IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Markov chains
A simple random walk on the integers starts at 0 and each step is +1 with probability p and -1 with probability 1-p, independently. Which statement about the walk is correct?
The walk is irreducible with period 2 for any p strictly between 0 and 1. A return to a state requires equal numbers of up and down steps, so returns occur only at even times, and all states communicate.
- AIt is a Markov chain with a finite state space
- BIt is irreducible with period 2 for any 0<p<1Correct
- CIt is aperiodic because it can move in either direction
- DIt is positive recurrent whenever p = 1/2
- It is transient when p = 1/2
Explanation
Returning to any state needs equal up and down steps, so only even numbers of steps work: period 2. Every state communicates with every other, so it is irreducible. The state space is infinite. At p=1/2 the walk is null recurrent, not positive recurrent, and not transient.
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