Help understanding a Poisson Process

Question

just a quick (presumably) one. I'm just trying to get my head around a few things:

Let $N_{t},t\geq 0$ be a counting process. then we say that a counting process is a poisson process of intensity $\lambda$ if

• the random variables $N_{t+s} - N_{t}$ is independent of $\{N_u, 0 \leq u \leq t\}$ for all $s,t \geq 0$
• The random variable $(N_{t+s} - N_{t}) \sim Pois(\lambda s)$

This definition is equivalent to: $\forall t \geq 0:$

• $P(N_{t+h} - N_{t} = 1) = \lambda h + o(h)$
• $P(N_{t+h} - N_{t} = 0) = 1 - \lambda h + o(h)$
• $P(N_{t+h} - N_{t} = 2) = o(h)$

which from my understanding tells us that the process can only increment by one during each time step. (third bullet point). That the probability of increasing by one is given by our intensity multiplied by how long we're in this time interval. and the second bullet point is obviously our compliment.

Then, we talk about increment times (which i believe are the same as holding times) $T_{1},T_{2},\cdots$ then we say that $T_n = \inf\{t\geq 0: N_{t} \geq n \}$ That is, the first time we get above some specified value (n)

then the interarrival times $(T_{n+1}-T_{n}) \sim \exp(\lambda)$

So...from my understanding then; $T_{n}$ is the time in when we've counted some number larger than n.and $T_{n+1}-T_{n}$ is how long it took to get there from the previous state?

If so, then lets say we specify some process. would we take some value from our poisson process, then wait an amount of time determined by an exponential process, then take some value from our poisson process.

so it'll be. Exponential time value 1: - count 1 - exponential time value 2 - count 2...and so on.

Answer 1

It is at best rather imprecise to say "the process can only increment by one during each time step", because

• this is a continuous-time process, so there are no "time steps", and
• there is no time interval so short that the processes cannot increment by more than one during that time interval; rather, incrementing by more than one is highly improbable, even by comparison to the very improbable event of an increment during a tiny tiny time interval.

However, the process can increment by only one at each time when it increments.

One should know the difference between two different words: "compliment" (with an "i") and "complement" (with an "e"). If I say your posting is brilliant that's a compliment (with an "i"). The complement (with an "e") of XYZ is whatever needs to be added to XYZ to make it complete. The fact that an "e" is used in the same position within the word in both "complement" and "complete" is not a mere coincidence.