- Email: [email protected]
Distributions of selfsimilar and semi-selfsimilar processes with independent increments
Statistics & Probability Letters 47 (2000) 395 – 401
Distributions of selfsimilar and semi-selfsimilar processes with independent increments Makoto Maejimaa;∗, Ken-iti Satob , Toshiro Watanabec a Department
of Mathematics, Keio University, Hiyoshi, Yokohama 223-8522, Japan 1101-5-103, Tenpaku-ku, Nagoya 468-0074, Japan c Center for Mathematical Sciences, The University of Aizu, Aizu-Wakamatsu, Fukushima 965-8580, Japan b Hachiman-yama
Received February 1999
Abstract Relationships between marginal and joint distributions of selfsimilar processes with independent increments are shown in terms of the Urbanik–Sato-type nested subclasses of the class of selfdecomposable distributions. Similar results are also c 2000 Elsevier Science B.V. All rights reserved shown for semi-selfsimilar processes with independent increments. Keywords: Selfsimilar process; Semi-selfsimilar process; Selfdecomposable distribution; Semi-selfdecomposable distribution
1. Introduction An Rd -valued stochastic process {X (t); t¿0} is called selfsimilar if there exists H ¿ 0 such that for any a¿0 d
{X (at); t¿0} ={aH X (t); t¿0}; d
(1.1)
where = denotes the equality in all joint distributions, and H is called the exponent of selfsimilarity. We may call {X (t)} H -selfsimilar. Selfsimilar processes have been attracting interest in the theory of stochastic processes and its applications. However, we know only little about distributions of general selfsimilar processes. If selfsimilar processes {X (t)} have independent and stationary increments, then they are strictly stable Levy processes and thus we have full knowledge of their distributions. Namely, X (1) is strictly stable
∗
Corresponding author. E-mail address: [email protected] (M. Maejima)
c 2000 Elsevier Science B.V. All rights reserved 0167-7152/00/$ - see front matter PII: S 0 1 6 7 - 7 1 5 2 ( 9 9 ) 0 0 1 8 4 - 4
396
M. Maejima et al. / Statistics & Probability Letters 47 (2000) 395 – 401
and its characteristic function ˆ satis es that for some ∈ (0; 2], ˆ 1= z) (z) ˆ a = (a
for all a ¿ 0:
(1.2)
But, this is not the case any more for selfsimilar processes without independent and stationary increments. Actually, as mentioned in Barndor -Nielsen and Perez-Abreu (1998), there is no simple characterization of the possible families of marginal distributions of selfsimilar processes {X (t)} with stationary increments. However, there are several works on this problem. Namely, O’Brien and Vervaat (1983) studied in R1 the concentration function of log X (1)+ and the support of X (1), gave some lower bounds for the tails of the distribution of X (1) in case H ¿ 1, and showed that X (1) cannot have atoms except in certain trivial cases. Also, Maejima (1986) studied the relationship between moments and exponents of selfsimilarity. One of interesting questions is: Is the distribution of X (1) outside 0 absolutely continuous if H 6= 1? This question was raised by O’Brien and Vervaat (1983), but it is still open. Now, how about selfsimilar processes with independent increments? Then the situation is better. Let us use the words marginal and joint in the following way. For an Rd -valued process {X (t)}, a marginal distribution of {X (t)} is the distribution of X (t) on Rd for any t; for any n, an n-tuple joint distribution of {X (t)} is the distribution of (X (t1 ); : : : ; X (tn )) on Rnd for any choice of distinct t1 ; : : : ; tn . It is shown in Sato (1991), henceforth referred to as [S91], that marginal distributions of selfsimilar processes with independent increments are selfdecomposable. Its de nition will be given later. It is also shown that any selfdecomposable distribution can appear in this way. Many distributions are known to be selfdecomposable, and their importance has been increasing, for instance, in mathematical nance and other elds (see, e.g. Barndor -Nielsen, 1998; Jurek, 1997). In studying stochastic processes, we want to know their joint distributions. Thus, the next problem is how much we can know about joint distributions of such processes. This will be answered in Theorem 1 in Section 2. On the other hand, non-Gaussian stable Levy processes are important in stochastic modeling for phenomena governed by distributions with heavy tails. Their generalizations are semi-stable distributions. Strictly semi-stable distributions with index are de ned by relation (1.2) with the replacement of “for all a ¿ 0” by “for some a ∈ (0; 1) ∪ (1; ∞)”. If = 2, they are Gaussian. If 0 ¡ ¡ 2, they have tail behaviors similar to those of strictly stable distributions, and make a much larger class than the class of strictly stable distributions. Hence, strictly semi-stable Levy processes o er higher exibility in stochastic modeling than strictly stable Levy processes. Also, checking strict semi-stability from data is much easier than strict stability, because it is enough to check (1.2) for some a ∈ (0; 1) ∪ (1; ∞). It is easily seen that a strictly semi-stable Levy process {X (t)} with index satis es (1.1) with H = 1= for some a ∈ (0; 1) ∪ (1; ∞). Motivated by this, the notion of semi-selfsimilarity has been introduced in Maejima and Sato (1999) (henceforth this paper is referred to as [MS]). Namely, an Rd -valued process {X (t); t¿0} is called semi-selfsimilar if there exist a ∈ (0; 1) ∪ (1; ∞) and b ¿ 0 such that d
{X (at); t¿0} = {bX (t); t¿0}:
(1.3)
This scaling property is much weaker than selfsimilarity. Di usions on Sierpinski gaskets {X (t)} on Rd constructed by Goldstein (1987), Kusuoka (1987), and Barlow and Perkins (1988) are not selfsimilar in the sense that (1.1) holds for all a ¿ 0, but they have semi-selfsimilarity, and actually they satisfy {X ((d + d
3)n t)} ={2n X (t)} for all n ∈ Z. The semi-selfsimilarity is thus an important notion in mathematical physics and it allows us again higher exibility in stochastic modeling. We shall see in Theorem 2 in Section 3 that, although those processes have weaker scaling property, we can prove some distributional properties similar to selfsimilar case, if they have independent increments. We use the following notation. P(Rd ) and I (Rd ) are, respectively, the class of all probability distributions on Rd and the class of all in nitely divisible distributions on Rd . L(X ) is the distribution of a random ˆ variable or vector X . The characteristic function of ∈ P(Rd ) is denoted by (z).
M. Maejima et al. / Statistics & Probability Letters 47 (2000) 395 – 401
397
2. Joint distributions of selfsimilar processes with independent increments The class of selfdecomposable distributions on Rd is called the class L and denoted by L0 (Rd ). A sequence of its subclasses Lm (Rd ); m = 0; 1; : : : ; ∞; was introduced by Urbanik (1972) and its study was continued by Sato (1980), which we refer to as [S80]. A description of the classes is as follows. A distribution ∈ P(Rd ) belongs to L0 (Rd ) if and only if, for any b ∈ (0; 1), there is b ∈ P(Rd ) such that (z) ˆ = (bz) ˆ ˆb (z);
∀z ∈ Rd :
(2.1)
If ∈ L0 (Rd ), then ∈ I (Rd ); b is uniquely determined by and b, and b ∈ I (Rd ). Let m be a positive d integer. A distribution ∈ P(Rd ) belongs to Lm (Rd ) if and only T if ∈ L0 (R ) and, for every b ∈ (0; 1); b in (2.1) belongs to Lm−1 (Rd ). The class L∞ (Rd ) is de ned by m¿0 Lm (Rd ). Then we have I (Rd ) ⊃ L0 (Rd ) ⊃ L1 (Rd ) ⊃ · · · ⊃ L∞ (Rd ) ⊃ S(Rd ); where S(Rd ) is the class of all stable distributions on Rd . A necessary and sucient condition for distributions to be in Lm (Rd ) in terms of Levy measures is given in [S80] so that examples of distributions in Lm (Rd ) are given in terms of Levy measures. But the following simpler examples in L0 (R) ∩ L1 (R)c and in L1 (R) are worthwhile to remark. Example 1. Let d = 1. Consider the case where Z ∞ k(x) dx (eizx − 1) (z) ˆ = exp x 0 R1 R∞ with k(x)¿0 satisfying 0 k(x) d x + 1 (k(x)=x) d x ¡ ∞. Let h(x) = k(e−x ). Then ∈ L0 (R) if and only if k(x) is nonincreasing. A necessary and sucient condition for to be in L1 (R) is the convexity of h(x) (see Theorem 3:2 of [S80]). (i) Let k(x) = ce−ax with a; c ¿ 0. Then is a gamma distribution. It is in L0 (R) but not in L1 (R), since h(x) is not convex. Thus Theorem 1 below tells us that some joint distributions (in fact, all n-tuple joint distributions with n¿2) of the corresponding selfsimilar process {X (t)} with independent increments are outside the class L0 (R). More properties of this process are studied in [S91] and Watanabe (1996). (ii) Let k(x) = cx− e−ax with a; c ¿ 0 and 0 ¡ ¡ 2. It is easy to nd the condition for the convexity of h(x). Thus ∈ L1 (R) if and only if ¿ 14 . Hence inverse Gaussian distributions (that is, = 12 ) are in L1 (R). (As to inverse Gaussian distributions, see, e.g. Seshadri (1993).) As mentioned in Section 1, it is shown in [S91] that if {X (t); t¿0} is a stochastically continuous selfsimilar process with independent increments on Rd , then, its marginal distributions are selfdecomposable. However, the n-tuple joint distribution for n¿2 is not always selfdecomposable (Proposition 4:2 of [S91]). In the following Theorem 1, we shall give conditions for joint distributions to be selfdecomposable, and furthermore, conditions for them to belong to the smaller classes Lm (Rd ). Theorem 1. Let {X (t); t¿0} be a stochastically continuous selfsimilar process with independent increments. Let m be a positive integer or ∞. Then the following four conditions are equivalent. We understand m−1=∞ if m = ∞. (i) L(X (t)) ∈ Lm (Rd ); ∀t¿0. nd (ii) L((X Pn(t1 ); : : : ; X (tn ))) ∈ Lm−1d (R ); ∀n; ∀t1 ; : : : ; tn ¿0. (iii) L( k=1 ck X (tk )) ∈ Lm−1 (R ); ∀n; ∀t1 ; : : : ; tn ¿0; ∀c1 ; : : : ; cn ∈ R. (iv) L(X (t) − X (s)) ∈ Lm−1 (Rd ); ∀s; t¿0.
398
M. Maejima et al. / Statistics & Probability Letters 47 (2000) 395 – 401
Lemma 1. Let m ∈ {0; 1; : : : ; ∞}. Let d1 ; : : : ; dn be positive integers. If ∈ Lm (Rd1 ) and if T is a linear transformation from Rd1 to Rd2 ; then T −1 ∈ Lm (Rd2 ); where (T −1 )(B) = (T −1 (B)). If k ∈ Lm (Rdk ) for k = 1; : : : ; n; then 1 × · · · × n ∈ Lm (Rd ) with d = d1 + · · · + dn . This lemma is essentially found in Theorem 2:4 of [S80]. The proof is based on the decomposition (2.1). Proof of Theorem 1. Let 06t1 6 · · · 6tn . Note that X (0) = 0 a.s. by the de nition (1.1). Let Y1 = X (t1 ) and Yk =X (tk )−X (tk−1 ) for k=2; : : : ; n. Then X (tk )=Y1 +· · ·+Yk . By Lemma 1, we see that L((X (t1 ); : : : ; X (tn ))) ∈ Lm−1 (Rnd ) if and only if L((Y1 ; : : : ; Yn )) ∈ Lm−1 (Rnd ). Since Y1 ; : : : ; Yn are independent, Lemma 1 shows that L((Y1 ; : : : ; Yn )) ∈ Lm−1 (Rnd ) if and only if L(Yk ) ∈ Lm−1 (Rd ) for k = 1; : : : ; n. Hence we see that (ii) and (iv) are equivalent. By Lemma 1, (ii) implies (iii). Obviously (iii) implies (iv). Hence (iii) is equivalent to (ii) and to (iv). Let us prove the equivalence of (i) and (iv). Let H ¿ 0 be the exponent of {X (t)}, and let 06s6t. Denote t = L(X (t)) and s; t = L(X (t) − X (s)). Then ˆt (z) = ˆs (z)ˆs; t (z) = ˆt ((s=t)H z)ˆs; t (z);
(2.2)
where we have used the independent increments property and (1.1) with a = s=t. On the other hand, as we have already mentioned, t ∈ L0 (Rd ). Thus, for any b ∈ (0; 1), there exists t; b ∈ I (Rd ) such that ˆt (z) = ˆt (bz)ˆt; b (z);
∀z ∈ Rd :
(2.3)
Since ˆt (z) 6= 0, it follows from (2.2) and (2.3) that t; (s=t)H = s; t : Hence t; (s=t)H ∈ Lm−1 (Rd ) if and only if s; t ∈ Lm−1 (Rd ), concluding that (i) and (iv) are equivalent. Note that if L(X (1)) ∈ Lm (Rd ), then (i) is true. This is because L(X (t)) = L(t H X (1)) by (1.1). 3. Joint distributions of semi-selfsimilar processes with independent increments We now consider semi-selfsimilar processes de ned by (1.3). A process {X (t)} is called trivial if, for each t; P{X (t) = const:} = 1. Otherwise, it is called nontrivial. It is proved in [MS] that if {X (t)} is a nontrivial, stochastically continuous semi-selfsimilar process, then there exists a unique exponent H ¿0 such that b = aH for all a, b satisfying (1.3), and that if, in addition, it has independent increments, then their marginal distributions are semi-selfdecomposable, whose de nition will be given below. For any xed b ∈ (0; 1) a sequence of the subclasses Lm (b; Rd ); m = 0; 1; : : : ; ∞, of I (Rd ) has recently been introduced in Maejima and Naito (1998). This paper is referred to as [MN]. A distribution ∈ P(Rd ) is in L0 (b; Rd ) if and only if (2.1) holds with some b ∈ I (Rd ). If ∈ L0 (b; Rd ) with some b ∈ (0; 1), it is called semi-selfdecomposable. These distributions are de ned in [MN] as limiting distributions of some subsequences of normalized partial sums of independent random vectors with in nitesimal condition. Again, if ∈L0 (b; Rd ), then ∈I (Rd ) and b is uniquely determined. A distribution ∈P(Rd ) belongs to Lm (b; Rd ) d d d with T a positived integer m if and only if ∈L0 (b; R ) and b ∈Lm−1 (b; R ). The class L∞ (b; R ) is de ned by L (b; R ). Then we have m¿0 m I (Rd ) ⊃ L0 (b; Rd ) ⊃ L1 (b; Rd ) ⊃ · · · ⊃ L∞ (b; Rd ) ⊃ SS(b; Rd ); ihc; zi where SS(b; Rd ) is the class of all semi-stable distributions ∈I (Rd ) with the property that (z) ˆ a = (bz)e ˆ for some a ∈ (0; 1) and c ∈ Rd . A necessary and sucient condition for distributions to be in Lm (b; Rd ) in
M. Maejima et al. / Statistics & Probability Letters 47 (2000) 395 – 401
399
terms of Levy measures is given in [MN] and examples of such distributions are found in Maejima et al. (1999). Furthermore, we have \ Lm (b; Rd ); m = 0; 1; : : : ; ∞: Lm (Rd ) = 0¡b¡1
This follows from the characterization of Levy measures of distributions in Lm (Rd ) and Lm (b; Rd ), namely, from the combination of Theorem 3:2 of [S80] and Theorem 4:2 of [MN]. Let {X (t); t ¿ 0} be a nontrivial, stochastically continuous, semi-selfsimilar process on Rd having exponent H ¿ 0 with independent increments. We assume that it is not selfsimilar. Let be the set of a ¿ 0 such that there is b ¿ 0 satisfying (1.3). Then it is shown in [MS] that = {an0 : n∈Z} for some a0 ¿ 1 and that d L(X (t))∈L0 (a−H 0 ; R ) for every t. We now have the following theorem. Theorem 2. Let {X (t); t¿0}; H; and a0 be as above. Let m be a positive integer or ∞. Then the following four conditions are equivalent: d (i) L(X (t))∈Lm (a−H 0 ; R ); ∀t¿0. nd (a−H (ii) L((X 0 ; R ); ∀n; ∀t¿0; ∀u1 ; : : : ; un ∈ . Pn(u1 t); : : : ; X (un t)))∈Lm−1 −H d (iii) L( k=1 ck X (uk t))∈Lm−1 (a0 ; R ); ∀n; ∀t¿0; ∀u1 ; : : : ; un ∈ ; ∀c1 ; : : : ; cn ∈R: d (iv) L(X (u2 t) − X (u1 t))∈Lm−1 (a−H 0 ; R ); ∀t¿0; ∀u1 ; u2 ∈ : Lemma 2. Let 0 ¡ b ¡ 1. The statement of Lemma 1 remains true if we replace Lm (Rdk ) and Lm (Rd ) by Lm (b; Rdk ) and Lm (b; Rd ); respectively. This is proved similarly to Lemma 1. Proof of Theorem 2. Using Lemma 2 in place of Lemma 1, we can prove the equivalence of (ii) – (iv) in the same way as in the proof of Theorem 1. To show the equivalence of (i) and (iv), let u; v∈ with 0 ¡ u ¡ v and let t¿0. Then, since u=v∈ , ˆvt (z) = ˆut (z)ˆut; vt (z) = ˆvt ((u=v)H z)eihc; zi ˆut; vt (z) z)eihc; zi ˆut; vt (z) = ˆvt (a−nH 0 with some c∈Rd and some positive integer n. To see that (iv) implies (i), it is enough to choose v = 1 and in the identity above. Conversely, suppose that (i) is satis ed. Then u = a−1 0 ˆ ˆvt (z) = ˆvt (a−H 0 z)(z) d with ∈Lm−1 (a−H 0 ; R ). Hence,
z)(a ˆ −H ˆ ˆvt (z) = ˆvt (a−2H 0 0 z)(z) = ˆvt (a−nH z)(a ˆ −(n−1)H z) : : : (z): ˆ 0 0 Since ˆvt (z) 6= 0, it follows that −ihc; zi ˆut; vt (z) = (a ˆ −(n−1)H z) : : : (z)e ˆ ; 0 −H d where u = va−n 0 . Since Lm−1 (a0 ; R ) is closed under convolution (Theorem 3:3 of [MN]), we see that −H d ut; vt ∈Lm−1 (a0 ; R ). That is, we get condition (iv). d We note that (i) is true if L(X (t))∈Lm (a−H 0 ; R ) for 16t ¡ a0 .
400
M. Maejima et al. / Statistics & Probability Letters 47 (2000) 395 – 401
Comparing Theorems 1 and 2, one might ask whether in Theorem 2 condition (i) implies that all joint nd distributions of {X (t)} are in Lm−1 (a−H 0 ; R ). We show, by an example, that the answer is negative. Example 2. Let d = 1 and 0 ¡ b ¡ 1. Consider an in nitely divisible distribution with Levy measure X = kn b−n ; n ∈Z
where kn ¿0; if
P
n¿0 kn
kn − kn+1 ¿0;
+
P n¡0
b−2n kn ¡ ∞, and b−n is the unit mass at b−n . Then, ∈L0 (b; R) if and only
∀n∈Z
(3.1)
and ∈L1 (b; R) if and only if, in addition to (3.1), (kn − kn+1 ) − (kn+1 − kn+2 )¿0;
∀n∈Z;
(3.2) −H
(see [MN]). Suppose that ∈L0 (b; R). Choose a ¿ 1 and H ¿ 0 such that b = a . If gn (t); n∈Z, are chosen to be nondecreasing continuous functions on [1; a) satisfying gn (1) = kn and limt↑a gn (t) = kn−1 , then we can construct a stochastically continuous, H -semi-selfsimilar process {X (t); t¿0} with independent increments satisfying L(X (1)) = such that L(X (t)) has Levy measure X gn (t)b−n (3.3) t = n ∈Z
for 16t ¡ a (Theorem 6:2 of [MS]). Now assume that the given kn ; n∈Z, satisfy (3.1) and (3.2) with strict inequalities (e.g. kn = bn=H with H ¿ 12 ). Let a−t t−1 + kn−1 ; 16t ¡ a: a−1 a−1 Then, for any xed t ∈ [1; a); hn (t); n ∈ Z, satisfy the inequalities corresponding to (3.1) and (3.2). Choose j ¿ 0 so small that hn (t) = kn
kj − 2hj+1 (1 + 2) + kj+2 ¿ 0
for j = 0; −1; −2:
This is possible, since kj − 2hj+1 (t) + kj+2 → kj − 2kj+1 + kj+2 ¿ 0 as t ↓ 1. Next choose gn (t); n ∈ Z, as follows: gn (t) = hn (t);
16∀t ¡ a; ∀n 6= 0;
g0 (t) = h0 (t);
1 + 26∀t ¡ a;
g0 (t) = k0 ;
16∀t61 + ;
and g0 (t) is chosen to be continuous and nondecreasing for 1 + 6t61 + 2. We claim that gn (t) − gn+1 (t)¿0;
16∀t ¡ a; ∀n∈Z;
(gn (t) − gn+1 (t)) − (gn+1 (t) − gn+2 (t))¿0; g0 (t) − k0 ¡ g1 (t) − k1 ;
(3.4) 16∀t ¡ a; ∀n∈Z;
1 ¡ ∀t61 + :
(3.5) (3.6)
In fact, (3.4) follows from that gn (t)¿kn ¿gn+1 (t). If n 6= 0; −1; −2 and t ∈ [1; a), or if n ∈ {0; −1; −2} and t ∈[1 + 2; a), then (3.5) is identical with the corresponding inequality for hn (t); n∈Z. If j ∈{0; −1; −2} and t ∈[1; 1 + 2), then (gj (t) − gj+1 (t)) − (gj+1 (t) − gj+2 (t))¿kj − 2hj+1 (1 + 2) + kj+2 ¿ 0: Hence (3.5) is true. Inequality (3.6) is obvious. Consider the process {X (t)} that corresponds to these gn (t); n ∈ Z. Thus, for 16t ¡ a; L(X (t)) has Levy measure (3.3). Now, we have L(X (t)) ∈ L1 (b; R) for
M. Maejima et al. / Statistics & Probability Letters 47 (2000) 395 – 401
401
16t ¡ a by (3.4) and (3.5), and also for all other t by the semi-selfsimilarity. However, L(X (t) − X (1))6∈ L0 (b; R) for 1 ¡ t61 + by virtue of (3.6), because X (gn (t) − kn )b−n 1; t = n ∈Z
for the Levy measure 1; t of L(X (t) − X (1)). Hence L((X (t); X (1)))6∈ L0 (b; R2 ) for 1 ¡ t61 + . References Barlow, M.T., Perkins, E.A., 1988. Brownian motion on the Sierpinski gasket. Probab. Theory Related Fields 79, 543–623. Barndor -Nielsen, O.E., 1998. Processes of normal inverse Gaussian type. Finance Stochastics 2, 41–68. Barndor -Nielsen, O.E., Perez-Abreu, V., 1998. Stationary and selfsimilar processes driven by Levy processes. Research Report no. 1, Centre for Mathematical Physics and Stochastics, University of Aarhus. Goldstein, S., 1987. Random walks and di usions on fractals. In: Kesten, H. (Ed.), Percolation Theory and Ergodic Theory of In nite Particle Systems, IMA Vol. Math. Appl., Vol. 8. Springer, Berlin, pp. 121–128. Jurek, Z.J., 1997. Selfdecomposability: an exception or a rule? Annales Univer. M. Curie-Sklodowska, Lublin-Polonia 51, Sect. A, 93–107. Kusuoka, S., 1987. A di usion process on a fractal. In: Itˆo, K., Ikeda, N. (Eds.), Probabilistic Models in Mathematical Physics, Proceedings Taniguchi Symposium, Katata 1985. Kinikuniya – North-Holland, Amsterdam, pp. 251–274. Maejima, M., 1986. A remark on selfsimilar processes with stationary increments. Canad. J. Statist. 14, 81–82. Maejima, M., Naito, Y., 1998. Semi-selfdecomposable distributions and a new class of limit theorems. Probab. Theory Related Fields 112, 13–31. Maejima, M., Sato, K., 1999. Semi-selfsimilar processes. J. Theor. Probab. 12, 347–383. Maejima, M., Sato, K., Watanabe, T., 1999. Operator semi-selfdecomposability, (C; Q)-decomposability and related nested classes. Tokyo J. Math., to appear. O’Brien, G.L., Vervaat, W., 1983. Marginal distributions of selfsimilar processes with stationary increments. Z. Wahrsch. Verw. Geb. 64, 129–138. Sato, K., 1980. Class L of multivariate distributions and its subclasses. J. Multivar. Anal. 10, 207–232. Sato, K., 1991. Selfsimilar processes with independent increments. Probab. Theory Related Fields 89, 285–300. Seshadri, V., 1993. The Inverse Gaussian Distribution, Oxford University Press, Oxford. Urbanik, K., 1972. Slowly varying sequences of random variables. Bull. Acad. Pol. Sci. Ser. Sci. Math. Astronom. Phys. 20, 679–682. Watanabe, T., 1996. Sample function behavior of increasing processes of class L. Probab. Theory Related Fields 104, 349–374.
Distributions of selfsimilar and semi-selfsimilar processes with independent increments Makoto Maejimaa;∗, Ken-iti Satob , Toshiro Watanabec a Department
of Mathematics, Keio University, Hiyoshi, Yokohama 223-8522, Japan 1101-5-103, Tenpaku-ku, Nagoya 468-0074, Japan c Center for Mathematical Sciences, The University of Aizu, Aizu-Wakamatsu, Fukushima 965-8580, Japan b Hachiman-yama
Received February 1999
Abstract Relationships between marginal and joint distributions of selfsimilar processes with independent increments are shown in terms of the Urbanik–Sato-type nested subclasses of the class of selfdecomposable distributions. Similar results are also c 2000 Elsevier Science B.V. All rights reserved shown for semi-selfsimilar processes with independent increments. Keywords: Selfsimilar process; Semi-selfsimilar process; Selfdecomposable distribution; Semi-selfdecomposable distribution
1. Introduction An Rd -valued stochastic process {X (t); t¿0} is called selfsimilar if there exists H ¿ 0 such that for any a¿0 d
{X (at); t¿0} ={aH X (t); t¿0}; d
(1.1)
where = denotes the equality in all joint distributions, and H is called the exponent of selfsimilarity. We may call {X (t)} H -selfsimilar. Selfsimilar processes have been attracting interest in the theory of stochastic processes and its applications. However, we know only little about distributions of general selfsimilar processes. If selfsimilar processes {X (t)} have independent and stationary increments, then they are strictly stable Levy processes and thus we have full knowledge of their distributions. Namely, X (1) is strictly stable
∗
Corresponding author. E-mail address: [email protected] (M. Maejima)
c 2000 Elsevier Science B.V. All rights reserved 0167-7152/00/$ - see front matter PII: S 0 1 6 7 - 7 1 5 2 ( 9 9 ) 0 0 1 8 4 - 4
396
M. Maejima et al. / Statistics & Probability Letters 47 (2000) 395 – 401
and its characteristic function ˆ satis es that for some ∈ (0; 2], ˆ 1= z) (z) ˆ a = (a
for all a ¿ 0:
(1.2)
But, this is not the case any more for selfsimilar processes without independent and stationary increments. Actually, as mentioned in Barndor -Nielsen and Perez-Abreu (1998), there is no simple characterization of the possible families of marginal distributions of selfsimilar processes {X (t)} with stationary increments. However, there are several works on this problem. Namely, O’Brien and Vervaat (1983) studied in R1 the concentration function of log X (1)+ and the support of X (1), gave some lower bounds for the tails of the distribution of X (1) in case H ¿ 1, and showed that X (1) cannot have atoms except in certain trivial cases. Also, Maejima (1986) studied the relationship between moments and exponents of selfsimilarity. One of interesting questions is: Is the distribution of X (1) outside 0 absolutely continuous if H 6= 1? This question was raised by O’Brien and Vervaat (1983), but it is still open. Now, how about selfsimilar processes with independent increments? Then the situation is better. Let us use the words marginal and joint in the following way. For an Rd -valued process {X (t)}, a marginal distribution of {X (t)} is the distribution of X (t) on Rd for any t; for any n, an n-tuple joint distribution of {X (t)} is the distribution of (X (t1 ); : : : ; X (tn )) on Rnd for any choice of distinct t1 ; : : : ; tn . It is shown in Sato (1991), henceforth referred to as [S91], that marginal distributions of selfsimilar processes with independent increments are selfdecomposable. Its de nition will be given later. It is also shown that any selfdecomposable distribution can appear in this way. Many distributions are known to be selfdecomposable, and their importance has been increasing, for instance, in mathematical nance and other elds (see, e.g. Barndor -Nielsen, 1998; Jurek, 1997). In studying stochastic processes, we want to know their joint distributions. Thus, the next problem is how much we can know about joint distributions of such processes. This will be answered in Theorem 1 in Section 2. On the other hand, non-Gaussian stable Levy processes are important in stochastic modeling for phenomena governed by distributions with heavy tails. Their generalizations are semi-stable distributions. Strictly semi-stable distributions with index are de ned by relation (1.2) with the replacement of “for all a ¿ 0” by “for some a ∈ (0; 1) ∪ (1; ∞)”. If = 2, they are Gaussian. If 0 ¡ ¡ 2, they have tail behaviors similar to those of strictly stable distributions, and make a much larger class than the class of strictly stable distributions. Hence, strictly semi-stable Levy processes o er higher exibility in stochastic modeling than strictly stable Levy processes. Also, checking strict semi-stability from data is much easier than strict stability, because it is enough to check (1.2) for some a ∈ (0; 1) ∪ (1; ∞). It is easily seen that a strictly semi-stable Levy process {X (t)} with index satis es (1.1) with H = 1= for some a ∈ (0; 1) ∪ (1; ∞). Motivated by this, the notion of semi-selfsimilarity has been introduced in Maejima and Sato (1999) (henceforth this paper is referred to as [MS]). Namely, an Rd -valued process {X (t); t¿0} is called semi-selfsimilar if there exist a ∈ (0; 1) ∪ (1; ∞) and b ¿ 0 such that d
{X (at); t¿0} = {bX (t); t¿0}:
(1.3)
This scaling property is much weaker than selfsimilarity. Di usions on Sierpinski gaskets {X (t)} on Rd constructed by Goldstein (1987), Kusuoka (1987), and Barlow and Perkins (1988) are not selfsimilar in the sense that (1.1) holds for all a ¿ 0, but they have semi-selfsimilarity, and actually they satisfy {X ((d + d
3)n t)} ={2n X (t)} for all n ∈ Z. The semi-selfsimilarity is thus an important notion in mathematical physics and it allows us again higher exibility in stochastic modeling. We shall see in Theorem 2 in Section 3 that, although those processes have weaker scaling property, we can prove some distributional properties similar to selfsimilar case, if they have independent increments. We use the following notation. P(Rd ) and I (Rd ) are, respectively, the class of all probability distributions on Rd and the class of all in nitely divisible distributions on Rd . L(X ) is the distribution of a random ˆ variable or vector X . The characteristic function of ∈ P(Rd ) is denoted by (z).
M. Maejima et al. / Statistics & Probability Letters 47 (2000) 395 – 401
397
2. Joint distributions of selfsimilar processes with independent increments The class of selfdecomposable distributions on Rd is called the class L and denoted by L0 (Rd ). A sequence of its subclasses Lm (Rd ); m = 0; 1; : : : ; ∞; was introduced by Urbanik (1972) and its study was continued by Sato (1980), which we refer to as [S80]. A description of the classes is as follows. A distribution ∈ P(Rd ) belongs to L0 (Rd ) if and only if, for any b ∈ (0; 1), there is b ∈ P(Rd ) such that (z) ˆ = (bz) ˆ ˆb (z);
∀z ∈ Rd :
(2.1)
If ∈ L0 (Rd ), then ∈ I (Rd ); b is uniquely determined by and b, and b ∈ I (Rd ). Let m be a positive d integer. A distribution ∈ P(Rd ) belongs to Lm (Rd ) if and only T if ∈ L0 (R ) and, for every b ∈ (0; 1); b in (2.1) belongs to Lm−1 (Rd ). The class L∞ (Rd ) is de ned by m¿0 Lm (Rd ). Then we have I (Rd ) ⊃ L0 (Rd ) ⊃ L1 (Rd ) ⊃ · · · ⊃ L∞ (Rd ) ⊃ S(Rd ); where S(Rd ) is the class of all stable distributions on Rd . A necessary and sucient condition for distributions to be in Lm (Rd ) in terms of Levy measures is given in [S80] so that examples of distributions in Lm (Rd ) are given in terms of Levy measures. But the following simpler examples in L0 (R) ∩ L1 (R)c and in L1 (R) are worthwhile to remark. Example 1. Let d = 1. Consider the case where Z ∞ k(x) dx (eizx − 1) (z) ˆ = exp x 0 R1 R∞ with k(x)¿0 satisfying 0 k(x) d x + 1 (k(x)=x) d x ¡ ∞. Let h(x) = k(e−x ). Then ∈ L0 (R) if and only if k(x) is nonincreasing. A necessary and sucient condition for to be in L1 (R) is the convexity of h(x) (see Theorem 3:2 of [S80]). (i) Let k(x) = ce−ax with a; c ¿ 0. Then is a gamma distribution. It is in L0 (R) but not in L1 (R), since h(x) is not convex. Thus Theorem 1 below tells us that some joint distributions (in fact, all n-tuple joint distributions with n¿2) of the corresponding selfsimilar process {X (t)} with independent increments are outside the class L0 (R). More properties of this process are studied in [S91] and Watanabe (1996). (ii) Let k(x) = cx− e−ax with a; c ¿ 0 and 0 ¡ ¡ 2. It is easy to nd the condition for the convexity of h(x). Thus ∈ L1 (R) if and only if ¿ 14 . Hence inverse Gaussian distributions (that is, = 12 ) are in L1 (R). (As to inverse Gaussian distributions, see, e.g. Seshadri (1993).) As mentioned in Section 1, it is shown in [S91] that if {X (t); t¿0} is a stochastically continuous selfsimilar process with independent increments on Rd , then, its marginal distributions are selfdecomposable. However, the n-tuple joint distribution for n¿2 is not always selfdecomposable (Proposition 4:2 of [S91]). In the following Theorem 1, we shall give conditions for joint distributions to be selfdecomposable, and furthermore, conditions for them to belong to the smaller classes Lm (Rd ). Theorem 1. Let {X (t); t¿0} be a stochastically continuous selfsimilar process with independent increments. Let m be a positive integer or ∞. Then the following four conditions are equivalent. We understand m−1=∞ if m = ∞. (i) L(X (t)) ∈ Lm (Rd ); ∀t¿0. nd (ii) L((X Pn(t1 ); : : : ; X (tn ))) ∈ Lm−1d (R ); ∀n; ∀t1 ; : : : ; tn ¿0. (iii) L( k=1 ck X (tk )) ∈ Lm−1 (R ); ∀n; ∀t1 ; : : : ; tn ¿0; ∀c1 ; : : : ; cn ∈ R. (iv) L(X (t) − X (s)) ∈ Lm−1 (Rd ); ∀s; t¿0.
398
M. Maejima et al. / Statistics & Probability Letters 47 (2000) 395 – 401
Lemma 1. Let m ∈ {0; 1; : : : ; ∞}. Let d1 ; : : : ; dn be positive integers. If ∈ Lm (Rd1 ) and if T is a linear transformation from Rd1 to Rd2 ; then T −1 ∈ Lm (Rd2 ); where (T −1 )(B) = (T −1 (B)). If k ∈ Lm (Rdk ) for k = 1; : : : ; n; then 1 × · · · × n ∈ Lm (Rd ) with d = d1 + · · · + dn . This lemma is essentially found in Theorem 2:4 of [S80]. The proof is based on the decomposition (2.1). Proof of Theorem 1. Let 06t1 6 · · · 6tn . Note that X (0) = 0 a.s. by the de nition (1.1). Let Y1 = X (t1 ) and Yk =X (tk )−X (tk−1 ) for k=2; : : : ; n. Then X (tk )=Y1 +· · ·+Yk . By Lemma 1, we see that L((X (t1 ); : : : ; X (tn ))) ∈ Lm−1 (Rnd ) if and only if L((Y1 ; : : : ; Yn )) ∈ Lm−1 (Rnd ). Since Y1 ; : : : ; Yn are independent, Lemma 1 shows that L((Y1 ; : : : ; Yn )) ∈ Lm−1 (Rnd ) if and only if L(Yk ) ∈ Lm−1 (Rd ) for k = 1; : : : ; n. Hence we see that (ii) and (iv) are equivalent. By Lemma 1, (ii) implies (iii). Obviously (iii) implies (iv). Hence (iii) is equivalent to (ii) and to (iv). Let us prove the equivalence of (i) and (iv). Let H ¿ 0 be the exponent of {X (t)}, and let 06s6t. Denote t = L(X (t)) and s; t = L(X (t) − X (s)). Then ˆt (z) = ˆs (z)ˆs; t (z) = ˆt ((s=t)H z)ˆs; t (z);
(2.2)
where we have used the independent increments property and (1.1) with a = s=t. On the other hand, as we have already mentioned, t ∈ L0 (Rd ). Thus, for any b ∈ (0; 1), there exists t; b ∈ I (Rd ) such that ˆt (z) = ˆt (bz)ˆt; b (z);
∀z ∈ Rd :
(2.3)
Since ˆt (z) 6= 0, it follows from (2.2) and (2.3) that t; (s=t)H = s; t : Hence t; (s=t)H ∈ Lm−1 (Rd ) if and only if s; t ∈ Lm−1 (Rd ), concluding that (i) and (iv) are equivalent. Note that if L(X (1)) ∈ Lm (Rd ), then (i) is true. This is because L(X (t)) = L(t H X (1)) by (1.1). 3. Joint distributions of semi-selfsimilar processes with independent increments We now consider semi-selfsimilar processes de ned by (1.3). A process {X (t)} is called trivial if, for each t; P{X (t) = const:} = 1. Otherwise, it is called nontrivial. It is proved in [MS] that if {X (t)} is a nontrivial, stochastically continuous semi-selfsimilar process, then there exists a unique exponent H ¿0 such that b = aH for all a, b satisfying (1.3), and that if, in addition, it has independent increments, then their marginal distributions are semi-selfdecomposable, whose de nition will be given below. For any xed b ∈ (0; 1) a sequence of the subclasses Lm (b; Rd ); m = 0; 1; : : : ; ∞, of I (Rd ) has recently been introduced in Maejima and Naito (1998). This paper is referred to as [MN]. A distribution ∈ P(Rd ) is in L0 (b; Rd ) if and only if (2.1) holds with some b ∈ I (Rd ). If ∈ L0 (b; Rd ) with some b ∈ (0; 1), it is called semi-selfdecomposable. These distributions are de ned in [MN] as limiting distributions of some subsequences of normalized partial sums of independent random vectors with in nitesimal condition. Again, if ∈L0 (b; Rd ), then ∈I (Rd ) and b is uniquely determined. A distribution ∈P(Rd ) belongs to Lm (b; Rd ) d d d with T a positived integer m if and only if ∈L0 (b; R ) and b ∈Lm−1 (b; R ). The class L∞ (b; R ) is de ned by L (b; R ). Then we have m¿0 m I (Rd ) ⊃ L0 (b; Rd ) ⊃ L1 (b; Rd ) ⊃ · · · ⊃ L∞ (b; Rd ) ⊃ SS(b; Rd ); ihc; zi where SS(b; Rd ) is the class of all semi-stable distributions ∈I (Rd ) with the property that (z) ˆ a = (bz)e ˆ for some a ∈ (0; 1) and c ∈ Rd . A necessary and sucient condition for distributions to be in Lm (b; Rd ) in
M. Maejima et al. / Statistics & Probability Letters 47 (2000) 395 – 401
399
terms of Levy measures is given in [MN] and examples of such distributions are found in Maejima et al. (1999). Furthermore, we have \ Lm (b; Rd ); m = 0; 1; : : : ; ∞: Lm (Rd ) = 0¡b¡1
This follows from the characterization of Levy measures of distributions in Lm (Rd ) and Lm (b; Rd ), namely, from the combination of Theorem 3:2 of [S80] and Theorem 4:2 of [MN]. Let {X (t); t ¿ 0} be a nontrivial, stochastically continuous, semi-selfsimilar process on Rd having exponent H ¿ 0 with independent increments. We assume that it is not selfsimilar. Let be the set of a ¿ 0 such that there is b ¿ 0 satisfying (1.3). Then it is shown in [MS] that = {an0 : n∈Z} for some a0 ¿ 1 and that d L(X (t))∈L0 (a−H 0 ; R ) for every t. We now have the following theorem. Theorem 2. Let {X (t); t¿0}; H; and a0 be as above. Let m be a positive integer or ∞. Then the following four conditions are equivalent: d (i) L(X (t))∈Lm (a−H 0 ; R ); ∀t¿0. nd (a−H (ii) L((X 0 ; R ); ∀n; ∀t¿0; ∀u1 ; : : : ; un ∈ . Pn(u1 t); : : : ; X (un t)))∈Lm−1 −H d (iii) L( k=1 ck X (uk t))∈Lm−1 (a0 ; R ); ∀n; ∀t¿0; ∀u1 ; : : : ; un ∈ ; ∀c1 ; : : : ; cn ∈R: d (iv) L(X (u2 t) − X (u1 t))∈Lm−1 (a−H 0 ; R ); ∀t¿0; ∀u1 ; u2 ∈ : Lemma 2. Let 0 ¡ b ¡ 1. The statement of Lemma 1 remains true if we replace Lm (Rdk ) and Lm (Rd ) by Lm (b; Rdk ) and Lm (b; Rd ); respectively. This is proved similarly to Lemma 1. Proof of Theorem 2. Using Lemma 2 in place of Lemma 1, we can prove the equivalence of (ii) – (iv) in the same way as in the proof of Theorem 1. To show the equivalence of (i) and (iv), let u; v∈ with 0 ¡ u ¡ v and let t¿0. Then, since u=v∈ , ˆvt (z) = ˆut (z)ˆut; vt (z) = ˆvt ((u=v)H z)eihc; zi ˆut; vt (z) z)eihc; zi ˆut; vt (z) = ˆvt (a−nH 0 with some c∈Rd and some positive integer n. To see that (iv) implies (i), it is enough to choose v = 1 and in the identity above. Conversely, suppose that (i) is satis ed. Then u = a−1 0 ˆ ˆvt (z) = ˆvt (a−H 0 z)(z) d with ∈Lm−1 (a−H 0 ; R ). Hence,
z)(a ˆ −H ˆ ˆvt (z) = ˆvt (a−2H 0 0 z)(z) = ˆvt (a−nH z)(a ˆ −(n−1)H z) : : : (z): ˆ 0 0 Since ˆvt (z) 6= 0, it follows that −ihc; zi ˆut; vt (z) = (a ˆ −(n−1)H z) : : : (z)e ˆ ; 0 −H d where u = va−n 0 . Since Lm−1 (a0 ; R ) is closed under convolution (Theorem 3:3 of [MN]), we see that −H d ut; vt ∈Lm−1 (a0 ; R ). That is, we get condition (iv). d We note that (i) is true if L(X (t))∈Lm (a−H 0 ; R ) for 16t ¡ a0 .
400
M. Maejima et al. / Statistics & Probability Letters 47 (2000) 395 – 401
Comparing Theorems 1 and 2, one might ask whether in Theorem 2 condition (i) implies that all joint nd distributions of {X (t)} are in Lm−1 (a−H 0 ; R ). We show, by an example, that the answer is negative. Example 2. Let d = 1 and 0 ¡ b ¡ 1. Consider an in nitely divisible distribution with Levy measure X = kn b−n ; n ∈Z
where kn ¿0; if
P
n¿0 kn
kn − kn+1 ¿0;
+
P n¡0
b−2n kn ¡ ∞, and b−n is the unit mass at b−n . Then, ∈L0 (b; R) if and only
∀n∈Z
(3.1)
and ∈L1 (b; R) if and only if, in addition to (3.1), (kn − kn+1 ) − (kn+1 − kn+2 )¿0;
∀n∈Z;
(3.2) −H
(see [MN]). Suppose that ∈L0 (b; R). Choose a ¿ 1 and H ¿ 0 such that b = a . If gn (t); n∈Z, are chosen to be nondecreasing continuous functions on [1; a) satisfying gn (1) = kn and limt↑a gn (t) = kn−1 , then we can construct a stochastically continuous, H -semi-selfsimilar process {X (t); t¿0} with independent increments satisfying L(X (1)) = such that L(X (t)) has Levy measure X gn (t)b−n (3.3) t = n ∈Z
for 16t ¡ a (Theorem 6:2 of [MS]). Now assume that the given kn ; n∈Z, satisfy (3.1) and (3.2) with strict inequalities (e.g. kn = bn=H with H ¿ 12 ). Let a−t t−1 + kn−1 ; 16t ¡ a: a−1 a−1 Then, for any xed t ∈ [1; a); hn (t); n ∈ Z, satisfy the inequalities corresponding to (3.1) and (3.2). Choose j ¿ 0 so small that hn (t) = kn
kj − 2hj+1 (1 + 2) + kj+2 ¿ 0
for j = 0; −1; −2:
This is possible, since kj − 2hj+1 (t) + kj+2 → kj − 2kj+1 + kj+2 ¿ 0 as t ↓ 1. Next choose gn (t); n ∈ Z, as follows: gn (t) = hn (t);
16∀t ¡ a; ∀n 6= 0;
g0 (t) = h0 (t);
1 + 26∀t ¡ a;
g0 (t) = k0 ;
16∀t61 + ;
and g0 (t) is chosen to be continuous and nondecreasing for 1 + 6t61 + 2. We claim that gn (t) − gn+1 (t)¿0;
16∀t ¡ a; ∀n∈Z;
(gn (t) − gn+1 (t)) − (gn+1 (t) − gn+2 (t))¿0; g0 (t) − k0 ¡ g1 (t) − k1 ;
(3.4) 16∀t ¡ a; ∀n∈Z;
1 ¡ ∀t61 + :
(3.5) (3.6)
In fact, (3.4) follows from that gn (t)¿kn ¿gn+1 (t). If n 6= 0; −1; −2 and t ∈ [1; a), or if n ∈ {0; −1; −2} and t ∈[1 + 2; a), then (3.5) is identical with the corresponding inequality for hn (t); n∈Z. If j ∈{0; −1; −2} and t ∈[1; 1 + 2), then (gj (t) − gj+1 (t)) − (gj+1 (t) − gj+2 (t))¿kj − 2hj+1 (1 + 2) + kj+2 ¿ 0: Hence (3.5) is true. Inequality (3.6) is obvious. Consider the process {X (t)} that corresponds to these gn (t); n ∈ Z. Thus, for 16t ¡ a; L(X (t)) has Levy measure (3.3). Now, we have L(X (t)) ∈ L1 (b; R) for
M. Maejima et al. / Statistics & Probability Letters 47 (2000) 395 – 401
401
16t ¡ a by (3.4) and (3.5), and also for all other t by the semi-selfsimilarity. However, L(X (t) − X (1))6∈ L0 (b; R) for 1 ¡ t61 + by virtue of (3.6), because X (gn (t) − kn )b−n 1; t = n ∈Z
for the Levy measure 1; t of L(X (t) − X (1)). Hence L((X (t); X (1)))6∈ L0 (b; R2 ) for 1 ¡ t61 + . References Barlow, M.T., Perkins, E.A., 1988. Brownian motion on the Sierpinski gasket. Probab. Theory Related Fields 79, 543–623. Barndor -Nielsen, O.E., 1998. Processes of normal inverse Gaussian type. Finance Stochastics 2, 41–68. Barndor -Nielsen, O.E., Perez-Abreu, V., 1998. Stationary and selfsimilar processes driven by Levy processes. Research Report no. 1, Centre for Mathematical Physics and Stochastics, University of Aarhus. Goldstein, S., 1987. Random walks and di usions on fractals. In: Kesten, H. (Ed.), Percolation Theory and Ergodic Theory of In nite Particle Systems, IMA Vol. Math. Appl., Vol. 8. Springer, Berlin, pp. 121–128. Jurek, Z.J., 1997. Selfdecomposability: an exception or a rule? Annales Univer. M. Curie-Sklodowska, Lublin-Polonia 51, Sect. A, 93–107. Kusuoka, S., 1987. A di usion process on a fractal. In: Itˆo, K., Ikeda, N. (Eds.), Probabilistic Models in Mathematical Physics, Proceedings Taniguchi Symposium, Katata 1985. Kinikuniya – North-Holland, Amsterdam, pp. 251–274. Maejima, M., 1986. A remark on selfsimilar processes with stationary increments. Canad. J. Statist. 14, 81–82. Maejima, M., Naito, Y., 1998. Semi-selfdecomposable distributions and a new class of limit theorems. Probab. Theory Related Fields 112, 13–31. Maejima, M., Sato, K., 1999. Semi-selfsimilar processes. J. Theor. Probab. 12, 347–383. Maejima, M., Sato, K., Watanabe, T., 1999. Operator semi-selfdecomposability, (C; Q)-decomposability and related nested classes. Tokyo J. Math., to appear. O’Brien, G.L., Vervaat, W., 1983. Marginal distributions of selfsimilar processes with stationary increments. Z. Wahrsch. Verw. Geb. 64, 129–138. Sato, K., 1980. Class L of multivariate distributions and its subclasses. J. Multivar. Anal. 10, 207–232. Sato, K., 1991. Selfsimilar processes with independent increments. Probab. Theory Related Fields 89, 285–300. Seshadri, V., 1993. The Inverse Gaussian Distribution, Oxford University Press, Oxford. Urbanik, K., 1972. Slowly varying sequences of random variables. Bull. Acad. Pol. Sci. Ser. Sci. Math. Astronom. Phys. 20, 679–682. Watanabe, T., 1996. Sample function behavior of increasing processes of class L. Probab. Theory Related Fields 104, 349–374.