Denote by TL the time until the cube [0, L]d is totally inhibited. The investigation of TL was motivated by a study10 of the replication time of a DNA molecule in higher animals. Such a molecule is topologically linear and its replication is one of the most important process inside the bodies of animals. The replication commences at specific sites called 'origins
of replication'. These origins are randomly scattered along the molecule.
Thus, we get Poisson seeds on the real line. Each origin of replication is recognised, after some random delay time, by an enzyme-complex which then binds to the site. The complex immediately initiates, via its influence on other enzymes, a bi-directional movement along the DNA molecule.
At this moving 'frontier', the helical structure unwinds and separates into two single strands. Replication of these single-stranded regions then takes place (see Kornberg17 for biological details). Therefore, an enzyme-complex approaches an origin that has been passed over by a moving frontier will not initiate any more unwinding and separation. In other words, the locations that have been passed over by moving frontiers are inhibited. Such a process can be modelled well by the germination-growth process described above with d = 1. Then the time TL in this context is the time until a molecule of length L is completely separated.
Let TX be the inhibition time of the location x S Rd, then TL = max-fri : x £ [0, L]d} is the maximum of a random field and so when A satisfies certain regularity conditions, TL has an extreme value distribution:
T h e o r e m 2 . 1 . For each real u,
P{aLTL -bL<u}^ e~9e~u as L -> oo,
where expressions for CLL, hi, and 6 can be explicitly found when A is given.
Theorem 2.1 generalises the results in Vanderbei and Shepp26 and Cowan et al.10 from A(i) = A and A(£) = Aae~a t, respectively, to a very general class of A, as well as from d = 1 to an arbitrary d. This theorem has been proved in Chiu2, in which the extreme value theory has not been used because of the strong correlation between TX. Instead, the problem was expressed in terms of a coverage problem as follows. Consider a union U(t) of balls; the centres of the balls are scattered according to a stationary Poisson process {xi} with intensity A(t) and the radius of the ball centred at Xi is v(t — t%), where U is a random variable having the distribution function A(-)/A(i) and is independent of {XJ} and {tj : j ^ i}. The event {[0, L]d C U(t)} is the same as {TL < t} and so the distribution of TL can be obtained from the probability of complete coverage13.
Letting u — ±E6L in Theorem 2.1 leads to an approximation of TL for large L:
Corollary 2 . 1 . As L —ằ oo,
— • 1 in probability.
bL
In particular, consider the case that d = 1. The inhibition time {TX} forms a bi-directional growth process: Bi-directional growths start from germinated seeds. The growth of a line in a direction stops whenever it meets another line coming from the other direction. See Figure 1(a) for a realisation. If we take the shear transformation (x, t) i—> (x + vt, t), the bi- directional process will be transformed to a uni-directional growth process {T£}, see Figure 1(b), which is actually a Markov process.
(a) A realisation of {TX}. (b) A realisation of {r^.}.
Figure 1. A realisation of the germination-growth process {TX} for d = 1 and the uni- directional process {T'X} obtained by taking the shear transformation of the realisation in (a).
Denote by £z the value of x (on the line representing that locations) at which the process {TX} first hits the level z on the time axis. Because of stationarity, the event {Tj, < z} in the bi-directional process has the same probability as the event {£z > L} in the uni-directional Markov process.
The Laplace transform of £z can be obtained from the transition semi- group {%} of operators, where
%f{t) : = E t / ( & )
= 11 - J A(t + u)du 1 f(t + x)
+ { I A{t + u)du\ [ f{u)^-du + o(x)
o A(t)
= {1 - xA{t + 5X)} f(t + x)+ xA(t + Sx) f f(u)^$-du + o(x),
J0 A W
for some 5X in (0,:r), where Et denotes the conditional expectation given TQ =t and / is a bounded measurable real-valued function on [0, oo). Thus,
the infinitesimal generator A is given by
Af(t) = f'(t)-A(t)f(t) + [ f(u)X(u)du.
Jo
Using this generator, Chiu and Yin9 obtained the exact distribution of Ti,.
Theorem 2.2. For t > 0, let Pt denote the distribution of the Markov process {r'x} with initial value t. For z > t, we have
Pt( TL <z) = Y,eakiz)LCk U + ak(z) J ea"^u+A^du\ , where a\(z) > a2{z) > 03(2;) > ••• are all negative zeros of g(a) := 1 + a /0 Zea u + A ( u )a X A(u) = /0" A(t)dt and
a\{z) J0Z Uea*(*)"+*Wdu - 1'
Also, they9 strengthened the convergence in probability in Corollary 2.1 to strong convergence.
However, this approach does not work for an arbitrary d because we do not know what transformation can be used in general. So, the exact distribution of TL for an arbitrary d remains unknown. We will also see that usually stronger results can be obtained when d — 1.