- Email: [email protected]
Some self-similar processes related to local times
STATISTICS& PROBABILITY lETTERS Statistics & Probability Letters 24 (1995) 213-218
ELSEVIER
Some self-similar processes related to local times Narn-Rueih Shieh* Department of Mathematics, National Taiwan University, TaipeL Taiwan Received April 1994; revised August 1994
Abstract
We construct some self-similar processes with continuous paths by using the local time on hyperplanes of a d-dimensional symmetric s-stable L6vy process, d/> 2 and 1 < ~t ~< 2, and its stochastic integral with respect to Gaussian white noise. Our construction gives a certain higher-dimensional extension of the previous work of Kesten and Spitzer (1979). AMS 1991 Subject Classifications: 60G18
Keywords." Self-similar processes; Local times; L6vy processes; Gaussian white noise
1. Introduction A real-valued stochastic process X(t), t t> 0, is called self-similar with exponent H (H-ss) if it satisfies the scaling condition that X(c.) ~ cHx(") for all c > 0, w h e r e " d,, denotes equality of the finite-dimensional distributions of two processes. There have been huge literatures to investigate various classes of ss processes; we refer to Taqqu (1979), Vervaat (1987) and Maejima (1989) for intensive surveys. In this aspect, Kesten and Spitzer (1979) constructed a class of ss processes with continuous paths in the following way. Let Y(t) and Z(x), t >~0 and x e ~, be two independent Levy processes which are, respectively, symmetric ~t-stable (S0tS) and symmetric fl-stable, 1 < ac ~< 2 and 0 < fl ~< 2. Let L,(x) be the local time of Y at x over [0, t]. It was proved in Kesten and Spitzer (1979) that
fx~ R L,(x) dZ(x)
(1.1)
defines a continuous H-ss process with H = 1 - (l/a) + 1/cc[3; moreover, the process is of stationary increments and is the limit of a certain random walk in random sceneries. Since the local time at points of a multidimensional Lbvy process does not exist, (1.1) cannot make sense directly for x ~ R a, d ~> 2. Lang and
* E-mail address: [email protected]. 0167-7152/95/$9.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 1 6 7 - 7 1 5 2 ( 9 4 ) 0 0 1 7 3 - 1
N.-R. Shieh / Statistics & Probability Letters 24 (1995) 213-218
214
Nguyen (1983) considered the extension of (1.1) to x ~ ~n by using the Fourier transform of occupation-time measure to replace local time. The purpose of this note is to consider a different point-of-view extension. We use the local time on hyperplanes, a concept developed by Bass (1984), and its stochastic integrals with respect to Gaussian white noise (GWN) on the unit sphere S~-1 := {x ~ •d: IIxll = 1}(11' Ildenotes the Euclidean norm) or on the whole space ~a to construct some continuous ss processes. We shall proceed the constructions in Sections 2 and 3. Here we firstly recall the local time on hyperplanes. Let Y (t), t ~> 0, be an SetS L6vy processes in R a, 1 < ct ~< 2 and d >/2(Y(t) is the Brownian motion when ct = 2). For each x~Rdo:= R d \ { o } , we let Yx(t):= Y(t)'(x/l[xl[), w h e r e , denotes the inner product. Note that Yx is real-valued S~S Levy with characteristic functions EeiOrx(o = e-tlox/ll~ill" = e-tl0t',
0 ~ R.
(1.2)
Thus, Y~(.) has a local time L(t,u,x) at u ~ R over [0,t]. Bass (1984) proved that L(t,u,x) can be chosen so that it is almostly surely jointly continuous in (t, u, x), and he called L(t, u, x) as the local time of Y(. ) on the hyperplane y. (x/ll x II ) = u. In the following, we shall make use of 4 ( x ) : = L ( t , 0 , x),
x ~ S a-t,
~(x):= L(t, Ilxlhx),
x e R~.
Note that l,(x), resp. Tt(x), represents the local time of Y(-) on the hyperplane which passes through O, resp. x, and is orthogonal to ~xx.
2. The first case Let W (dx) be the G W N on S~- 1, i.e. it is the Gaussian random measure on Sa- 1 with covariance function cov(W (A), W (B)) = a(Ac~B),
A, B e ~ ( S d- 1),
where or(. ) denotes the Lebesgue surface measure on Sd- '. We assume that W (.) and Y (.) are independent and are defined on the same probability space (f2, 5, P). Theorem 2.1. Let Y ( . ) be SetS Ldvy, 1 < ct <<.2. Then A(t):= f l,(x) W ( d x ) , ds d - I
t/>0,
defines an H-ss process with H = 1 - 1/~. Moreover, it has a modification of which almost every path is uniformly Holder continuous with any 9iven exponent 6:5 < H. Proof, We proceed in three steps. (1) Well-defined. Since Y(.) and W(.) are independent, it suffices to prove that lt(x) = 4(x, w) E L2(da ® d P ) for each t > 0. By the definition of Yx(t) in (1.2), it has a bounded continuous density pr(z), z ~ •, of which values do not depend on x. By the definition of lt(x) and an explicit formula for one-dimensional local times in Berman (1985, Lemma 2.1), we have El](x) = 2
;OlD
ps(O)ps,_~(O)dsds'.
(2.1)
Here, we have used the Markov property to express the bivariate density function as a product of transition density functions. Since p~(0) ,-~ s- 1/~ and e > 1, (2.1) is of finite value which again do not depend on x. Thus, 4(x) e L2(d~r ® dP), noting that a is a finite measure on S a- 1.
N.-R. Shieh / Statistics & Probability Letters 24 (1995) 213-218
215
(2) Self-similarity. We adopt the arguments in the proof of Vervaat 0987, Theorem 7.1). We show that L(t, u, x) defined in Section 1, regarded as a random function of (t, u, x ) • R + ® ~ ® S d- 1, has the scaling
L(ct, cl/~u,x) dcl-1/~L(t,u,x)
for all c > 0.
(2.2)
By the definition of one-dimensional local times, for each nonnegativef(t, u) and x • S d- 1,
fofRL(dt,u,x)f(t,u)dU=fof(t,Y(t).x)dt,
(2.3)
where L(dt, u, x) is the measure induced by t-~ L(t, u, x). Recall that Y (c.) d c lily(.) as a d-dimensional process. Thus, considered jointly for finitely many f and x, (2.3) is equal in distribution to
f o f ( t , cl/~ Y @ )" x ) d t = c f o f (Ct',c'/~ Y (t')" x)dt'
=CfofRL(dC,u,x)f(cC,c'~u)du ~cl-1/afO JRIL(dt,¢-l/~tu, ¢ x)f(t,u') from which (2.2) follows. Taking u = 0 in (2.2), we have
la(x) a=c l- 1/~l,(x), regarded as a random function of (t, x) • R+ ® S d- 1. Now by the independence of Y(-) and W (.),
A (ct) = [ .Is
d 1l,(x) W (dx)
d cl-1/~ fsa , lt(x) W (dx) = cl-1/~ d(t)" This shows that A is ss with H = 1 - 1/0c (3) Continuous modification. For all even integer k and 0 < s < t, we define k
Os.t(xl, ..-, Xk):= E I-I (l,(xi) - ls(xi)),
X1 ' "",
Xk
• S d- 1.
i=1
Note that g~.t is symmetric in its arguments. Since k is an even integer, by independence of Y, W and the orthogonality of W (.) over disjoint sets, we have E(,a (t) - ~ (s)) k
= ck" fx
1ES d - !
"'" fx
k/2E~ d - I
gs.t(Xl,Xl, ... ,Xk/2,Xk/2)f(dx1) "" a(dXk/2),
where Ck = (k -- 1)(k - 3) "'" 3"1. By H61der inequality for the product of k random variables, k
g,.,(xl . . . . . Xk) <<.H [E(l,(xi) - l,(xi))k] 1/k. i=1
N.-R. Shieh / Statistics & Probability Letters 24 (1995) 213-218
216
Using Berman (1985, Lemma 2.1) and p,(0) ~ t-1/~ again, we have
E(l,(xi) - l~(xi))k
=k!
f...fp~,(O)p~_~,(O)...p~-U~_l(O)du~...du~ s<
= O((t
-
...
<~uk<~t
s) k ~ -
"/~)
for all xi • S a- 1. Here we have also used the Markov property. Therefore,
E(A(t) - A(s)) k <<.Ck(t - s)k(~- 1~/~
(2.4)
for some positive constant Ck, for each even integer k. Hence, the last assertion of Theorem 2.1 follows from (2.4) and Kolmogorov's criterion. []
3. The second case
The G W N itself has the scaling property; this scaling does not appear in the ss exponent of A (t) in Theorem 2.1 since we have integration over Sd- 1 there. Professor H. Tanaka (1993) suggests the process Z(t) defined below, which does involve the scaling of GWN. Let ff'(dx) be the G W N on R d, i.e. it is the Gaussian random measure on E d with covariance function cov(I~(A), ff'(B)) = m(Ac~B),
A , B • ~(~a),
where m(') denotes the Lebesgue volume measure on R d. Again, we assume that if'(.) and Y(.) are independent and are defined on the same probability space (f2, 5, P). Theorem 3.1. Let Y (. ) be the Brownian motion in ~d. Then
,~(t):= fRg ~(x) ff'(dx),
t ~> O,
where ~(x) is the local time of Y defined in Section 1, defines an H-ss process with H = ½(1 + d/2). Moreover, it has also a continuous modification. Proof. The idea and the procedure are similar to those of Theorem 2.1. (1) Well-defined. We prove again that ~(x) • L2(dp ® dm) for each t > 0. Using the density function p,(z) in the proof of Theorem 2.1 and Berman's Lemma again, we have
where ps(z) = ( 1 / x / ~ ) e -~2/2', z • E; since we are now considering the Brownian motion. Integrating the above display over x • ~ao:= Ea\ {O} with respect to m(dx), we have a finite value. Note that, if we consider the S~S Levy process with 1 < ~ < 2, we may obtain a finite value only when d = 2, since we merely have a certain polynomial decay in case ~ < 2. (2) Self-similarity. Taking ~ = 2 and u = c-1/2 ]1x II in (2.2), we have Ta(x) =d cl/2 L( t, c- 1/2 rlx II, x),
N.-R. Shieh / Statistics & Probability Letters 24 (1995) 213-218
217
regarded as a random function of (t, x). Integrating the above display with respect to ff'(dx) over R~, by the change of variable c- ~/2x = y and the scaling lg' (aA) = a d/2 lg'(A), we have the desired self-similarity. Note that the scaling of I~ follows from the definition of its covariance function. (3) Continuous modification. As it has been proceeded in the proof of Theorem 2.1, we estimate k
l-I [Eff,(x,) - Ts(x,))k] '/~ =
O ( ( t - s) ~/~)
i=1
and then integrate it over x~ ~ ~ . By Berman's lemma and the argument in Theorem 2.1, each kth moment in the above display is equal to ? ? ] "'" |
= k!
Pu,(llxll)Pu2-u,(O) "'" Put - Uk-,(O)du, ... auk
s<~ul<~ ..-<~uk<~t Here p~(z) denotes now the Gaussian density as that in (1). Integrating the multiple integral in the above display with respect to xi, and using again the independence of Y, W and the orthogonality of W, we have again (note that k is still an even integer)
E(Z(t) - ~(s)) ~ = Ck"
,ER~
~<
-
O((t
k/2eRoa
gs,t(X1,X1 . . . . . Xk/2,Xk/2)m(dx1)
m(dXk/2)
s)k/2),
where Ck = (k - 1)(k - 3) ... 3.1 and gs,, is defined analoguely to the g~,, of Section 2. Then we see again that zT(t) has a modification of which sample paths are a.s. uniformly H61der continuous of order < ½. []
4. Some remarks
(1) Stationary increments. We cannot proceed as the proof of Vervaat (1987, Theorem 7.2) to assert that our A and zt are of stationary increments. In fact, from the proof of Theorem 2.1, we see that t*t t*u
E(A(t)-
A(S)) 2 = 2o'(S d-l)
Js Js pu(O)pu'-u(O)dudu"
which cannot depend on t - s only, thus A is not of stationary increments. The situation is the same for z~. (2) Non-Gaussianity. Using the independence of Y and W, we see that the joint distribution of A(tl), ..., A(tk) is determined by
Eei(°'~('"+
+°~('~'=Eexp{-~
fx~sd
,[i=~ Ofl,,(x)J2a(dx)};
the situation is same for A. Note that A is of finite variance. Therefore, we have a continuous ss non-Gaussian process with finite variance which is not included in the general scheme proposed in Major (1981). Certainly, our A is also different from Kestern-Spitzer processes since the ss parameters are in complementary ranges; this can also be seen by the fact that our process does not have stationary increments while theirs does have. (3) Domain of attraction. According to Lamperti (1962, Theorem 2), A should have a nonvoid domain of attraction. To characterize it, we proposed a scheme in Shieh (1993) in case Y is planar Brownian motion. Unfortunately, we are unable to give a rigorous proof even for that simplest case.
218
N.-R. Shieh / Statistics & Probability Letters 24 (1995) 213-218
Acknowledgements This work is completed while the author visited Fac. Sci. and Tech., Keio Univ., from September to December 1993; the hospitality of the Math. Dept. is deeply appreciated. I also thank the referee for his thoughtful reading of the manuscript.
References Bass, R.F. (1984), Joint continuity and representation of additive functionals of d-dimensional Brownian motion, Stochastic Processes Appl. 17, 211-227. Berman, S.M. (1985), Joint continuity of the local times of Markov processes, Z. Wahrsch. Verw. Geb. 69, 37-46. Kesten, H. and F. Spitzer (1979), A limit theorem related to a new class of self similar processes, Z. Wahrsch. Verw. Geb 50, 5-25. Lamperti, J. (1962), Semi-stable stochastic processes, Trans. Amer. Math. Soc. 104, 62-68. Lang, R. and X.X. Nguyen (1983), Strongly correlated random fields as observed by a random walker, Z. Wahrsch. Verw. Geb. 64, 327-340. Maejima, M. (1989), Self-similar processes and limit theorems, Sugaka Exp. 2, 103-123. Major, P. (1981), Multiple Wiener-lto integrals, Lecture Notes in Math., Vol. 849 (Springer, Berlin). Shieh, N.R. (1993), A class of self-similar processes related to planar random sceneries, Handout of the Annual Meeting of Japanese Probabilist, Keio University, 21-24 December 1993. Tanaka, H. (1993), private communication. Taqqu, M. (1979), Self-similar processes and related ultraviolet and infrarer catastrophes, in Random Fields, Colloq. Math. Soc. Janos Bolyai, vol. 27, 1057-1096. Vervaat, W. (1987), Properties of general self-similar processes, 46th Session of ISI at Tokyo, Japan, 1987.
ELSEVIER
Some self-similar processes related to local times Narn-Rueih Shieh* Department of Mathematics, National Taiwan University, TaipeL Taiwan Received April 1994; revised August 1994
Abstract
We construct some self-similar processes with continuous paths by using the local time on hyperplanes of a d-dimensional symmetric s-stable L6vy process, d/> 2 and 1 < ~t ~< 2, and its stochastic integral with respect to Gaussian white noise. Our construction gives a certain higher-dimensional extension of the previous work of Kesten and Spitzer (1979). AMS 1991 Subject Classifications: 60G18
Keywords." Self-similar processes; Local times; L6vy processes; Gaussian white noise
1. Introduction A real-valued stochastic process X(t), t t> 0, is called self-similar with exponent H (H-ss) if it satisfies the scaling condition that X(c.) ~ cHx(") for all c > 0, w h e r e " d,, denotes equality of the finite-dimensional distributions of two processes. There have been huge literatures to investigate various classes of ss processes; we refer to Taqqu (1979), Vervaat (1987) and Maejima (1989) for intensive surveys. In this aspect, Kesten and Spitzer (1979) constructed a class of ss processes with continuous paths in the following way. Let Y(t) and Z(x), t >~0 and x e ~, be two independent Levy processes which are, respectively, symmetric ~t-stable (S0tS) and symmetric fl-stable, 1 < ac ~< 2 and 0 < fl ~< 2. Let L,(x) be the local time of Y at x over [0, t]. It was proved in Kesten and Spitzer (1979) that
fx~ R L,(x) dZ(x)
(1.1)
defines a continuous H-ss process with H = 1 - (l/a) + 1/cc[3; moreover, the process is of stationary increments and is the limit of a certain random walk in random sceneries. Since the local time at points of a multidimensional Lbvy process does not exist, (1.1) cannot make sense directly for x ~ R a, d ~> 2. Lang and
* E-mail address: [email protected]. 0167-7152/95/$9.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 1 6 7 - 7 1 5 2 ( 9 4 ) 0 0 1 7 3 - 1
N.-R. Shieh / Statistics & Probability Letters 24 (1995) 213-218
214
Nguyen (1983) considered the extension of (1.1) to x ~ ~n by using the Fourier transform of occupation-time measure to replace local time. The purpose of this note is to consider a different point-of-view extension. We use the local time on hyperplanes, a concept developed by Bass (1984), and its stochastic integrals with respect to Gaussian white noise (GWN) on the unit sphere S~-1 := {x ~ •d: IIxll = 1}(11' Ildenotes the Euclidean norm) or on the whole space ~a to construct some continuous ss processes. We shall proceed the constructions in Sections 2 and 3. Here we firstly recall the local time on hyperplanes. Let Y (t), t ~> 0, be an SetS L6vy processes in R a, 1 < ct ~< 2 and d >/2(Y(t) is the Brownian motion when ct = 2). For each x~Rdo:= R d \ { o } , we let Yx(t):= Y(t)'(x/l[xl[), w h e r e , denotes the inner product. Note that Yx is real-valued S~S Levy with characteristic functions EeiOrx(o = e-tlox/ll~ill" = e-tl0t',
0 ~ R.
(1.2)
Thus, Y~(.) has a local time L(t,u,x) at u ~ R over [0,t]. Bass (1984) proved that L(t,u,x) can be chosen so that it is almostly surely jointly continuous in (t, u, x), and he called L(t, u, x) as the local time of Y(. ) on the hyperplane y. (x/ll x II ) = u. In the following, we shall make use of 4 ( x ) : = L ( t , 0 , x),
x ~ S a-t,
~(x):= L(t, Ilxlhx),
x e R~.
Note that l,(x), resp. Tt(x), represents the local time of Y(-) on the hyperplane which passes through O, resp. x, and is orthogonal to ~xx.
2. The first case Let W (dx) be the G W N on S~- 1, i.e. it is the Gaussian random measure on Sa- 1 with covariance function cov(W (A), W (B)) = a(Ac~B),
A, B e ~ ( S d- 1),
where or(. ) denotes the Lebesgue surface measure on Sd- '. We assume that W (.) and Y (.) are independent and are defined on the same probability space (f2, 5, P). Theorem 2.1. Let Y ( . ) be SetS Ldvy, 1 < ct <<.2. Then A(t):= f l,(x) W ( d x ) , ds d - I
t/>0,
defines an H-ss process with H = 1 - 1/~. Moreover, it has a modification of which almost every path is uniformly Holder continuous with any 9iven exponent 6:5 < H. Proof, We proceed in three steps. (1) Well-defined. Since Y(.) and W(.) are independent, it suffices to prove that lt(x) = 4(x, w) E L2(da ® d P ) for each t > 0. By the definition of Yx(t) in (1.2), it has a bounded continuous density pr(z), z ~ •, of which values do not depend on x. By the definition of lt(x) and an explicit formula for one-dimensional local times in Berman (1985, Lemma 2.1), we have El](x) = 2
;OlD
ps(O)ps,_~(O)dsds'.
(2.1)
Here, we have used the Markov property to express the bivariate density function as a product of transition density functions. Since p~(0) ,-~ s- 1/~ and e > 1, (2.1) is of finite value which again do not depend on x. Thus, 4(x) e L2(d~r ® dP), noting that a is a finite measure on S a- 1.
N.-R. Shieh / Statistics & Probability Letters 24 (1995) 213-218
215
(2) Self-similarity. We adopt the arguments in the proof of Vervaat 0987, Theorem 7.1). We show that L(t, u, x) defined in Section 1, regarded as a random function of (t, u, x ) • R + ® ~ ® S d- 1, has the scaling
L(ct, cl/~u,x) dcl-1/~L(t,u,x)
for all c > 0.
(2.2)
By the definition of one-dimensional local times, for each nonnegativef(t, u) and x • S d- 1,
fofRL(dt,u,x)f(t,u)dU=fof(t,Y(t).x)dt,
(2.3)
where L(dt, u, x) is the measure induced by t-~ L(t, u, x). Recall that Y (c.) d c lily(.) as a d-dimensional process. Thus, considered jointly for finitely many f and x, (2.3) is equal in distribution to
f o f ( t , cl/~ Y @ )" x ) d t = c f o f (Ct',c'/~ Y (t')" x)dt'
=CfofRL(dC,u,x)f(cC,c'~u)du ~cl-1/afO JRIL(dt,¢-l/~tu, ¢ x)f(t,u') from which (2.2) follows. Taking u = 0 in (2.2), we have
la(x) a=c l- 1/~l,(x), regarded as a random function of (t, x) • R+ ® S d- 1. Now by the independence of Y(-) and W (.),
A (ct) = [ .Is
d 1l,(x) W (dx)
d cl-1/~ fsa , lt(x) W (dx) = cl-1/~ d(t)" This shows that A is ss with H = 1 - 1/0c (3) Continuous modification. For all even integer k and 0 < s < t, we define k
Os.t(xl, ..-, Xk):= E I-I (l,(xi) - ls(xi)),
X1 ' "",
Xk
• S d- 1.
i=1
Note that g~.t is symmetric in its arguments. Since k is an even integer, by independence of Y, W and the orthogonality of W (.) over disjoint sets, we have E(,a (t) - ~ (s)) k
= ck" fx
1ES d - !
"'" fx
k/2E~ d - I
gs.t(Xl,Xl, ... ,Xk/2,Xk/2)f(dx1) "" a(dXk/2),
where Ck = (k -- 1)(k - 3) "'" 3"1. By H61der inequality for the product of k random variables, k
g,.,(xl . . . . . Xk) <<.H [E(l,(xi) - l,(xi))k] 1/k. i=1
N.-R. Shieh / Statistics & Probability Letters 24 (1995) 213-218
216
Using Berman (1985, Lemma 2.1) and p,(0) ~ t-1/~ again, we have
E(l,(xi) - l~(xi))k
=k!
f...fp~,(O)p~_~,(O)...p~-U~_l(O)du~...du~ s<
= O((t
-
...
<~uk<~t
s) k ~ -
"/~)
for all xi • S a- 1. Here we have also used the Markov property. Therefore,
E(A(t) - A(s)) k <<.Ck(t - s)k(~- 1~/~
(2.4)
for some positive constant Ck, for each even integer k. Hence, the last assertion of Theorem 2.1 follows from (2.4) and Kolmogorov's criterion. []
3. The second case
The G W N itself has the scaling property; this scaling does not appear in the ss exponent of A (t) in Theorem 2.1 since we have integration over Sd- 1 there. Professor H. Tanaka (1993) suggests the process Z(t) defined below, which does involve the scaling of GWN. Let ff'(dx) be the G W N on R d, i.e. it is the Gaussian random measure on E d with covariance function cov(I~(A), ff'(B)) = m(Ac~B),
A , B • ~(~a),
where m(') denotes the Lebesgue volume measure on R d. Again, we assume that if'(.) and Y(.) are independent and are defined on the same probability space (f2, 5, P). Theorem 3.1. Let Y (. ) be the Brownian motion in ~d. Then
,~(t):= fRg ~(x) ff'(dx),
t ~> O,
where ~(x) is the local time of Y defined in Section 1, defines an H-ss process with H = ½(1 + d/2). Moreover, it has also a continuous modification. Proof. The idea and the procedure are similar to those of Theorem 2.1. (1) Well-defined. We prove again that ~(x) • L2(dp ® dm) for each t > 0. Using the density function p,(z) in the proof of Theorem 2.1 and Berman's Lemma again, we have
where ps(z) = ( 1 / x / ~ ) e -~2/2', z • E; since we are now considering the Brownian motion. Integrating the above display over x • ~ao:= Ea\ {O} with respect to m(dx), we have a finite value. Note that, if we consider the S~S Levy process with 1 < ~ < 2, we may obtain a finite value only when d = 2, since we merely have a certain polynomial decay in case ~ < 2. (2) Self-similarity. Taking ~ = 2 and u = c-1/2 ]1x II in (2.2), we have Ta(x) =d cl/2 L( t, c- 1/2 rlx II, x),
N.-R. Shieh / Statistics & Probability Letters 24 (1995) 213-218
217
regarded as a random function of (t, x). Integrating the above display with respect to ff'(dx) over R~, by the change of variable c- ~/2x = y and the scaling lg' (aA) = a d/2 lg'(A), we have the desired self-similarity. Note that the scaling of I~ follows from the definition of its covariance function. (3) Continuous modification. As it has been proceeded in the proof of Theorem 2.1, we estimate k
l-I [Eff,(x,) - Ts(x,))k] '/~ =
O ( ( t - s) ~/~)
i=1
and then integrate it over x~ ~ ~ . By Berman's lemma and the argument in Theorem 2.1, each kth moment in the above display is equal to ? ? ] "'" |
= k!
Pu,(llxll)Pu2-u,(O) "'" Put - Uk-,(O)du, ... auk
s<~ul<~ ..-<~uk<~t Here p~(z) denotes now the Gaussian density as that in (1). Integrating the multiple integral in the above display with respect to xi, and using again the independence of Y, W and the orthogonality of W, we have again (note that k is still an even integer)
E(Z(t) - ~(s)) ~ = Ck"
,ER~
~<
-
O((t
k/2eRoa
gs,t(X1,X1 . . . . . Xk/2,Xk/2)m(dx1)
m(dXk/2)
s)k/2),
where Ck = (k - 1)(k - 3) ... 3.1 and gs,, is defined analoguely to the g~,, of Section 2. Then we see again that zT(t) has a modification of which sample paths are a.s. uniformly H61der continuous of order < ½. []
4. Some remarks
(1) Stationary increments. We cannot proceed as the proof of Vervaat (1987, Theorem 7.2) to assert that our A and zt are of stationary increments. In fact, from the proof of Theorem 2.1, we see that t*t t*u
E(A(t)-
A(S)) 2 = 2o'(S d-l)
Js Js pu(O)pu'-u(O)dudu"
which cannot depend on t - s only, thus A is not of stationary increments. The situation is the same for z~. (2) Non-Gaussianity. Using the independence of Y and W, we see that the joint distribution of A(tl), ..., A(tk) is determined by
Eei(°'~('"+
+°~('~'=Eexp{-~
fx~sd
,[i=~ Ofl,,(x)J2a(dx)};
the situation is same for A. Note that A is of finite variance. Therefore, we have a continuous ss non-Gaussian process with finite variance which is not included in the general scheme proposed in Major (1981). Certainly, our A is also different from Kestern-Spitzer processes since the ss parameters are in complementary ranges; this can also be seen by the fact that our process does not have stationary increments while theirs does have. (3) Domain of attraction. According to Lamperti (1962, Theorem 2), A should have a nonvoid domain of attraction. To characterize it, we proposed a scheme in Shieh (1993) in case Y is planar Brownian motion. Unfortunately, we are unable to give a rigorous proof even for that simplest case.
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N.-R. Shieh / Statistics & Probability Letters 24 (1995) 213-218
Acknowledgements This work is completed while the author visited Fac. Sci. and Tech., Keio Univ., from September to December 1993; the hospitality of the Math. Dept. is deeply appreciated. I also thank the referee for his thoughtful reading of the manuscript.
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