Suppose that (N t)) is a homogeneous Poisson process with intensity (lambda=12), then we know the instantaneous average rate of claims is (7.2-4.8 e^{-24t}rho) fRemarkmark 3.
Proposition 4.3 Let ( X ∗ ( t ), t ≥ 0 ) be a compound Poisson process, that is, X ∗ ( t ) = ∑ i = 1 N ∗ ( t ) X i, in which ( N ∗ ( t ), t ≥ 0 ) is a homogeneous Poisson process, and ( X n ) n = 1, 2, … is a sequence of independent, identically distributed non-negative random variables, having finite mean and being independent of the process.
Assume that the riskless interest rate (delta=0.05) and the log returns of some stock are modeled by L t)=0.15t+0.2B t)+suM ti=1}^{M(t)} zeta_{i}, where ({B t), tgeq0 }) is a standard Brownian motion, (M t)) is a homogeneous Poisson process with intensity 10, and the generic r.v.
The counting process N t)=max{n: V_{1}+cdots+ V_{n}leq t} is a homogeneous Poisson process with intensity (lambda>0), where (V_{1}) is the time until the first claim arrival, and for (igeq2), (V_{i}) is the inter-claim time between the ((i-1))th claim and the ith claim.
For instance, assuming that a spike train is an homogeneous Poisson process, is equivalent to parameterizing the intensity by one parameter, namely the fixed constant firing rate.
The time-rescaling theorem, in its simplest version, states therefore that if N is a Poisson process with intensity λ , observed on [ 0, T max ], then N = { X = Λ ( T ) : T ∈ N } is an homogeneous Poisson process on [ 0, Λ ( T max ) ] with intensity 1, fact which can be tested by practitioners.
For illustration, assume for example that we wish to test the hypothesis H 0 "the X i 's are exponential with unknown parameter λ." Note that this hypothesis is often tested on the interspike time intervals (ISI) [24] in order to test whether the observed spike process is an homogeneous Poisson process with unknown intensity λ.
Generally, rate variability was higher than would be predicted by a homogeneous Poisson process, as indicated by variance that exceeded the mean (Figure 2, top panels).
Due to this empty space, the fraction of centroids having their nearest neighbor at short distances was lower than would be expected in a homogeneous Poisson process, while at longer distances the observed curves tended to overlap or slightly exceed the Poisson curves.
The simplest form of radioactive decay is one in which all the components have an identical decay probability, which can be modelled as a homogeneous Poisson process.
