Let {N(t), t ≥ 0} be a homogeneous Poisson process with intensity λ and let {T 1 , T 2 , ...} be the associated random point process. Derive trend function m(t) and covariance function C(s, t) of the...

Let {N(t), t ≥ 0} be a homogeneous Poisson process with intensity λ and let {T₁, T₂, ...} be the associated random point process. Derive trend function m(t) and covariance function C(s, t) of the shot noise process {X(t), t ≥ 0} defined by

by partitioning the positive half axis [0,∞) into intervals of length Δx and making use of the homogeneity and independence of the increments of a homogeneous Poisson process.

Note that {X(t), t ≥ 0} is the same process as the one analyzed in example 3.4 with another technique.