where II is a finite Borel measure on the line and f is a Lebesgue integrable function on the line. .... W e alw ays assume H : 0 < H < 1. Note that there .... (Proposition 2.2) and the P- invariance of 41(H- self- similarity), ..... J., 27 (1978) , 309-330.
641
J. Math. Kyoto Univ. (JMKYAZ) 36 4 (1996) 641-652 -
Limit theorems for local times of fractional Brownian motions and some other self-similar processes By Narn Rueih SHIEH -
1.
Introduction and Main Results 0, is called self-similar with ex-
A real valued stochastic process X (t), -
ponent H (H ss for b re v ity ) if X (cl) =
HX (t) for each c >O . It is called of stad tionary increments (si fo r b re v ity ) if X (t+ b ) X(b) =X (t) — X (0) for each b> -
c
—
d
0. T he notation = in t h e above m eans th e finite dim ensional equivalence of two processes. In th is paper, we consider the exponent H : 0 O. T he equality extends to all (t, x ) since both sides a re known to be jointly continuous i n (t, x). Now, w e illustrate specific exam ples. Propositons 2.1 a n d 2.2 a re known to be applicable to the following. 1 Brownian motion, 11=-- — -
2
;
• a-atable Levy process, I < a< 2 and H = 1 / a ; • fractional Brownian motion of exponent H, 0 < 11 0 a n d each x e R . T h is proves the finite dimensional convergence. As for tightness, by the fact that x — *L (t, x ) is HOlder continuous and com pactly supported, we see that, for each a > 1 and 13 : -
T El ,
L (1, x) . T his proves
the assertion.
5. Multi - parameter and multi - dimensional extensions F o r a re al-v alu ed N param eter p ro c ess X (t) , t E W.Y_ and
2 , the
-
d
self-similarity is defined exactly as one param eter case, namely X (ct) =c HX(t) for all c > O . F o r the stationary increm ents property, it is assum ed that X (t) is invariant not only under translations but also under rotations; th u s the distrib u tio n of X (t) — X (s) depends only o n lit — s i where II • Hdenotes the N-dimensional Euclidean norm. Now let X = (X1, •••, X a), X i=X i (t) and t -
-
IV,f, ta k e s values in R s o t h a t X i s an N-parameter, d-dimensional-valued vector field (w e call it a n (N, d) f ie ld for b re v ity ). W e assume th a t the components X i are independent and each X i s H ,-ss si (for si, i t i s in the above stronger sense). The exponents 111 are assumed to be 0 0,
regarded a s th e equivalence in distribuation o f two d - dimensional processes. The canonical setting for X is then to consider the path space -
cod) : coi = co, (t) , t e R_I `Fr , continuous and co.; (0) =O L
= w = -
-
cylinder a - algebra,
a n d a p ro b a b ility m e a su re P o n (D , strong - translation transformations wi
(Jaw ) (t)
(at)c
Hi a
(Joito) (t) : = w (OA),
a) s u c h t h a t t h e sc a lin g a n d th e
o d
(at) a"
),
a > 0
: rigid - body motion on R N ,
are both P invariant and moreover that the component processes coi, j=1,••., d, are P independent, w e call X to be ergodic if any A G w hich is Zia -invariant ( j a A CA V a >0) i s of P (A ) = 0 o r 1. T h is definition indicates that, for each -
a
-
(est)
fixed t, the one - param eter stationary process defined by s X
es
H ' scR, is
ergodic. W e assume that each X , is AII on every compact box Jc R N bounded a w a y f r o m t h e origin ( t h e d e f in itio n of A l l f o l l o w s t h e previous one-param eter case, w ith som e attentions on the ordering of the param eters t, see Pitt (1978) and Nolan (1989)). Assume that
Let f (x ) , x e R , be a Lebesgue integrable function on and let the field F (f, t ) , be defined by d
F(f ,
t) = f • •• o
IV with nonzero ff (x) dx,
f rN 0 f (X (s )) ds, t = (t 1 , •••, tN )
G
T h e f o llo w in g tw o r e s u lts c a n b e p r o v e d i n t h e s a m e w a y a s th e one param eter case in §3, 4. -
Theorem 4.
As 2 . —
0 0
, t, 21s
Hi j t
converges in the law of the space of continuous N parameter (N variable) functions -
to
-
651
Fractional Brownian motions
i ff (x )d x • L (t, 0 ) t.
Theorem 5.
Assume that X is ergodic, then almost surely
1 r In T
.
dt s u p x L ((t , •••, t),
Jt
,1) ,
= E sup L
.
1
In the above, L (t, x ), t E R I,v_ and x E R , denotes the local tim e of X ( • ) at x on the box [0, te], t= (ti, •••, tN). W e rem ark that, fo r th e (N, d) field X, w e h a v e th e e x te n s io n o f P r o p o s itio n 2.1, w h e n it is u n d e r s u it a b le parameter dim ension m odifications a n d w e a ls o h a v e th e following scaling property of L (t, d
-
L (t, x, Jaw) =
L
(at, (a
f f
ix 1 ,
a
•••,
al i d xd) , co)
N-
T he m ost im portant case in the above consideration is fractional Brownian v ector field, in w h ic h e a c h X , is a mean zero G aussian fie ld w ith covariance function -
E X (s) X i ( t ) = s 1
2H
t ± t 11
211t
t —
T h e re g u larity of local tim es fo r th is im portant vector fie ld w as studied by Pitt (1978); Theorems 4, 5 then show a certain lim it behavior of this Gaussian local time.
Acknowledgement. T he author thanks to Professor S. W atanabe and P rofessor N . Kôno f o r th e helfpful d isc u ssio n o n th e c o n te n t o f th is paper. This w ork is supported in p a rt by a grant NSC 199516 2121 M 002 012. -
-
-
-
D EPA RTM EN T O F MATHEMATICS
NATIONAL TAIWAN UNIVERSITY T A IP E I, TAIWAN
e mail : shiehnr@math. ntu. etu. tw -
References T. Bedford and A. M. Fisher, Analogues of Lebesgue density theorem for fractal sets of re a l and integers, Proc. London. Math. Soc., 64 (1992), 95-124. S. M. Berman, Local nondeterm inism and local tim es of G aussian processes, Indiana U niv. Math. J., 23 (1973), 69-94. S. M. Berman, Joint continuity of local tim es of M arkov processes, Z. W ahr. verw . G eb., 69 (1985), 37-46.
[4] [5]
P. Billingsley, Convergence of Probability M easures, W iley, 1968. G . A . B rosam ler, The asym ptotic behaviour of certain additive funct onals of B row nian motion, Inventiones Math., 20 (1973), 87-96.
652 [6] [7]
Narn-Rueih Shieh I. P. Cornfeld, S.V. Fomin and Ya G. Sinai, Ergodic Theory, Springer-Verlag, 1980. S . Cambanis, C . D . H ardin a n d A . Weson, Ergodic properties o f stationary stable processes, Stochastic Processes and Appl., 24 (1987), 1 18. D. A. Darling and M. Kac, On occupation times for Markoff processes, Trans. Amer. Math. Soc., -
[8]
84 (1957), 444-458. P. J. Fitzsimmons and R. K. Getoor, Limit theorems and variation properties for fractional derivatives of the local time of a stable process, Ann. Inst. Henri Poincaré, 28 (1992), 311-333. [10] D. Geman and J. Horowitz, Occupation densities, Ann. Probab., 8(1980), 1-67. [11] N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes 2nd Edition, North Holland & Kodansha, 1989. [12] Y. Kasahara, Limit theorems of occupation times fo r Markov processes, Publ. RIMS Kyoto Univ., [9]
-
12 (1977), 801-818.
[15] [16]
N. M ilo and N. R. Shieh, Local times and related sample path properties of certain self-similar processes, J. Math. Kyoto Univ., 33 (1993), 5 1 64. N. K6no, Kallianpur Robbins law for fractional Brownian motion, Abstract for Japan-Russian Symposium on Probability and Math. Statistics, July 26-30,1995. M. Mae,jima, Self-similar processes and limit theorems, Sugaku Exposition, 2(1989), 103-123. J. P. Nolan, Local nondeterminism and local times for stable processes, Probab. Th. Rel. Fields,
[17] [18] [19] [20]
J. P. Nolan, Path properties of index /3 processes, Ann. Probab., 16 (1988), 1596-1607. L. D. Pitt, Local times for Gaussian vector fields, Indiana Univ. Math. J., 27 (1978) , 309-330. G. Samorodnitsky and M. Taqqu, Stable non Gaussian random processes, Chapman Hall, 1994. K . Takashima, Sam ple paths properties o f ergodic self sim ilar processes, Osaka J. Math., 26
[21]
W . Vervaat, Properties of general self similar processes, 46th session of ISI a t Tokyo Japan Sept. 8-16,1987. T . Yamada, On some limit theorems for occupation times of one-dimensional Brownian motion and its continuous additive functionals locally of zero energy. J. Math. Kyoto Univ., 26 (1986)
[13]
-
[14]
-
82 (1989), 387-410. -
-
-
-
(1989), 159-189.
[22]
-
309-322,
