Wold's theorem

The Woldsche decomposition refers to a special decomposition in the time series analysis, a branch of mathematical statistics. The decomposition is named after Herman Wold, who showed in 1938 that the random variables of a discrete-time kovarianzstationären, stochastic process can be decomposed into two parts:

• In a deterministic, not random component and
• In a purely non-deterministic component which is caused by smoothing of random variables:

The random variables have the expected value of zero and a constant variance and are pairwise uncorrelated:

The result of smoothing is

• Possibly infinitely long (but can also be finite )
• Square limited to:
• " Causal" (there are no terms )
• Are constant (that is, regardless of the time )

Usually, we set:

Pure non-deterministic means that all linear deterministic components were subtracted from above. Such a time series is called white noise. A linear deterministic component as can be predicted perfectly because of their own past values ​​. This is the case for a constant mean, periodic, polynomial or exponential effects in the times.

The required quadratic convergence of the series which guarantees the existence of the second moments of the process. For the validity of this decomposition no distributional assumptions need to be made and need not be independent; it is sufficient uncorrelated.

For the expectation value is obtained

That is, it applies:

The variance is calculated as follows:

Because of this expression simplifies to

The variance is therefore finite and independent of time. Accordingly, one obtains with the autocovariances

With. It can be seen that the autocovariance is only a function of the time difference. All the conditions for the covariance are met. The autocorrelation function can be written as follows:

For example, can be brought into the Woldsche representation ARMA models. This representation is more of theoretical interest, because in practical applications, models with infinitely many parameters unusable.

Wold decomposition in functional analysis

There is also a Wold'sche decomposition in functional analysis, see Shift Operator.