Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 1 / 28K / c

Table of contents
- i c
  - Solution i
- ii c
  - Solution ii

i

Solution

The jump chain of

(X_{t})

is exactly

(Y_{m})

: from

i

it visits

j

next with probability

p_{ij}

. Consequently the two chains have exactly the same directed paths of positive-probability transitions. Thus

X is irreducible ⟺ Y is irreducible .

ii

Solution

Because

J_{m} \to \infty

almost surely, no infinite portion of the jump chain is hidden inside a finite time interval. After either chain leaves a state

i

, the event that it ever returns is therefore the same event for both chains. Hence irreducibility and recurrence pass between a CTMC and its jump chain gives

X is recurrent ⟺ Y is recurrent .

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