Skip to content

IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Markov chains

A simple random walk on the integers starts at 0 with X_n = X_{n-1} + Z_n, where Z_n = +1 with probability 0.5 and -1 with probability 0.5, independently. A Markov chain approach is used to study this process. Which statement is correct?

The symmetric walk has period 2 and is null recurrent. Returns happen only at even times, the return probability is one, but the mean return time is infinite, so no stationary distribution exists. All integer states communicate, so the chain is irreducible.

  1. AThe chain is positive recurrent, with a stationary distribution
  2. BThe chain is transient because the up and down probabilities are equal
  3. CThe chain is aperiodic because it can move in both directions
  4. DThe chain has period 2 and is null recurrent, so it has no stationary distributionCorrect
  5. The chain is reducible because states of different parity cannot communicate

Explanation

Returning to any state requires an even number of steps, so the period is 2. The symmetric walk on the integers returns to its start with probability 1, so it is recurrent, but the expected return time is infinite, so it is null recurrent. Null recurrent chains have no stationary distribution. Every integer is reachable from every other, so it is irreducible, and the parity claim confuses communication with periodicity.

Did you get it right without looking?

One question tells you little. A timed set on Markov chains shows your real accuracy, how long you take and where you lose marks.

Practise Markov chainsFree diagnostic mock

More Markov chains questions