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Arcsine Laws and Interval Partitions Derived from a Stablesubordinator

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Arcsine Laws and Interval Partitions Derived from a Stablesubordinator Arcsine laws and interval partitions derived from a stable subordinator by Jim Pitman and Marc Yor Technical Report No. 189 November 1990 Research supported by NSF grant DMS-88-01808 Department of Statistics University of California Berkeley, Califomia 94720 Arcsine laws and interval partitions derived from a stable subordinator by Jim Pitman and Marc Yor. November 28, 1990 Abstract. Levy discovered that the fraction of time a standard one-dimensional Brownian motion B spends positive before time T has arcsine distribution, both for T a fixed time when BT * 0 almost surely, and for T an inverse local time, when BT = 0 almost surely. This identity in distribution is extended from the fraction of time spent positive to a large collection of functionals derived from the lengths and signs of excursions of B away from 0. Similar identities in distribution are associated with any process whose zero set is the range of a stable subordinator, for instance a Bessel pro- cess of dimension d for 0 < d < 2. Keywords. Brownian motion, Bessel process, local time, occupation time, excursions, Poisson point process, Palm distribution, length/size biased sampling 1. Introduction. Let (B,,t > 0) be a Brownian motion on the line, starting at 0. Let J+(t) be the time B spends above 0 up to time t: r+(t) = f1(Bv >0)ds. Levy (1939) showed that for each t > 0 the variable F+(t)/t has the arcsine law: (l.a) P (I+(t)/t E du) = -dudu_ , for 0 < u < 1. ICu (1 - u On the way to this result, Levy showed that the same arcsine distribution is obtained if the fraction of time spent positive, F+(t)/t, is considered at the random time t = T, where (TS, s > 0) is the inverse of the continuous local time process (S,, t 2 0) which Levy associated with the random set of zeros of B. We denote this identity in distri- bution by (l.b) ~~~~~~r+(t)d -r+(Ts) (1.b) t TS As Levy noted in the third paragraph of page 326 of his 1939 paper, it is remarkable that the same law for the fraction of time spent positive should appear both at a fixed time t when B, . 0 almost surely, and for the random time t = TS when B, = 0 almost surely. Our aim is to expose as best we can what lies behind the identity of laws (l.b). This enables us to extend the identity to a large collection of functionals derived Pitman-Yor: Arcsine laws and interval partitions. November 28, 1990 1.2 from the lengths and signs of excursions of B away from 0. We find similar identities for a process whose zero set is the range of a stable subordinator. These results are closely related to the multidimensional arcsine laws of Barlow, Pitman and Yor (1989b) referred to below as BPYb. We start by recalling how Levy derived the arcsine distribution of J7(T3)/T,. He first showed that (Ts, s > 0) is an increasing process with independent increments, or subor- dinator, which is stable with index a = 1/2. There is a one to one correspondence beteen jumps of this subordinator and maximal open intervals during which the Brownian motion is away from zero. From the Poisson character of the jumps of the subordinator, and the fact that each of these jumps contributes to time positive with probability 1/2, independently of all others, Levy argued that (l.c) ( r+(TS), TS ) =(Ts2d Ts ), so that (1ld) Jr+(Ts) d Ts__2 Ld TS = Ts 2 +Ts2 where T'/,2 = - Ts12 is independent of Tsa2 with the same stable distribution with index 1/2. Levy showed the distribution of the right hand variable in (l.d) is arcsine by a two dimensional integration. This can also be seen more simply as in paragraph (1.1) of BPYb. Using (l.c) and the formula (l.e) P (T, E dt)/dt = S t-3/2e-s2/ s >0 t>0 Levy gave an explicit formula for the density of the conditional distribution (l.f) P(FP(T3) E - I T=,=t P(r+(t) e j B, =0,SI =s). By integration with respect to the conditional distribution of S, given B, = 0, he then obtained his famous result for the time spent positive by a Brownian bridge: (l.g) the conditional distribution of + given B, = 0 is uniform on [0, 1]. t Finally, Levy pointed out that the unconditional distribution of T+(t)/t could be found by conditioning on the time G, of the last zero before time t. Levy showed that G, / t too has arcsine distribution and that (1.h) given G, = g the pre-g process (B., 0 < u < g) is a Brownian bridge of length g. He then derived the arcsine law (l.a) for F,(t)/t by applying (l.g) to the bridge in Pitman-Yor: Arcsine laws and interval partitions. November 28, 1990 1.3 (1.h), and using the fact that independently of what happens before g, on the final zero free interval of length 1 - g, the Brownian path is equally likely to go positive or negative. The most important ingredient in these results of Levy is the fact that the random zero set Z of the Brownian motion B is the closure of the range of a stable subordinator of index a = 1/2. Somewhat more generally, suppose B is replaced by a diffusion on the line whose zero set is the closed range of a subordinator (Ti, s 2 0), which is stable with index a, for some 0 < a < 1, meaning that for every a > 0, d (l.i) (T(s), s 2 0) (aT(s/aa), s . 0). (We often use notation like T(s) instead of T, for typographical convenience.) A diffusion B with such an inverse local time process at zero can be constructed as in BPYa using a Bessel process of dimension d = 2(1 - a) for the modulus, and coin tossing for the signs. Levy's formulae (l.c) and (l.d) adapt at once to this setting to describe the law of r+(Ts)/Ts. And as shown by a Laplace transform calculation in BPYb, the identity in law (l.b) between 17+(t)/t and r+(Ts)/Ts continues to hold, even though Levy's approach sketched above is blocked in this setting by the lack of any analogue of the explicit formula (l.e) for a * 1/2. Puzzled by why (l.b) should hold so generally, we were led to analyze the interval partition generated by the zeros up to time t. By this we mean the collection of lengths of maximal intervals comprising Z'C (0, t), without regard to the order in which these lengths appear. We describe the interval partition by the sequence of ranked lengths of intervals, as in the following theorem: Theorem 1.1. Fix a with 0 < a < 1. Suppose Z is the closure of the range of a stable subordinator (Ti) of index a. Let V(t) be the infinite sequence of lengths of the maxi- mal open subintervals of Zc n (0, t), arranged in descending order: (1j) V(tO VI(t), V2(0) VP)t),..**. where V1(t) 2 V2(t) . V3(t) > ... Then for t > 0 and s > 0, V(t) d V(T.Y) t s To illustrate this result in the Brownian case, a= 1/2, Z is the zero set of the Brownian motion, and each interval of Zc is given a sign by a fair coin tossing process independent of interval length. So the ith longest interval of Zc r) (0, t), that is Vi (t), contributes with probability 1/2 to the sum of positive interval lengths r+(t), indepen- dently as i varies and independently of V(t). The same can be said at the random time Pitman-Yor: Arcsine laws and interval partitions. November 28, 1990 1.4 Ts instead of the fixed time t. So L&vy's identity (l.b) follows at once from (l.k). Notice that for a fixed time t, one of the lengths appearing in the sequence V(t) represents the age A, of the excursion in progress at time t: (1.m) At = t - G,, where G, = sup {Z n ( 0, t)} is the last time in Z before t. More precisely, A, = V#(,)(t), where # (t) - 1 is the number of excursions completed before time t with lifetime longer than A,. As a com- plement to Theorem 1.1 we show: Theorem 1.2. Condionally given V(t), the length of the final interval is picked by length biased sampling. In symbols: P (# (t) = j I V(t)) = Vj (t)/t, j = 1, 2, In contrast to a fixed time t, the random time T, falls in Z almost surely, so GT(S) = Ts and AT(S) = 0 almost surely. This means that every length appearing in V(T5) represents the lifetime of a complete excursion. And # (TS) is undefined. So despite the equality in distribution of V(t)/t and V(T3)I/T it is nonsense to substitute the ran- dom time T, instead of the fixed t in Theorem 1.2. But suppose V* is the length of the complete excursion containing a random time point T.*, which given the subordina- tor is distributed uniformly on the interval (0, Ts). Then, according to Theorems 1.1 and 1.2, the joint distribution of V(t)/t and A, ft is the same as that of V(Ts)fTs and Vs I Ts .
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