= Random affine recursion
{title2=$X_n=A_nX_{n-1}+B_n$}
A random affine recursion updates a real <random variable> by $X_n=A_nX_{n-1}+B_n$. 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.
Back to article page
