 Random affine recursion

A random affine recursion updates a real random variable by $X_n=A_n{X}_{n-1}+B_n$. Assume the coefficient pairs $(A_n,B_n)$ are independent and identically distributed random variables, independent of the initial value. The two coordinates within a pair may be dependent. Fresh coefficients make the recursion a time-homogeneous Markov process: its next conditional distribution depends on the current value alone.

The competition between the random multiplier and the fresh input determines whether initial conditions are forgotten. A synchronous coupling makes this particularly visible by using the same coefficient sequence for two initial values.