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10
AlicjaGanczarek-Gamrot
where:
ρ
(
k
)
=
cov(
s
2
y
(
t
,
y
y
t
)
t
−
k
)
−(ACF(k))isaautocorrelationfunction,C
ρ
>0(constant),
α
∈(0,1).
Theautocorrelationfunctionoflongmemoryprocessdecaysslowlyat
ahyperbolicrate.Sotheautocorrelationsarenotsummable:
∑
∞
ρ
(
k)
=
∞
.
k
=
−∞
Forastationaryprocessautocorrelationfunctioncontainsequivalentinfor-
mationtoitsspectraldensity:
f
(
ω
)
=
2
1
π
k
∑
=
∞
−∞
ρ
(
k
)
e
ik
ω
,
where
ω
istheFourierfrequency
2(c.f.Hamilton,1994).
(2)
Thespectraldensityoflongmemoryprocesstendstoinfinityatzerofre-
quency:
f
(
ω
)
ω
→
→
0
C
f
ω
α
−
1
,
whereCf>0(constant),
α
∈(0,1).
(3)
Twoconvergences(1)and(3)areequivalent.Inpracticeveryoftenthe
Hurstcoefficient(H)isused
3insteadof
α
.FortimeserieswithlongmemoryH
isarealnumberfromthe(0.5;1)interval.ThelargerHthelongermemoryofthe
stationaryprocess.CoefficientsHand
α
satisfytherelation:
H
=
1
−
α
2
.
(4)
2J.D.Hamilton:TimeSeriesAnalysis.PrincetonUniversityPress,NewJersey1994.
3H.E.Hurst:LongTermStorageCapacityofReservoirs."TransactionsoftheAmericanSo-
cietyofCivilEngineers”1951,No.116,p.770-799.
