I am reading a proof concerning a conditional density f ( t <mrow class="MJX-TeXAtom-OR

Emmy Dillon

Emmy Dillon

Answered question

2022-06-16

I am reading a proof concerning a conditional density f ( t | G n ) where G n = σ ( T 1 , T 2 , . . . , T n ) is a sub σ-algebra generated by random variables T 1 , . . . , T n . In the proof, they express this conditional density in terms of the probability of T n + 1 being in some infinitesimal interval ds around t. They write:
f ( t | G n ) = P ( T n + 1 [ t , t + d s ] | G n ) d s
Intuitively, this makes sense to me, but I'm not really sure how to understand this more rigorously. I know that a conditioal probability given a σ-algebra is the same as a conditional expectation of an indicator function, but how do we make sense of this equality and the left hand side?
Edit: f ( t | G n ) is the conditional density of event times in a point process given the first n points.

Answer & Explanation

Sydnee Villegas

Sydnee Villegas

Beginner2022-06-17Added 22 answers

You can define f ( t G n ) as
lim k P ( t < T n + 1 t + δ k G n ) δ k ,
where ( δ k ) is a deterministic sequence converging to 0, provided that the limit exists and is independent of the choice of the sequence ( δ k ).

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