Another interesting characteristic is the number NL of germinations in the cube [0, L]d. This investigation was motivated by a study23 of release of neurotransmitters at synapses. The terminal of a neuronal axon at the neuromuscular junction has branches consisting of strands containing many randomly scattered sites. Thus, we have Poisson seeds on intervals. At a synapse an action potential triggers the release of neurotransmitters at these sites. Such a release is regarded as a germination. Each quantum released is assumed to cause release of an inhibitory substance which diffuses along the terminal at a constant rate preventing further releases in the inhibited region. In other words, a spherical inhibited region grows with a constant rate. Therefore, we have a germination-growth process with d = 1.
It has been shown that the distribution of NL, after normalisation, converges weakly to the standard normal distribution.
Theorem 2.3. Let
H= J exp i - J u)dvd{t - u)ddA(u) 1 dA(t),
L-oo Ld
where LUJ — VT^/T(1 + d/2) is the volume of a unit ball inRd. If fi < oo and a > 0, then
NL-nLd „ . , .L .
— , > Z in distribution,
where Z has the standard normal distribution.
Such a central limit theorem was first proved by Quine and Robinson23 for d = 1 and A(i) = At and then generalised to more general A by Chiu3 and Hoist et o/.14. Chiu and Quine6 established the asymptotic normality for a general class of A and d > 2, and in particular, when d = 1, they showed that NL satisfies the functional central limit theorem, meaning that after suitable normalisation and interpolation, NL behaves asymptotically like a Brownian motion. If A(t) ~ KV for some positive K and 1 < j < oo or A(i) = A/0 ya~1e~y/T(a)dy for some positive finite a and A, then the rate of convergence of the central limit theorem is 0(L~d/2 logd Ld) and that of the functional central limit theorem is 0(L~l^4+£) where e > 0.
The tool used in Chiu and Quine6 was strong mixing, which means that the dependence between the numbers of germinated seeds in two sets are vanishing when the distance between the two sets is increasing. This fact comes from two model assumptions. The first one is that seeds form a Poisson process, which implies that the numbers of seeds in two sets are independent whenever they are disjoint. The second one is that a seed located at x and stimulated at t will germinate if and only if there is no seed in the cone {(y, s) G Rd x [0, oo) : \\y - x\\ < v(t — s),s < t}, since if there is a seed in the interior of this cone, the seed at x will be prohibited from germination at some time before t by the growth of the inhibited region from the seed in the interior. From these two properties, the numbers of germinated seeds located in A and B are not independent only if such cones corresponding to seeds in A intersect those corresponding to seeds in B. When the distance between A and B is large, this event happens only if at least one of these cones has a large height. However, the probability that there is no seed in a cone tends to zero as the height of the cone goes
to infinity, and thus the dependence between the numbers of germinated seeds located in A and B is diminishing when their distance is increasing.
Under such a setting, if the word "intersect" is replaced by the phrase
"are within a distance m of", the above argument remains valid whenever m is finite, because we are talking about infinite distance, and adding a finite constant to infinity does not change anything. Therefore, "Poisson" can be replaced by "spatially m-dependent", which means that the numbers of seeds located in two disjoint sets are independent if the distance between the two sets is at least m. That is, the asymptotic normality of Ni still holds for spatially m-dependent processes of seeds (Chiu and Quine7). Such a model allows some short-range dependence between seed locations. For example, the locations can be a Poisson clustered process with bounded cluster radius or a dependently thinned Poisson process with bounded thinning radius.
It is very natural to conjecture that the m-dependence condition can be further relaxed to strong mixing. I believe it is true but have not yet got a proof. Quine and Szczotka25 considered more general point processes of seeds but their approach works only for d = 1.
For central limit theorems obtained by using strong mixing, the positiv- ity of the normalised variance a2 is usually assumed instead of proved. The reason is that the calculation of the variance requires a thorough knowl- edge of the dependence structure, which is typically difficult to obtain. Chiu and Quine6 showed this positivity numerically for d = 1, 2, 3 and 4, and A(£) — Xt. By considering the Markov process obtained by taking the shear transformation, Chiu and Lee4 showed that if d = 1, u2 is always positive for a very general class of A. Moreover they established several other strong limits such as the strong invariance principle and the strong law of large numbers for NL in the case d = 1.
3. S t a t i s t i c s
The statistical problem discussed here is the estimation of the parameters, namely A and v, from a realisation of the germinated seeds observed in a bounded window. Such an estimation problem was motivated by the same application in neurobiology5,8'19'24 described in Section 2.2, in which d = 1.
Nevertheless, all methods discussed below can be applied for d > 1.
3.1. Estimation of the speed
To estimate the speed v is the same as to estimate the absolute value of the slopes of the cones arising along the time axis from the germinated seeds
{(xi,ti)} observed, i.e. {(y,s) 6 Mdx[0,oo) : | | y - i i | | < v(s—ti),s > t}, the lower envelope of the union of which forms the sample path of the process {Tx}> which are not observable except the local minima, at which we have germinated seeds. For d = 1, the sample path is piecewise linear and v is the absolute value of the slopes of the lines that form the sample path. The estimation can be done by maximising the likelihood.
The likelihood of such a realisation is the product of the likelihood ri(x u •) dA(£jj) of having spatial-temporal Poisson points at the location Xij and germination time Uj of the observed seeds and the probability exp{— fB(v\ dxdA(t)} that there are no points below the sample path, so that the observed seeds are germinated seeds, where B(v) is the space-time region which lies beneath the sample path.
Although the sample path is not observable, when the germinated seeds are known, the sample path is uniquely determined by the parameter v, which appears only in the probability exp{— JBM dxdA(t)} and such that the probability increases as v increases, because as v increases, the cones are bigger, or the lines of the sample path are flatter for d = 1, leading to a smaller region B(v). Thus, no matter what A is, the probability increases with v, and so does the likelihood as far as v < Umax, where vmax is the maximum possible speed of a given collection of germinated seeds. The value vmax is the maximum possible speed if at least one germinated seed lies above the sample path corresponding to v whenever v > vmax. The likelihood drops to zero if v > vmax, because when a seed is above the sample path, it should not have germinated. So the maximum likelihood estimator v of v is just umax, which is calculated as the minimum of the ratios ||XJ — Xj\\/\ti — tj\ over all distinct pairs of germinated seeds (xi,U) and (xj,tj) and is simply the reciprocal of the maximum slope among the lines joining adjacent germinated seeds if d = 1. If there are n independent realisations with at least two germinations, the maximum likelihood estimator is the minimum of these n individual estimates.
With an unnormalised gamma density
dA(t) = ^f c - 1e -7 td i , (1)
where A = 5, 7 = 2 and k — 4, a series of simulations on an interval of length L have been performed and the estimates of v are given in Table 1, from which we can see that the maximum likelihood estimation works fine and the biases are small for moderate n and L.
Table 1. Maximum likelihood estimates of v, where the true value is v = 0.2.
Tl = l
5 10 50 100
L = 5 0.2694858 0.2083324 0.2060151 0.2000026 0.2006374
10 0.2107414 0.2031187 0.2002706 0.2001489 0.2000903
50 0.2071159 0.2000478 0.2005522 0.2000029 0.2000017
100 0.2001104 0.2001629 0.2000303 0.2000076 0.2000141