Entries into a measurable set for sequences with stationary
independent values, with an application to renewal processes
Matija Vidmar
March 4, 2016
Let (Ω, G, P) be a probability space, F = (Fn )n∈N0 a filtration thereon. Let furthermore Z =
(Zi )i∈N be a sequence of random elements with values in a measurable space (E, E), with common
law L, and with Zi+1 independent of Fi , and Fi+1 -measurable, for each i ∈ N0 . Let also A ∈ E,
L(A) > 0. Let τ0 := 0 and then recursively
τk+1 := inf{i > τk : Zi ∈ A} for k ∈ N0 .
Then (τk )k∈N0 is a sequence of F-stopping times and τk < ∞ for all k ∈ N0 a.s. Assume then that
the latter holds not just a.s. but with certainty. Then for all k ∈ N0
(Zτk+1 , τk+1 − τk ) is independent of Fτk
and
P? (Zτk+1 , τk+1 − τk ) = L(·|A) × geomN (L(A)).
In particular, (Zτk )k∈N and (τk − τk−1 )k∈N are independent iid sequences.
Proof. That each τk is a stopping time follows by induction from the adaptedness of Z to F≥1 .
Their a.s. finiteness follows from the Borel-Cantelli lemma: P(lim supn→∞ {Zn ∈ A}) = 1. Now
assume they are finite with certainty. Since for each k ∈ N0 , τk+1 − τk = inf{i ∈ N : Zτk +i ∈ A}, it
follows at once from the strong Markov property for iid sequences that (Zτk +τk+1 −τk , τk+1 − τk ) is
independent of Fτk and equal in distribution to (Zτ1 , τ1 ). Denote τ := τ1 for short. Then, from the
fact that Z is iid, for B ∈ E|A and l ∈ N,
P(Zτ ∈ B, τ = l) = P(Zl ∈ B, Z1 ∈
/ A, . . . , Zk−1 ∈
/ A) = L(B|A)L(A)(1 − L(A))k−1 .
The equality P? (Zτk+1 , τk+1 − τk ) = L(·|A) × geomN (L(A)) now follows by an application of the
π/λ-lemma. The final claim follows from the observation that each Zτk and τk − τk−1 are Fτk measurable, induction and a π/λ-argument.
Example. Let T := (Ti )i∈N be iid [0, ∞)-valued random variables, P(T1 = 0) ∈ (0, 1), let N be the
associated renewal process. Take for F the natural filtration of T and for A = (0, ∞). Then we see
that the following are independent:
• the inter-arrival times of the process N , which are i.i.– with law P(T1 ∈ ·|T1 > 0) – d.;
• N0 , whose distribution is geomN0 (P(Ti > 0)) and
• the sequence of the jump sizes of N (excluding N0 ) which is itself i.i.– with law geomN (P(T1 >
0))–d..
In this precise sense a renewal process which has P(T1 = 0) > 0 can be reduced to a renewal process
with P(T1 = 0) = 0 /modulo the independent sequence of jump sizes/.
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