# How to calculate the coefficient of absolute regularity?

I am reading the paper and try to understand the example solution in p. 14.

In particular, if the Markov chain has stationary distribution $$\pi$$ and $$a$$-step transition distribution $$P^a$$, then $$β(a) = \int π(dx)||P^a(x) − π||_{TV}. \tag{18}$$ Consider first the two-state Markov chain $$S_t$$ pictured in Figure 2. By direct calculation using (18), the mixing coefficients for this process are $$β(a) = \frac {4}{9} (\frac{1}{2})^a$$.

Also, I cited the

Definition 1 ($$β$$-mixing). For each $$a \in N$$ and any $$t \in Z$$, the coefficient of absolute regularity, or $$β$$-mixing coefficient, $$β(a)$$, is $$β(a) :=||P^t_{−∞} ⊗ P^∞_{t+a} - P_{t,a}||_{TV} \tag{1}$$ where $$|| · ||_{TV}$$ is the total variation norm, and $$P_{t,a}$$ is the joint distribution of $$(X^t_{−∞}, X^∞_{t+a})$$. A stochastic process is said to be absolutely regular, or $$β$$-mixing, if $$β(a) → 0$$ as $$a → ∞$$.

My question. How to obtain the $$β(a) = \frac {4}{9} (\frac{1}{2})^a$$?

My attempt is.

library(markovchain)
P = matrix(c(1/2, 1, 1/2, 0),,2) # transition matrix
MyStates <- c("A", "B")
M <- matrix(data = P, byrow = FALSE, nrow = 2,
dimnames = list(MyStates, MyStates))

mcM <- new("markovchain", transitionMatrix = P)
is.irreducible(mcM) # TRUE