![]() Here we will see another definition, by looking at what happens in a very small time period of length \(\tau\). We have seen two definitions of the Poisson process so far: one in terms of increments having a Poisson distribution, and one in therms of holding times having an exponential distribution. 22 End of Part II: Continuous time Markov jump processes.20 Long-term behaviour of Markov jump processes.19.3 Hitting probabilities and expected hitting times.18.2 Transition semigroup and the forward and backward equations.18.1 Transitions in infinitesimal time periods.17 Continuous time Markov jump processes.16.2 Time inhomogeneous Poisson process.15.3 Forward equations and proof of equivalence.15.2 Example: sum of two Poisson processes.15.1 Definition 3: increments in infinitesimal time.15 Poisson process in infinitesimal time periods.14.3 Markov property in continuous time.14.2 Definition 2: exponential holding times.14 Poisson process with exponential holding times.13.3 Summed and marked Poisson processes.13 Poisson process with Poisson increments.Part II: Continuous time Markov jump processes.12 End of of Part I: Discrete time Markov chains.11.4 Proofs of the limit and ergodic theorems.11.2 Examples of convergence and non-convergence.11 Long-term behaviour of Markov chains.10.1 Definition of stationary distribution.8.3 Hitting and return times for the simple random walk.8.1 Hitting probabilities and expected hitting times.6.3 A no-claims discount model with memory.5.1 Time homogeneous discrete time Markov chains.4.4 Expected duration for the gambler’s ruin.4.3 Inhomogeneous linear difference equations.4.2 Probability of ruin for the gambler’s ruin.4.1 Homogeneous linear difference equations.2.3 Exact distribution of the simple random walk.1 Stochastic processes and the Markov property. ![]()
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