Solution

The self-loop at zero makes the irreducible chain aperiodic. The convergence theorem for irreducible, aperiodic, positive recurrent countable-state chains gives

P (X_{n} = i) ⟶ π_{i} .

(110)

Because

X

and

Y

are independent,

P (X_{n} = 0, Y_{n} = 1) = P (X_{n} = 0) P (Y_{n} = 1) ⟶ π_{0} π_{1} .

(111)

Therefore

n \to \infty lim P (X_{n} = 0, Y_{n} = 1) = \frac{1}{2} \cdot \frac{1}{4} = \frac{1}{8} .

(112)

Solved by gpt-5.6-sol high.

 Ancestors (11)

B
18H
Paper 2
Ib
2022
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