Example time series I FTSE 100 I Meteorology: temperature, pressure... I Seismology

I A stochastic process is a family of random variables X (t), t ∈ T indexed by a parameter t in an index set T I We will consider discrete-time stochastic processes where T = Z (the integers) I A time series is said to be strictly stationary if the joint distribution of X (t1),...,X (tn) is the same as the joint distribution of X (t1 + τ),...,X (tn + τ) for all t1,..., tn, τ I A time series is said to be weakly stationary if its mean is constant and its autocovariance function depends only on the lag, i.e. E [X (t)] = µ ∀ t Cov[X (t)X (t + τ)] = γ(τ) I A Gaussian process is a family of random variables, any finite number of which have a joint Gaussian distribution I The ARMA models we will study are stationary Gaussian processe