Wold's theorem

Wold's theorem is the statement of mathematical statistics , according to which each weakly stationary time series can be represented as a moving average of infinite order $\mathrm {MA} (\infty )$ ${\ displaystyle \ mathrm {MA} (\ infty)}$ ${\ displaystyle \ mathrm {MA} (\ infty)}$ . This representation is called the moving average representation for time series.

Established by Herman Wold .

Formally:

Y_{t}=\sum _{j=0}^{\infty }b_{j}\varepsilon _{t-j}+\eta _{t}

{\ displaystyle Y_ {t} = \ sum _ {j = 0} ^ {\ infty} b_ {j} \ varepsilon _ {tj} + \ eta _ {t}}

{\ displaystyle Y_ {t} = \ sum _ {j = 0} ^ {\ infty} b_ {j} \ varepsilon _ {t-j} + \ eta _ {t}}

,

Where:

$Y_{t}$ ${\ displaystyle Y_ {t}}$ ${\ displaystyle Y_ {t}}$ - considered time series ,
$\varepsilon _{t}$ ${\ displaystyle \ varepsilon _ {t}}$ $\ varepsilon _ {t}$ - white noise at the input of the line filter $\{b_{j}\}$ ${\ displaystyle \ {b_ {j} \}}$ ${\ displaystyle \ {b_ {j} \}}$ ; also applies the term "innovation" ( eng. innovation ) ^[1]
$b_{j}$ ${\ displaystyle b_ {j}}$ $b_ {j}$ - a sequence of moving average coefficients (parameters or weights)
$\eta _{t}$ ${\ displaystyle \ eta _ {t}}$ $\ eta _ {t}$ - deterministic component; equals zero if y $Y_{t}$ ${\ displaystyle Y_ {t}}$ ${\ displaystyle Y_ {t}}$ no trends .

Odds $b_{j}$ ${\ displaystyle b_ {j}}$ $b_ {j}$ satisfy the following conditions:

$b_{0}=1$ ${\ displaystyle b_ {0} = 1}$ ${\ displaystyle b_ {0} = 1}$
row $b_{j}$ ${\ displaystyle b_ {j}}$ $b_ {j}$ absolutely converges : $\sum _{j=1}^{\infty }|b_{j}|<\infty$ ${\ displaystyle \ sum _ {j = 1} ^ {\ infty} | b_ {j} | <\ infty}$ ${\ displaystyle \ sum _ {j = 1} ^ {\ infty} | b_ {j} | <\ infty}$
no members with $j<0$ ${\ displaystyle j <0}$ ${\ displaystyle j <0}$
constant (independent of $t$ ${\ displaystyle t}$ $t$ )

Notes

↑ Diebold FX Elements of Forecasting. - 4. - South-Western College Pub, 2007. - P. 124. - 384 p. - ISBN 032432359X .

Literature

Anderson, TW (1971) The Statistical Analysis of Time Series . Wiley.
Wold, H. (1954) A Study in the Analysis of Stationary Time Series , Second revised edition, with an Appendix on “Recent Developments in Time Series Analysis” by Peter Whittle. Almqvist and Wiksell Book Co., Uppsala.
Scargle, JD (1981) Studies in astronomical time series analysis. I - Modeling random processes in the time domain ,, '1981, Astrophysical Journal Supplement Series, 45, pp. 1-71.

[1] Diebold FX Elements of Forecasting. - 4. - South-Western College Pub, 2007. - P. 124. - 384 p. - ISBN 032432359X .

Wold's theorem

Notes

Literature

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