Handbook of Discrete-Valued Time Series

87 Count Time Series with Observation-Driven Autoregressive Parameter Dynamics

4.2.3 Coupling Arguments

An alternative but overlapping approach to proving existence of a stationary measure pro-

ceeds via coupling arguments. This is the approach of Neumann (2011). See also Franke

(2010). Neumann works with the quite general nonlinear recursive system,

Y

t

|F

t

Y

−

,λ

1

≡ Poisson(λ

t

), λ

t

= f(λ

t−1

, Y

t−1

), (4.16)

which clearly is also related to the iterative system (4.10). The function f is supposed to

satisfy the contractive condition,

|f (λ, y) − f (λ



, y



)|≤α

1

|λ − λ



|+α

2

|y − y



|, (4.17)

so that α = α

1

+α

2

< 1. This is essentially the sort of condition mentioned for the nonlinear

additive system (4.15), and the same method as was used to prove ...