Let S be a metric space and λ a σ-finite measure on S. Construct a Poisson point process which has λ as the corresponding measure. Let   be independent Poisson processes with parameter   where aj is a...

Let S be a metric space and λ a σ-finite measure on S. Construct a Poisson point process which has λ as the corresponding measure.

Let
  be independent Poisson processes with parameter
  where aj is a sequence such that X_t
is finite, a.s. For

define Nt(A) to be the number of times before time t that X has a jump whose size is in A:

Prove that N_t
is a Poisson point process and determine λ.

Chapter 19