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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