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their cumulative inflow up to time n so that
represents the overall average over a period N . Note that mayas well represent the internet traffic at some specific local areanetwork and the average system load in some suitable time frame.
To study the long term behavior in such systems, define theextermal parameters
as well as the sample variance
In this case
}{ i y
N y
},{max1
N n N n
N yn su ! ee
.)(1 2
1 N
N
nn N y y N
D ! !
},{min1
N n N n N yn sv ! ee
(17-3)
(17-4)
(17-5)
N N N vu R ! (17-6)
N
s y
N y N
N
i i N
!! ! 1
1
(17-2)
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defines the adjusted range statistic over the period N , and the dimen-sionless quantity
that represents the readjusted range statistic has been used extensively by hydrologists to investigate a variety of natural phenomena.
To understand the long term behavior of whereare independent identically distributed random
variables with common mean and variance note that for large N by the strong law of large numbers
and
N
N N
N
N
Dvu
D! (17-7)
N N D R / N i yi .,2,1 , ! Q ,2W
),,(2
W Q nn N
sd
np
Q Q pp )/,( 2 N N y d N
2
W p d
N D
(17-8)
(17-9)
(17-10)
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with probability 1. Further with where 0 < t < 1, we have
where is the standard Brownian process with auto-correlationfunction given by min To make further progress note that
so that
Hence by the functional central limit theorem, using (17-3) and
(17-4) we get
, N t n !
- - )(limlim t B N
Nt s
N
n s d Nt N
n
N p !
gpgp W W (17-11))(t B
).,( 21 t t
)()(
)(
Q Q
Q Q
N s N
nn s
ynn s yn s
N n
N n
N n
!
!
(17-12)
( ) (1), 0 t 1.d n N n N s ny s n s N n
t t N N N N
Q Q! p (17-13)
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where Q is a strictly positive random variable with finite variance.Together with (17-10) this gives
a result due to Feller. Thus in the case of i.i.d. random variables therescaled range statistic is of the order of Itfollows that the plot of versus log N should be linear with slope H = 0.5 for independent and identically distributed
observations.
0 10 1max { ( ) (1) } min{ ( ) (1) } ,d N N
t t
u vt t t t Q
N W p |
,Q N vu
Dd N N
N
N p pW
N N D/ ).(2/1 N O
)/log( N N D
(17-14)
(17-15)
)/log( N N
D
N log
S lope=0.5
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The hydrologist Harold Erwin Hurst (1951) generatedtremendous interest when he published results based on water level
data that he analyzed for regions of the Nile river which showed thatPlots of versus log N are linear with slopeAccording to Fellers analysis this must be an anomaly if the flows arei.i.d. with finite second moment.
The basic problem raised by Hurst was to identify circumstancesunder which one may obtain an exponent for N in (17-15).The first positive result in this context was obtained by Mandelbrotand Van Ness (1968) who obtained under a stronglydependent stationary Gaussian model. The Hurst effect appears for
independent and non-stationary flows with finite second moment also.In particular, when an appropriate slow-trend is superimposed ona sequence of i.i.d. random variables the Hurst phenomenon reappears.To see this, we define the Hurst exponent fora data set to be H if
)/log( N N D R .75.0} H
2/1" H
2/1" H
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where Q is a nonzero real valued random variable.IID with slow Trend
Let be a sequence of i.i.d. random variables with common meanand variance and be an arbitrary real valued function
on the set of positive integers setting a deterministic trend, so that
represents the actual observations. Then the partial sum in (17-1)
becomes
where represents the running mean of the slow trend.From (17-5) and (17-17), we obtain
, , gpp N Q N D
d
H
N
N (17-16)
}{ n X Q ,2W n g
nnn g x y ! (17-17)
)(1
2121
nn
n
iinnn
g xn
g x x x y y y s
!
!! !
..
(17-18)
!! ni in g n g 1/1
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S ince are i.i.d. random variables, from (17-10) we getFurther suppose that the deterministic sequence
converges to a finite limit c. Then their Caesaro meansalso converges to c. S ince
applying the above argument to the sequence and
we get (17-20) converges to zero. S imilarly, since
).)((2
)(1
))((2)(1)(1
)(1
1
2
1
2
11 1
22
2
1
N n N n
N
n N
N
nn X
N n
N
n N n
N
n
N
n N n N n
N
N
nn N
g g x x N
g g N
g g x x N
g g N
x x N
y y N
D
!
!
!
!!
!! !
!
W (17-19)
}{ n x. 22 W W p d }{ n g
N N
n n N g g ! ! 11
,)()(1
)(1 2
1
22
1
c g c g N
g g N
N
N
nn N n
N
n
! !!
(17-20)
2)( c g n2)( c g N
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by S chwarz inequality, the first term becomes
But and the Caesaro meansHence the first term (17-21) tends to zero as and so does thesecond term there. Using these results in (17-19), we get
To make further progress, observe that
!!
! N
n N N nn N n N n
N
n
c g xc g x N
g g x x N 11
),)(())((1
))((1 Q Q
.)(1
)(1
))((1
1
22
1
2
1
!!!e
N
nn
N
nn
N
nnn c g N
x N
c g x N
Q Q
(17-21)
(17-22)
21
21 )( W p ! N n n N x .0)(1 21 p ! N
n n N c g ,gp N
. 2
W p pd
N n Dc g (17-23)
(17-24))}({max)}({max
)}()({max
}{max
00 N n N n N n N n
N n N n
N n N
g g n x xn
g g n x xn
g n su
e
!!
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and
Consequently, if we let
for the i.i.d. random variables, then from (17-6),(17-24) and (17-25)(17-26), we obtain
where
From (17-24) (17-25), we also obtain
)}.({min)}({min
)}()({min
}{min
00 N n N n N n N n
N n N n
N n N
g g n x xn
g g n x xn
g n sv
u
!
!
)}({min)}({max 00 N n N n N n N n N x xn x xnr !
(17-25)
(17-26)
N N N N N Gr vu e! (17-27)
)}({min)}({max00 N n N n N n N n N
g g n g g nG ! (17-28)
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From (17-27) and (17-31) we get the useful estimates
and
S ince are i.i.d. random variables, using (17-15) in (17-26) we get
a positive random variable, so that
)},({min)}({max00
N n N n N n N n N g g n x xnv e
)},({max)}({min00 N n N n N n N n N
g g n x xnu u
. N N N r Gu
(17-29)
(17-30)
(17-31)
, N N N r G e
. N N N Gr R e(17-32)
(17-33)
}{ n x
y probabilit inQ N
r N
r N X
N ,2
ppW W (17-34)
y. probabilit in Q N r N pW (17-35)
henceand)](max)(min})min{(max)}{(max [use iiiiiiiiiii y x y x y x !u
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Consequently for the sequence in (17-17) using (17-23)in (17-32)-(17-34) we get
if H > 1 /2. To summarize, if the slow trend converges to a finitelimit, then for the observed sequence for every H > 1 /2
in probability asIn particular it follows from (17-16) and (17-36)-(17-37) that
the Hurst exponent H > 1 /2 holds for a sequence if and onlyif the slow trend sequence satisfies
}{ n y
0/
2/1 ppp H H N H
N
N N N
Q N r
N DG
W W (17-36)
}{ n g },{ n y
0
pp H N H N
N H
N
N H
N
N
N G
N D N DG
N D W (17-37)
.gp N
}{ n y}{ n g
.2/1 ,0lim0
""!gp
H c N
G
H
N
N (17-38)
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In that case from (17-37), for that H > 1 /2 we obtain
where is a positive number.Thus if the slow trend satisfies (17-38) for some H > 1 /2,
then from (17-39)
Ex ample: Consider the observations
where are i.i.d. random variables. Here and thesequence converges to a for so that the above result applies. Let
, /0 gpp N as y probabilit inc N D
H N
N W (17-39)
0c}{ n g
. as ,loglog gpp N c N H D N
N (17-40)
1 , u! nbna x ynn
E
(17-41)n x ,
Ebna g n !,0E
.)(11
!! !! N
k
n
k N nn
k
N
nk b g g nM EE
(17-42)
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To obtain its max and min, notice that
if and negative otherwise. Thus max is achievedat
and the minimum of is attained at N =0. Hence from (17-28)and (17-42)-(17-43)
Now using the Reimann sum approximation, we may write
N ,)(/1
11 E
E !N k N k n
EE
/1
10
1
! ! N
k
k N n
(17-44)
01
1
1"
! !
N
k nn
k
N nb EE
(17-43)
0! N M
.1
0
1
0
!
!!
N
k
n
k N
k N
nk bG EE
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so that
and using (17-45)-(17-46) repeatedly in (17-44) we obtain
!
"
}
g ! 1 ,
1 ,log
1 ,)1(
/1
1
/1
0
E
E
EE
E
E
E
N
k
N
N
N
n
k
(17-46)
!
"
!}
g
!
!
1 ,
1 ,log
1 ,)1(
11
1
1
01
E
E
EE
E
E
EE
N
k
N N
N
dx x N
k N
k
N N
k (17-45)
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where are positive constants independent of N . From (17-47),notice that if then
where and hence (17-38) is satisfied. In that case
321 ,, ccc
,02/1 E
}
!}
"}
}
!
g
!
!!
1 ,1
1 ,loglog1
log1
1 ,)(1
11
31
0
20
0
0
110
0
1100
0
E
E
EE
E
EEE
EE
ck N n
b
N c N N
nn
bn
N c N nbn
k N
k n
bnG
k
N
k
n
k n
(17-47)
,~ 1 H
n N cG
12/1 H
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and the Hurst exponent Next consider In that case from the entries in
(17-47) we get and diving both sides of (17-33)with
so that
where the last step follows from (17-15) that is valid for i.i.d.
observations. Hence using a limiting argument the Hurst exponent
. 1)1( gpp N as y probabilit inc N D N N
E (17-48)
),( 2/1 N oG N !.2/1E
,2/1 N D N
y probabilit in N
N o N D
r R
N
N N 0
)(~ 2/1
2/1
2/1p
W
Q N r
N D N
N
N p2/12/1 W (17-49)
.2/11 "! E H
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H = 1 /2 if Notice that gives rise to i.i.d.observations, and the Hurst exponent in that case is 1 /2. Finally for
the slow trend sequence does not converge and(17-36)-(17-40) does not apply. However direct calculation shows
that in (17-19) is dominated by the second term which for large N can be approximated as so that
From (17-32)
where the last step follows from (17-34)-(17-35). Hence for from (17-47) and (17-50)
.2/1eE 0!E
}{ n g
N D
EE 20
21 N x N
N }
gpp N as N c D N 4 E (17-50)
014
pp} EW
N cQ N
N Dr
N DG
N
N
N
N N
0"E
,0"E
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as Hence the Hurst exponent is 1 if In summary,
4
11
4
11
cc
N c
N c N D
G
N D
R
N
N
N
N p}} E
E
.gp N .0"E
""!"
!
-1 /2 2/1
-1 /20 1
0 2/1
0 1
)(
EEEEE
E H
(17-51)
(17-52)
Fig.1 Hurst exponent for a process with superimposed slow trend
1/2
1
-1 /20
E
)(E H Hurst
phenomenon