Naval Research contract N00014-75-C-0809.
N~
(MOS) Subject Classification (1970): Primary 26A54, 26A27.
Key Words
&Phrases: local time, level set, energy integral, modulus of
continuity.
3
e
Introduction.
probabilistic
By and large~ the study of local times has been confined to
settings~
either as in
~~rkov
theoretic and stochastic analysis are
processes where the potential
f~scg~
or as in
[2]~
[3], [6]
[7] and
[8]~
'''here iVreal variable" results may be separately developed, but with an eye toward
applications, especially to sample function analysis.
of [8] and the references therein.)
Here
'VIe
(See the discussion in §O
deal exclusively with non-random
More specifically ~ we intend to further develop the observation of
functions.
S. M. Bennan that, loosely, "the more regular the local time, the more irregular
the
function~i1by amplifying
such as:
several earlier results of ours and J. Horowitz,
if a function has a local time ~ any iVapproximate local modulus!! grows
at least linearly, and grows faster than linearly if the local time is continuous
in its Iftimen parameter. (See the comments after the corollaries.)
Let 1\ denote the Borel sets in Rk (Euclidean Ie-space), Ak(dt)
Lebesgue measure in RIc (just dt
Bk(t,E)
be
for integrationL and let ~ = Ak{Bk(o,l)}~
being the open ball in Rk centered at
t
and of radius
E.
Further~
let EE: Bn be bounded and F: E+R m Borel measurable. The oaaupation measure of
F is ~(B) = An {F-l(B)}~ BE:Bm • If ~« 'Til
A (i.e. '11
Al(B) = 0 => ~(B) = 0,
BE:Bm ), then for each AE:B ~ the measure \n{tE:EnA: F(t)€dx) is also dominated
n
by \n (dx), and we may select versions a (x ,A) of the Radon-Nikodym derivatives
is Bm-measurable for each AE:Bn and (ii) <l(x~o) is a
finite measure on B Vx. We call this family cx.(x,dt) of measures the ZocaZ
n
time of F because it represents the litime spent n ·by F in the state x
such that
during dt.
(i)
a(Q~A)
4
By definition, for any Be:Bm , Ae:Bn ,
which extends to
fE H(t,F(t))dt = fnm fE H(t,~)~(x,dt)dx
(2)
for any non-negative, Borel measurable H on ~
\n --
~(x,r.~) = 0 for
a.e.
X"
Rrn •
x, where Mx = {te:E: F(t)=x}.
Consider the measures Ix(dsdt) = ~(x,ds)~(x,dt)
on
Bn~Bn;
It follows that
and
I(dsdt)
=~Ix(dsdt)dx
the latter is the measure H(ds,dt) which figures in [2].
For
o :5: k(s, t) Borel measurable, lITe write tx(k) and I(k) for the corresponding
integrals:
(3)
I(k) =
f rn Ix(k)dx = f . f
f k(s,t)~(x,dt)~(x,ds)dx.
IflEE
R
Let ¢ be a Hpotential kernel" on Rn, Le.
<I> is positive and continuous on
r:f/{O} and <1>(0)=00, and recall that Ae:Bn is said to have Ilpositive <I>-capaciti'
if there exists a non7zero, finite roeasure y(ds) concentrated on A such that
¢(s-t)
e:
LI(y(ds)y(dt)). Various authors (see e.g. [1], [2], [9], [10], [11])
\ have considered the capacity of the "level sets"
stochastic processes, and
~(x,dt)
Iv~
for Gaussian (and other)
has been the natural measure to use;
one obtains probabilistic conditions for f I(k)d.P <
00
typically~
for some kernel k(s,t)
<I> (s-t) . See also the "concluding remark. 'I
To approximate I(k) using the increments of F, we define, for each E>O,
(4)
T (k)
E
=__1__m
CrnE
fE fE k(s,t)~ (F(s)-F(t))dsdt ,
E
=
5
(u) = lB (0" ) (u), and consider the following question: for which funce:
m ,e:
tions k(s,t) does Te:(k)+I(k) as e:~O? To see this may fail, notice first
where
l;
that I ~ A2n because, with G = {(s,t)€ExE: F(s)=F(t)}, .we have G€B 2n ,
c
l(G ) = 0, whereas ~« Am implies that v = ~*-~ «~ and hence A2n(G)
v({O})=O.
= lG(s,t),
Consequently, with k(s,t)
I(k) =
f
=
we have Te:(k)=O but
2
a (x,E)dx > 0 •
ffi
R
(Other examples are easily constructed.)
Naturally we would like to have
for a wide class of potential
kenlels, for example for the Riesz potentials ~a(t) = C(n,a) Iltll a -n , O<a<n
(here I(~), Te:(~)
stand for l(k), TE(k), k(s~t) = ~(s-t)).
~€Llcdt)
kernel of "positive type," Le.
A
Te:(~)+l(~)
and
Suppose ~ is a
has a positive Fourier transfonn
m
A
cp. Let ax be the Fourier transfonn of the measure a(x,dt), xeR • Then
I(~) = f I (~)dx = f
ffl
x
dx
f~ ~(A)I~x(A)12dA
=f dA~(A)
Rn
([9, p. 141])
f m I~X(A)12dx
R
= f ~(A)I(eiAo(t-s))dA.
Rn
As "lill be seen in the course of the proof of the Theorem,I(k) = lim Te:(k)
e:
any continuous k,
and
hence
I(~) = f
Rn
~(A) linl T (eiAo(t-s))dA
e:
e:
'\'lhich, proceeding "formally, Vi
A
= lim f
e:
Rn
~(A)Te:(eiAo(t-s))dA = lim Te:(~(t-s)) = lim Te:($) •
€.
e:
for
6
Among other problems ~ however,
cf>
has no transform because ess sup cf> = 00
~ Ll(dt); but this docs suggest how to proceed.
implies ;
Main result.
oo
Let F
that cf>(t) = sup cf>n(t) where
n
OScf> l scf>2 .•. ,
cf>n
tains any even function in L1 (dt)
(0,00)
(approximate cf>
-po
theorem) • For n> 1,
Writing just a (x)
Theorem.
Suppose a
(5)
E
cf>(0)=, such
Ll(dt) 'fn, and each cf>n is the
€
Fourier transfonn of some Borel measurable 0 S f
on
cf> e: LlO"n) ~
denote the class of functions
€
L1 (dt) • For n=l,
rx'
con-
which is convex, continuous, and decreasing
by convex, continuous functions and use Polya 9 s
also contains the usual kernels.
and with Ao = {(s,t)e:ExE:
for a(x,E)
L2 (dx)
and
cf>
oo
€
F
•
Then
I(cf» = lim TE(cf» = sup TE(cf»
E
I Is-tl Is8},
S 00
E
Moreover, if I (cf»<00 ,
(6)
Corollaries 1 and 2 about the local growth of F depend on (6), which, in
turn, rests on (5).
I (k)
g
==
The proof of the Theorem will be split into several steps.
lim To~(k...::),--_fo_r_k__c_on_t_in_u_o_u_s.
For any bounded, complex-valued
E
on R11 ,
I I
!g(s)la(x,ds)dx =
RID E
and hence for
Ig(s)!ds < 00
E
m
A -- a.e.
IE g(s)a(y ~ds)
I
y~
==
J
liln ~
dx
g(s)a(x,ds) •
E CrnE Bm(y,E)
E
I
7
Using (2) and the fact that the exceptional set of y's
find that$ for
An -- a.e.
= lim
~
E ernE
I g(s)~
(F(s)-F(t))ds •
E
E
like g,
Consequently 7 for any f
= I~ dx IE
f(t){I g(s)a(x,ds)}a(x,dt)
E
= IE
f(t)
IE
=J
lim
~
E
E
g(s)a(F(t),ds)dt by (2)
C E
I f(t)g(s)~
E
m
= lim TE(f(t)g(s))
which lviIl
0, we
t$
JE g(s)a(F(t) ,ds)
r(f(t)g(s))
has wrneasure
(F(s)-F(t))dsdt ,
E
if we can show
E
sup ~
rn
E CmE
Let
~ (x)
IIE f(t)g(S)~E(F(S)-F(t))dsl
Ll(dt) ,an E.
be the Hardy-Littlewood rna.",imal function for
~(x) = sup ~
E
If f
~
C E
m
I
cx(y)dy,
Bm(x,r)
,
a(x):
x~lfl.
and g are b01.,mded by k,
f sup ~
E E CrnE
II
E
f(t)g(s)~
~
(F(s)-F(t))ds/dt
E
k
=k
which is finite because a
€
2
L (dx)
implies
Taking f=g::l, we find that TE(l)
measures
-+
f
JE supE CrnE1 m fE ~ E(F(s)-F(t))ds dt
I
Q (F(t))dt
E a
~
r:i.
€
=k f
ffl
Qo;(x)a(x)dx
2
L (dx) •
Consider the probability
$
8
=e
Choosing f(t)
II
Rn Rn
In other
e
iAl~t
~
get)
i(Al~A2)oCs~t)
=e
W(dsdt)
iAzot
~
= lim
Al~A2 €
II
e;
Rn Rn
e
n
R , we have,
i(Al~A2)o(s,t)
We;(dsdt) •
the measures We; converge weakly to W; since these measures are
words~
supported on ExE, which is compact~ we obtain 1°.
+ ¢(t)
Let ¢n(t)
Step 2°:
I(¢)
with {¢n} as described above.
lim Te;(¢).
S
This is easy:
€
= lim
I(¢)
IC¢n)
(by the monotone convergence theorem)
n
= lim lim Te;(¢n) (by 1°)
n
S
e;
lim Te;C¢L since ¢n
e;
S
¢ Vn •
Next, we introduce the followi."lg finite measures on Bm :
Aj(B)
A(B)
= IE
IE ¢j(s-t)lB(F(S)-F(t))dsdt,
= IE IE
j=1,2, ... ,
¢(s-t)lB(F(s)-F(t))dsdt ,
and recall that
Each of these is absolutely continuous with respect to
1/Jj (Y)
" ,
and let l/J.
J
then,
"
ljJ,
=: : (Y)
,
1/J(y)
=~
(y).
a(y)·
~.
Let
~ (y)
and "13 be the corresponding Fourier transfonns.
For example,
9
a
(6)
•
~. (A)
=
I
Rm
J
.~(y) = II
Step 3°:
eiAox tP· (x)dx
1(</»
~
J
Rm
=
iAox
2
e
a(x)dxI
(27T)-m
IJ
EE
~
eiAoF(s) e-iAoF'(t) ~. (s;.,t)dsd~ ,
.:j
°.
J ~(y)dy. To start~ ~j(A) ~ ° VA and j~l; this
Rffi
follows from (6) because the </>j'S are positive definite.
vergence theorem, for any A€Rm ,
~(A) = J
I
E E
Next, since a
e
That is,
1
A
S
€
€
L (dA).
</>(s-t) eiAoF(s) e-iAoF(t)dsdt
By the dominated con-
= lim ~'(A) ~
j
J
°.
L2 (dx), the Parseval relation yields
Irf IfIfl
e
iAox
2
a(x)dx/ dA <
00
•
It follows that S has a bounded (and continuous) version,
and recalling that each </>j
is bounded, we have, for any B€zf1,
Thus, each l/Jj has a bounded version a'ld a non-negative Fourier transform.
According to [4, p. 66], then~ ~j E: Ll(dA) and the inversion fonnula holds for
~
(7)
-- a.e. y:
l/Jj(y)
= (27T)-m
Jm e- iAoy ~j(A)dA
.
R
We now assume B and tPj
Vy.
The continuity of </> j
are bounded and continuous; in particular, (7) holds
gives
10
I(~.)
J
= lim
T
E
E
(~.)
J
f
=
lim ~
Wj (y)dy
E CrnE Bm(O,E)
=
W· (0)
J
= (2TI)-rn
I
~j(A)dA.
Rrn
Consequently,
I (~ ) =
l~n
I (~j )
J
=
lim (2TI) -m
j
~
(2TI)-m
fRID ~.J (A)dA
IR ~ ~j(A)dA
I ~(A)dA.
B
= (2TI)-m
"
Fatou's lemma, since Wj
(by
~
0)
J
RID
0 0 "
Step 4:
Assume
I(~) ~ s~(~).
already seen that
E"
tP~O.
As
ljJ(y)
I(~)<oo.
I
From 3, W€ L (dA) and we have
above, then, we can and do assume
= (2TI)-m
f
e-iAoY;(A)dA Vy.
RID
In particular, W(y)
~
W(D) Vy, so that
sup
E
TE(~)
f
=
sup ~
W(y)dy
E CrnE Bm(O,E)
=
ljJ (0)
=
(2TI)-m
f
Combining 20 and 40 we have (5).
~(A)dA ~ I(~) •
rfl
As
for (6), we'll assume, for notational
ease, that n=l and Ee[O,I]; the proof for arbitrary n and bounded E is
11
essentially the same.
A.
--j<
k k
k-2
k
. k
IJ, ~
. k
wJ ,
.
'+2
J
1(
~
~
k=2,3,... ,
j=O~ ... ,k-2 .
, etc. refer to the quantities
I, W, etc. but definE:~
D~ instead of E. Now~
J
Ij,k(~)
And
=
If ~(s-t)Ij,k(dsdt)
=
ID~ J
J
because, for any E' eE,
a (x, dtnE I)
Furthennore cj>(D)=oo and
a.e.
.
D. = E n (J_,~,
e R :: u (D.xD.),
-Ie
j=D
J J
In what follows,
relative to
Writing Ale instead of A
l/k
I(~)<oo
~(s-t)I(dsdt)
J
in the local time of
F restricted to Ei
together imply a(x,{t})=D YtE:E, for
y, which, in turn, implies that
from which it follows that
1
D:
I (dsdt)
has no atoms.
>m--
As a result,
•
12
oy,·re-a.rranging some tenns in the summation just above. Finally, putting
D = {(t,t)~
T~E},
~
lim sup Tg(<p olA )
k
e:
k
2 lim I I <P(s-t)l(dsdt)
k B1
~{
=2
InI
<P(s-t)ICdsdt)
(since
D=~Bk
and lC<P)<oo)
= 0,
because I(D)
=I
Ifl
r a 2Cx,{t})dx = o.
QED
t~E
Remark. As the proof shows,
l::n
l~
Tt/<Pn)
= IC<P) = l~
we can select <Pn's which converge uniformly to
example, when n=l and
Te:(<Pn)
-+-
<p
l~
Te:(<Pn).
Suppose
away from teO, as, for
<p
is convex, etc. as described before. Then, in fact,
I(<Pn) as e:+0 uniformly in n if I (<p)<oo.
Briefly, here's
"lhy~
write
given ~>O, the first righthand term is ~ ~Cl+fa2) Ve:>O and large n, using (6)
and the 1.ll1ifonn convergence of the
<P ' s;
n
the second term is
~ Z;
for all small
e: by (5); and of course, the last tenn vanishes as n:-+oo.
Applications.
a e L2Cdj).
t
-+
C;; (II t'll L
Throuehout this section and the next we assume a(x,dt) exists and
Let C;;: (0,00)
-+ (0,00)
be decreasing;
C;;(O+)=oo, and suppose
teRn, belongs to r" and (ii) Vet) ::
+
for t>O, with V(O )=0.
(For exmnple,
Corollary 1. Suppose Ix(C;;) < 00 for
c;;(t) = t
11 -
a.e.
-y
(i)
(tn~ (t)) 11m is increasing
.
, O<y<n.)
XE~Il. Then
13
(8)
An - a.e. on E.
Proof., We must show that for
An - a.e.
teE:
~
(9)
Denote the ratio in (9) by Tt(k~E)
Vk>O.
and Er
= F-l(HrL r=l,2~...
Obviously?
and define r~
H1cH 2c ••• ~
= {x€Bm(O~r):
Ix(~)~r}
u~ = {x: 1x(~)<oo} :: H,
and
Restricting
F to Er ,
JIt"TI J J
Er
Er
~(Ils-ti I)a(x,ds)a(x,dt)dx ~
m1
cm r + < 00 ,
1(7;)<00 in proving (8).
and therefore we may even assume
Now,
Tt(k,E) ~ ~ An{sE:EnBn(t,E): IIF(s)-F(t) II~kV(E)}
CnE
=
~
Hence? for any
(10)
7;; (E)
~{s€EnB (t,E): IIF(s)-F(t) lI~kV(E)}
cn(V(E))m
n
J
1 m
Cn(V(E))
EnBn(t,E)
0>0
and
~(Ils-tlj) ~1rV(E) (F(s)-F(t))ds
,..
k=1? 2 ~ ... ?
JE Tt(k~E)dt ~
C
Ifl
~n TkV(E)(~olAo) VE ~
+
Recalling that V(O )=0, (10) leads to
=0
by (6).
0
(since
7; +).
14
In -
lim 'It(k~e:) = 0 for
e:
An - a. e. tEE.
Finally, Fatou's lemma shows that
hence
lim 'It (k, e:)
e:
= 0 Vk,
for
a.e.
tEE, v.k, a11d
QED
We actually found that
(11)
If lim 'It(k,e:)
e:
exists, (11) then llJylies
(12)
ap lim
s~t
'~(fl)-~f,j'
I =
s-
00
lim 'It(k,e:)
e:
= 0 Vk,
a.e. on E, i.e.
A - a.e. on E .
n
We don't know, however, whether (12) is true asstmrlng only IxU,;) <
The (luaiting) case
~
= constant
00
11 - a.e.
should correspond to assuming only that
is a continuous measure Vx, and it does: in [6] for n=m=l, then in [7]
for, arbitrary n,m, we showed that (12) holds with VeE) = En / m • More generally,
cx(x,dt)
e
in fact;i su:(?pose cx.(x,dtr has.a k:.d:iJnenslona.l
~'l11<i.tgirlaJ",:di$tributio111',,(lomina,t~d
.,'
by' N , ... O<k<n,· \.iTitll the reT1:1.ill.ine--n-J::-dine!lsional Llarginal: distribu:tioncontil1~k
OtiS;
that'is, suppose
a(x, BxA) =
f
g(x,s,A)Ak(ds),
B
where g(x,s, a)
corresponds to
is a continuous measure on
a(x,dt)
continuous.)
Be:Bk ,
Bn+k
Ae:Bn _k
m
"if xe:R
,
SEE.
(The case k=O
Then (12) holds with V(e:) = e:(n-k)/m •
(See [7, Lenmm 3].)
For our second application, assume E is compact,
as above, and let w(e:)
be the modulus of
w(e:) =
sup
s, tEE
II s-t II s-e:
Also, define
F on E:
(IF(s)-F(t) II,
e:>0.
F is
continu~ls,
~
is
L(E) =
~(lloll)
Corollary
0:
~ implies L(E) <
€
Suppose
2.
15
00
1
t(rlrn-1dr]m, E'O;
VEe
lim ttE:~
EtO E
Proof.
Choose A
{x: Ix (7;) <00 }
c
=
00
•
such that A€ Bm
IA I x (Z;;;)dx
Let E'
Then
ll{X: IX(l;;)<oo} > O.
<
00
~
II (A) >0 ~ and
•
Restricting F to E', (6) holds with E replaced by E'~
= F-l(A).
Le.
(13)
II
lim lim ~
lBn(too)CS) z;;;(1 Is-tl
o E CmE E' B'
,
1)~e(F(S)-F(t))dsdt = o.
But the L.H.S. of (13) is
;;: lim lim
o
= lim
o
E
Ii'll
I
~I
~
c Eli
E'
m
E CmE
Jr
B'
Bn(O~o)
lB (t o)(s) z;;;Clls-tl Dl[O E](wClls-tl D)dsdt
n
~
An{E'nE'-s}z;;;Cllsl D1[0 e](wCllsl j.))ds
,
by making the change of variables
~{E'nE'-s}
gration. Notice that
As a
,
s-t-+s and then reversing the order of inte-+ "nCE') as 5+0 and that AnCE')
result~
o = lim
o
lim EE
m
J
BnCO,o)
Changing to polar coordinates,
z;;;Cllsl Dl[O?E] (wCllsl /))ds •
= ll(A)
> O.
16
since w(O+)=O~ which completes the proof.
(I,Iote: Here ~ the illimiting case l1
I;
==
B~,
constant should be compared to Theorems
B7 of [8J.)
Suppose F has a differentiable ~ global modulus 1jJ (1. e.
oo
W~W ) for which Z;;(t) == m(W(t))m-lw (t)t l - n :Ji;. as above, Le. 1;(11 1I)E:F and,
Concludil1.g Remark.
0
decreases on
11 - a.e.
x. But
'Then it follows from Corollary 2 that
this~
I (1;)
x
together with the special role of a(x,dt)
measures concentrated on the
I/I,/s~
",\,
conditiGns/' one ~.:ctually has
t", f
\
i
for
(Le.
\11 -, a.e.
Am « 11, i.e.
1!1ibh::': elen
const.
x
WeE)
'
p- a.e., Le.
CaPl}\ = 0
,C1ls- t ll)Y(ds)Y(dt) =
~ const. ES
x, where "dimi l
a(x»O
>m -
a.e.
for
among the
00
I"~ •
Generalizing a result of Kahane, Adler [1] proves that if F: [O,l]n
t3
00
leads ·0ne' to
,beJ-i3ve that,.'lll1der I1 suitable
..
for every non-trivial probability measure carried by
Lipschitz
=
) and
-+
Rill
i1;i
n-(3m:2: 0, then di::. J.\t~ ~ n-(3m for
stands for Hansdorff dimension.
Suppose that
In vieW":of the :rer.l8.Tl:s aOo'p:;,A::'l?r. 7s 'result
follow by choosing WeE) = const.
t - (n- SIn) ) and the fact that
x E(3
(giving
I;(t) =
dim UI.x is the supremum of the nurnbers
for which l/~ has positive capacity for
Iltll-
o.
0
17
References
1.
R. J. Adler ~
2.
S.
4.
and Gaussian fields, Ann. PJ:aobabiZity (to appear)"
M. Bennan, Gaussian processes with stationary increments: local times and
s~~le
'7
.).
Hausdor~f dimension
function
properties~
Ann. Math. Stat.
!1,
1260-1272 (1970).
, Gaussian sample functions: tmifonn dimension and }folder conditions
-----n-ow~h-e-r-e~ Nagoya Math. J. ~, 63-86 (1972).
S. Bochner 1 Leatures on Fourier Integpats ~ Ann. of ii/ath Studies, No. 42, Princeton
Univ.
P1~ess
(1959).
5.
H. Federer, Geometria Measure Theory, Berlin: Springer-Verlag (1969).
6.
D. Geman,
7I
_ _~."..~ On the approxllnate local grOlifth of multidimensional random fields, Z.
•
A note on the continuity of local t:ii"nes, PJ:aoa. Amer. Math. Soa• .§Z,
321-326 (1976).
Wahrsaheintiahkietstheorie v. Geb. (to appear).
n
".
••
9.
.1.
--~-
J.
~
and J. Horowitz, Local times for real and random functions, Duke Math.
Dec. 1976 (to appear).
J. P. Ka'mne, Some Random Series of Funations, D. C. Heath, Lexington,
rlA (1968).
M. B. Marcus, Capacity of level sets of certain stochastic processes, Z. Wahr,:"
saheintiahkeitstheorie v. Geb. ~, 279-284 (1976).
S. Orey, Gaussian sample functions and the
Z. Wahrsaheintiahkeitstheorie v. Geb.
Hausdor-~f
1§,
dimension of level crossings,
249-256 (1970).