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

Let Y_1, Y_2, ... be independent identically distributed random variables and S_n = Y_1 + ... + Y_n. Which statement about the process {S_n} is correct?

The partial-sum process is Markov because the next value equals the current value plus an independent new increment, so the conditional distribution of the future depends only on the present value, whatever the increment distribution.

  1. AIt is a Markov chain only if Y_i are normally distributed
  2. BIt is not Markov, since S_n depends on all earlier Y values
  3. CIt is Markov because S_{n+1} = S_n + Y_{n+1}, with Y_{n+1} independent of the pastCorrect
  4. DIt is Markov only if the Y_i have zero mean
  5. It is Markov only if the state space is finite

Explanation

Given S_n, the future S_{n+1} = S_n + Y_{n+1} depends only on S_n and the independent increment Y_{n+1}. So the Markov property holds for any iid increments, with no need for normality, zero mean or finite states. Dependence on the past is fully captured by S_n.

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