 Solution

ID: past-exam-of-the-probability-course-of-the-university-of-oxford/2014/1/c/i/solution

Write

S_{n + 1} = S_{n} + X_{n + 1}

X_{n + 1}

is independent of the entire history generated by

X_{1}, \dots, X_{n}

, hence of

(S_{1}, \dots, S_{n})

. This is a Markov chain obtained from independent innovations: conditional on the past, only the current sum enters the next state. Its time-homogeneous transition probabilities are

P_{s t} = ⎩ ⎨ ⎧ \frac{1}{( t - s ) ( t - s + 1 )}, 0, t > s, t \leq s, s, t \in {1, 2, \dots} .

Every jump is a positive integer, and each row sums to one by the probability mass function from part (b).

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