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Let (Ω, FY, P ) be a probability space. A zero mean stochastic process {yt} is said to

be strictly stationary if the finite dimension joint cumulative distribution function of {yt} at times t1+ s, · · · , tk+ s satisfies

FY(yt1+s, · · · , ytk+s) = FY(yt1, · · · , ytk)

for all k and s > 0. A simple example would be the white noise process with identical distribution. Following similar notations of Cheng et al. (2000), for each process {yt}

with yt∈ Lp(Ω), that is,

R

Ω|y| p_dF

Y(y) < ∞, we define the following subspaces:

Ht(Y ) = ¯sp{ys, s ≤ t}, and H−∞(Y ) =

\

t≤0

Ht(Y )

where ¯_{sp{· · · } represents the closed linear space spanned by the elements in the} bracket under the Lp _norm.

The process {yt} is said to be deterministic if

Ht−1(Y ) = Ht(Y )

for all t, and is called nondeterministic otherwise. If a nondeterministic process satisfies

H−∞= {0},

then it is said to be a pure nondeterministic process.

2.6.1. Weakly stationary process

In most situations, strict stationarity is too strong of an assumption in prediction theory of stationary process. A zero mean stochastic process {yt} is called a weakly

stationary process if

E|yt|2 < ∞, and cov(ys, yt) = γ(s − t),

for all s, t, where γ(·) is referred to as the covariance function.

The weakly stationary process has been extensively studied and it can be shown that, any weakly stationary process with a continuous spectral density can be approx- imated by a weakly stationary autoregressive model with a large order (Brockwell and Davis, 1991). In fact, it is shown in Pourahmadi (1988) that for a purely nondeter- ministic weakly stationary process {yt}, there exists a unique series {ak} such that

for all t, one has yt= ∞ X k=1 akyt−k+ ǫt, provided that P∞

k=1akyt−k is convergent in the L2 norm. A sufficient condition for

the convergence of P∞

k=1akyt−k is that

P∞

k=1|ak| < ∞.

Another nice property of this decomposition is that variables in the innovation process {ǫt} are orthogonal under the inner product induced by the L2 norm, i.e.,

they are uncorrelated. This is a very useful result which indicates that, for a general weakly stationary process, we can use an autoregressive model with a sufficiently large order to do one-step or multi-step predictions, without knowing the true probability structure of the process.

2.6.2. p-stationary process

The popularity of the autoregressive model in time series studies is largely due to the fact that any second order stationary process with symmetric continuous spectral density can be approximated by an autoregressive process (Brockwell and Davis, 1991). It would be very appealing if this type of approximation still holds for the infinite variance process, which can justify the use of autoregressive model to do predictions. However, even for the strictly stationary process with infinite variance, this is difficult to show.

Miamee and Pourahmadi (1988) established such a relationship for the p-stationary process. A discrete time stochastic process {yt} is said to be a p-stationary process if

E|yt|p < ∞, and E      n X k=1 ckytk+h      p = E      n X k=1 ckytk      p ,

(1 < p ≤ 2) for all integers n ≥ 1, t1, . . . , tn, h, and scalars c1, . . . , cn. Note that, when

variance. This class of processes includes the harmonizable stable processes of order α with α ∈ (1, 2] and strictly stationary processes with finite p-th moment. Miamee and Pourahmadi (1988) showed that for a purely nondeterministic p-stationary process {yt} with innovation {ǫt}, there exists a unique series {ak} such that for all t, one has

yt= ∞ X k=1 akyt−k+ ǫt, provided thatP∞

k=1akyt−k is convergent in the mean of order p. A sufficient condition

for the convergence of P∞

k=1akyt−k is that

P∞

k=1|ak| < ∞. For regularity conditions

and more recent advances in this area, see Cheng et al. (2000).

Compare to the weakly stationary process, the autoregressive representation above does not have the property that variables in the innovation process {ǫt} are

not uncorrelated for the case of 0 < p < 2. So the above representation does provide some insights for using an autoregressive model for predicting a general stationary infinite variance time series in that even though the underlying structure of the time series is not autoregressive, it can be approximated by an autoregressive model under certain conditions. However things are not as nicely done as in p = 2 case.

CHAPTER III

VARIABLE SELECTION FOR INFINITE VARIANCE AUTOREGRESSIVE MODELS