Holding times

[rajeev]

Hello Everyone...

In the computation of Holding times distribution for the continuous time
markov jump process there is a sequence of sets B's defined, which is a cantor
set and the set containg chain seems to be wrong i.e


B1 > B2 > B3....... is wrong it should be other way round........

and

{To > t} = lim Bn

needs an explanation........there seems to be the usage of density of cantor set..........



The derivation of holding time distribution for inhomogenuos case is
difficult..........if anybody is through with this please explain

 

[Leila]

Cantor Set

You're scaring me!! I've never heard of a Cantor set... What is it?!

 

[rajeev]

In chapter six in holding times section we have

{To > t } = Intersection B

how did we get this and moreover the way the sets are defined...it should not be intersection of B's.....it should lim B

 

[Julie Lewis]

B1 is the set of events for which X(t/2) = X(t) = X(0).

B2 is the set of events for which X(t/4) = X(2t/4) = X(3t/4) = X(t) = X(0).

B3 is the set of events for which X(t/8) = X(2t/8) = X(3t/8) = ... = X(8t/8) = X(0).

And so on.

So B2 lies wholly within B1, B3 lies wholly within B2, etc.

The intersection of all the B's is the same as the limit as n goes to infinity.

So the course notes are correct here. Having said that, I think it's much easier to show that the first holding time is exponentially distributed by setting up a differential equation and solving it. This is what the alternative proof does.