SOJOURN TIMES

Let ((u), u >_ 0} be a stochastic process with state space A U B where A and B are disjoint sets. Denote by/(t) the total time spent in state B in the interval (0, t). This paper deals with the problem of finding the distribution of/(t) and the asymptotic distribution of fl(t) as t-.oc for various types of stochastic processes. The main result is a combinatorial theorem which makes it possible to find in an elementary way, the distribution of (t) for homogeneous stochastic processes with independent increments. This article is dedicated to the memory of Roland L. Dobrushin.


Introduction
Let {(u), u >_ 0) be a stochastic process with state space A U B where A and B are disjoint sets. If (u)e A, then we say that the process is in state A at time u, and if (u)e B, then we say that the process is in state B at time u. Denote by c(t) the total time spent in state A in the time interval (0, t) and by (t) the total time spent in state B in the time interval (0, t). Clearly, c(t)+(t)-t for all t_> 0. Our aim is to find the distribution of fl(t) and the asymptotic distribution of (t) as t-oc for various types of stochastic processes.
Sojourn time problems have been studied extensively in the theory of probability. In 1939, P. Lvy [13,14] obtained some basic results for the sojourn time of Brownian motion. Let {(u),u >_ 0} be a standard Brownian motion process. We have e{(u) <: x} -(x/V for u > 0 where x 1 j 2/2 is the normal distribution function. We use the notation i(S) for the indicator variable of an event S, that is, (S) 1 if S occurs, and (S) 0 if S does not occur. Define 1 J < 0 that is, T(c) is the sojourn time of the process {(u), u >_ 0} spent in the set (-oc, c]  P{7(0) _< x} arcsinx/ (4) for 0 _< x _< 1. Formula (5) was found by P. Lvy [13,14,p. 303] in 1939, and is called the arcsine law. The more general result (3) was also found by P. Lvy [14, p. 326] but in a form more complicated than (3). The above form is given by M. Yor [24]. In 1949, M. Kac [11] gave a general method of finding the distribution of the random variable / 0 for the Brownian motion {(u),u 0} where V(x) is a given function subject to certain restrictions. If, in particular, Y(x) is the indicator function of the set (-c, a] and t= 1, then e(t) reduces to r(a) defined by (2). M. Kac [11] showed that the double Laplace transform of r(t) can be obtained by solving the differential equation subject to the conditions (x)0 as x-c, In 1957, D.A. Darling and M. Kac [5] considered the problem of finding the asymptotic distribution of or(t) for a Markov process {(u), u _> 0}.
Note 1: By using a combinatorial method, in 1961, A. Brandt [4] already determined the distribution of Wn(a for interchangeable real random variables. Actually, he considered the random variable Nn(a defined as the number of subscripts r-1,2,...,n for which r > a. In our notation Nn(a)-n+l-wn(a if a>_0 and Nn(a)-n-on(a if a<0. By the result ofA. nrandt P{Nn(a k) P{ max (i--/)+mink < a and min k > a) k<i<n 1<i< 1<i< (a9) max (i-/) + min k > a and max ((i-(k) < a} +P{k<i<n 1 <i< k<i<n for 0<k<nandany aE(-oo, oo). If, in particular, 1,2,'",n are independent and identically distributed random variables, then Theorem 2 is applicable and in this case the two random variables on the right-hand side of (38) are independent. Thus we can write that rl,j rlj, j + ,_ j, o (42) for 0 _< j _< n where Tj, j and n-j,o are independent and n-j,o has the same distribution as Tn j,O" Note 3: In the case of independent and identically distributed random variables 1, 2,"', n, relation (42) can also be deduced from a result of F. Pollaczek where p > 0, q > 0 and p + q 1. Then {(r, r >_ 0} describes a random walk on the real line and P{ 2j-n} ()pq'- for 0 _< j _< n. By the reflection principle we obtain that P{rln, n < k} P{(n < k}-P{4n < -k}
Note 4: If we assume that {X(u), 0 u 1} is a stochastic process with interchangeable increments, then Theorem 3 is still valid.
Example 2: Let {(u),u 0} be a standard Brownian motion process. We have P{(u) x} (x/) for u > 0 where (x) is defined by (1). Let us consider the process {(u)+ mxu, u 0} where m is a real number. Define 1 7(a, m) / 5((u) + mu a)du, 0 that is, 7(a, m) is the sojourn time of the process {(u) + mu, u 0} spent in the set , a] in the time interval (0, 1). We also define 7(x, m) inf{a: 7(a, m) > x} (66) for 0 < z < 1, that is, {7(z,m),O < z < 1} is the inverse process of {r(,m),-< < }. We hve P{r(, m) x} P{7(x, m) > } (67) for 0 < x < 1 and e (-, ). To find the distribution of 7(, m) or 7(x, m) by Theorem 3 it is sufficient to determine the following probability Several recent papers are concerned with the problem of finding the distribution function of r(a,m). By the results of J. Akahori [1] and A. Dassios [6] the distribution function of r(a,m) can be expressed in the form of a double integral. The above formula (74) is in agreement with their result. Both authors applied the method of M. Kac [11] in their papers. In finding the density function of 7(x, m), Dassios observed that this density function is the convolution of the density functions of two random variables. One of the variables is sup0 < u < xX(u) and the other has the same distribution as inf o < u < 1 xX(u) where X(u) (u) + mu -for ->_ 0. Thus Dassios coneluded that Theorem 3 is tru f the Brownian motion with drive. As we have seen, Theorem 3 is true more generally for homogeneous stochastic processes with independent increments. Recently P. Embrechts, L.C.G. Rogers and M. Yor [9] gave two different proofs for Dassio's result. By using formula (53) (76) for a _> 0 and m E (-oo, oo) where O(x)is defined by (1). See L. Takcs [23].

Exact Distributions
Let us consider again a stochastic process {(u),u >_ 0} with state space A U B where A and B are disjoint sets. Let us assume now that in any finite interval (0, t) the process changes states only a finite number of times with probability one. Let us suppose that P{(0) A}-1 and denote by c1, 1, c2, 2,'-" the lengths of the successive intervals spent in states A and B respectively in the interval (0, oc). Denote by c(t) the total time spent in state A in the time interval (0, t) and by /(t) the total time spent in state B in the time interval (0, t). Obviously, c(t) and /(t) are random variables and c(t) + (t) for all >_ 0. Our aim is to determine the distributions of c(t) and/(t) for >_ 0 and their asymptotic distributions as t---oc.
This follows from the identities < < < t} < t} < Here we used that a(t)+/(t)t for all t>_ 0, and that a(t) and fl(t) are nondecreasing continuous functions of t for 0 < t < c.
If for each t >_ 0 we define p(t) as a discrete random variable which takes on only nonnegative integers and satisfies the relation {p(t) < n} {Tn >-t} (84) for all t _> 0 and n 1, 2,..., then we can write P{/(t) _< x} P{hp(tx) <-x} (85) for0_<x_<t. We note that e{p(0)-0)-l. Now 5.(t is the sum of a random number of random variables. If we can determine the distribution ofo)o for all t_> 0, then by (85)  In what follows, we assume that the two sequences {an, n _> 0} and {flu, n >_ 0) are independent. If in addition, the random variables {an, n >_ 0) are identically distributed independent random variables and the random variables {n,n >_ 0) are also identically distributed independent random variables, then as an alternative we can determine the distribution of fl(t) by using

Limit Distributions
Let us assume that the two sequences {an, n >_ 0} and {n,n >_ 0} are independent. If we know the asymptotic distributions of 7n c1+ c2 +-" + Cn and 6n fll + 2 +"" + fin as n, then we expect that the asymptotic distribution of (t) for t is determined by these two distributions. This is indeed the case. For a detailed discussion, see L. Takcs [18,19]. Here we consider only particular case.
Let us assume that both 7n and 5n have an asymptotic normal distribution if n, namely, P 7an x (x) (91) and lim P/6n--< xwhere (x)is the normal distribution function defined by (1) and a,b, rr a constants. We can simply write that Vn N(na, nrr2a) as n---,c and 6 n By (84) we can prove that ,(t) N(t/a,(r2at/a3) (92) and r b are positive real N(nb, n) as n--c.
(93) as t---,cx. Now 5 can be interpreted as a sum of a random number of random variables. By P() working with characteristic functions, H. Robbins [16,17] determined the asymptotic distributions of such sums. By his results, we can conclude that 5p(t) N(bt/a, (a2r + b2a2a)t/a3) (94) as t-c. This result can be proved in a simple way by a result of R.L. Dobrushin [8] for compound random functions. The substance of Dobrushin's idea is that the asymptotic distribution of 5p(t) is independent of the particular choices of {Sn} and {p(t)}; it depends only on their asymp totic distributions. and p(t) by Consequently, we may replace 5 n by + p*(t) t/a -t-t0-bP/a 3/2 (95) (96) where and p are independent random variables having the same normal distribution defined by (1). Since o(t) has the same asymptotic distribution as p*(t) if tc, we can conclude that tli_,rnP P < x P/ :i7-2 < x (97) V/ This proves (94). Finally, by (85) and (94) we obtain that fl(t) g(bt/(a + b), (a20-+ b20-2a)t/(a / b)3) (98) as t---oo. For the asymptotic distribution of /3(t), many more examples can be found in L. Takcs [18,19].
By using a limit theorem of F.J. Anscombe [2], we can find the asymptotic distribution of 13(t) as tee for stochastic processes in which (cn,n) are independent vector random variables.
For details, see L. Takcs [19,20]. Example 4: Let us suppose that in the time interval (0, co) customers arrive at a counter in accordance with a Poisson process of intensity A and are served by one server. The server is always busy if there is at least one customer at the counter. The service times are assumed to be independent identically distributed random variables having a finite expectation a and a finite variance 0-2 and independent of the arrival times. It is also assumed that ,a < 1. Denote by /3(t), the total occupation time of the server in the time interval (0, t). Now the lengths of the successive idle periods, an(n= 1,2,...) and the lengths of the successive busy periods, fln(n 1,2,...) are independent sequences of independent and identically distributed random variables and by (98) /3(t) has an asymptotic normal distribution. The parameters in (98) where (I)(x)is defined by (1). For further details and extensions, see L. Takcs [19].