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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2010

Paper 39

For the <autoregressive process of order one>, the formula at $t=0$ reads $r_0=r_0$, with an empty sum. If it holds at time $t$, substitution into the recursion gives
$$
r_{t+1}=\beta\left(\beta^tr_0+\sum_{s=1}^t\beta^{t-s}\xi_s\right)+\xi_{t+1}
=\beta^{t+1}r_0+\sum_{s=1}^{t+1}\beta^{t+1-s}\xi_s.
$$
Thus <mathematical induction> proves
$$
\boxed{r_t=\beta^tr_0+\sum_{s=1}^t\beta^{t-s}\xi_s.}
$$
No restriction such as $|\beta|<1$ is needed for this finite-time identity.

Solution

Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 39 / 1 / a / Solution

Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 39 / 1 / a / Solution

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Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 39 / 1 / a / Solution

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