lectr17 - 1 17 Long Term Trends and Hurst Phenomena From...

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Unformatted text preview: 1 17. Long Term Trends and Hurst Phenomena From ancient times the Nile river region has been known for its peculiar long-term behavior: long periods of dryness followed by long periods of yearly floods. It seems historical records that go back as far as 622 AD also seem to support this trend. There were long periods where the high levels tended to stay high and other periods where low levels remained low 1 . An interesting question for hydrologists in this context is how to devise methods to regularize the flow of a river through reservoir so that the outflow is uniform, there is no overflow at any time, and in particular the capacity of the reservoir is ideally as full at time as at t. Let denote the annual inflows, and 1 A reference in the Bible says “ seven years of great abundance are coming throughout the land of Egypt, but seven years of famine will follow them ” ( Genesis ). t t + } { i y n i n y y y s + + + = " 2 (17-1) PILLAI 2 their cumulative inflow up to time n so that represents the overall average over a period N . Note that may as well represent the internet traffic at some specific local area network and the average system load in some suitable time frame. To study the long term behavior in such systems, define the “extermal” parameters as well as the sample variance In this case } { i y N y }, { max 1 N n N n N y n s u − = ≤ ≤ . ) ( 1 2 1 N N n n N y y N D − = ∑ = }, { min 1 N n N n N y n s v − = ≤ ≤ (17-3) (17-4) (17-5) N N N v u R − = (17-6) N s y N y N N i i N = = ∑ = 1 1 (17-2) PILLAI 3 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 where are 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 D v u D R − = (17-7) N N D R / N i y i " , 2 , 1 , = µ , 2 σ ), , ( 2 σ µ n n N s d n → µ σ µ → → ) / , ( 2 N N y d N 2 σ → d N D (17-8) (17-9) (17-10) PILLAI 4 with probability 1. Further with where 0 < t < 1, we have where is the standard Brownian process with auto-correlation function 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 , Nt n = ) ( lim lim t B N Nt s N n s d Nt N n N → − = − ∞ → ∞ → σ µ σ µ (17-11) ) ( t B ). , ( 2 1 t t ) ( ) ( ) ( µ µ µ µ N s N n n s y n n s y n s N n N n N n − − − = − − − = − (17-12) ( ) (1), t 1....
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lectr17 - 1 17 Long Term Trends and Hurst Phenomena From...

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