1 Continuous Time Processes - IPM

While we have eliminated the case of instantaneous transitions, there is
nothing in the definitions that precludes having an infinite number transitions
in a finite interval of time. It in fact a simple matter to construct
continuous time Markov chains where infinitely many transitions in finite
interval occur with positive probability. In fact, since the expectation of an
exponential random variable with parameter λ is 1 , it is intuitively clear
λ
that if λi’s increase sufficiently fast we should expect infinitely transitions in
a finite interval. In order to analyze this issue more closely we consider a
family T1, T2, . . . of independent exponential random variables with Tk having
parameter λk. Then we consider the infinite sum
k Tk. We consider the
events
[
Tk < ∞] and [
Tk = ∞].
k
The first event means there are infinitely many transitions in a finite interval
of time, and the second is the complement. It is intuitively clear that if
the rates λk increases sufficiently rapidly we should expect infinitely many
transitions in a finite interval, and conversely, if the rates do not increase
too fast then only finitely many transitions are possible infinite time. More
precisely, we have
Proposition 1.1.1 Let T1, T2, . . . be independent exponential random variables
with parameters λ1, λ2, . . .. Then
k 1
< ∞ (resp. λk k 1 = ∞)
λk
implies P [
k Tk < ∞] = 1 (resp. P [
k Tk = ∞] = 1).
Proof - We have
E[
Tk] = 1
,
k
and therefore if
k 1
λk < ∞ then P [ Tk = ∞] = 0 which proves the first
assertion. To prove the second assertion note that
Now
k
k
λk
E[e −
Tk ] = E[e −Tk ].
E[e −Tk ] =
k
∞
P [−Tk > log s]ds
◦
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