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On the behaviour of a long cascade of linear reservoirs

Published online by Cambridge University Press:  14 July 2016

John E. Glynn*
Affiliation:
Geological Survey of Canada
Peter W. Glynn*
Affiliation:
Stanford University
*
Postal address: Earth Sciences Sector, 601 Booth St., Ottawa, Ontario, K1A 0E8, Canada
∗∗Postal address: Department of Engineering-Economic Systems and Operations Research, Stanford University, Stanford, CA 94305-4023, USA

Abstract

This paper describes the limiting asymptotic behaviour of a long cascade of linear reservoirs fed by stationary inflows into the first reservoir. We show that the storage in the nth reservoir becomes asymptotically deterministic as n → ∞, and establish a central limit theorem for the random fluctuations about the deterministic approximation. In addition, we prove a large deviations theorem that provides precise logarithmic asymptotics for the tail probabilities associated with the storage in the nth reservoir when n is large.

Type
Research Papers
Copyright
Copyright © by the Applied Probability Trust 2000 

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Footnotes

This research was supported by the US Army Research Office under contract no. DAAG55-97-1-0377 and by the National Science Foundation under grant no. DMS-9704732.

References

Bhattacharaya, R. N., and Rao, R. R. (1976). Normal Approximation and Asymptotic Expansions. John Wiley, New York.Google Scholar
Brémaud, P. (1981). Point Processes and Queues: Martingale Dynamics. Springer, New York.Google Scholar
Bucklew, J. A. (1990). Large Deviations Techniques in Decision, Simulation, and Estimation. John Wiley, New York.Google Scholar
Bucklew, J. A., Ney, P., and Sadowsky, J. S. (1990). Monte Carlo simulation and large deviations theory for uniformly recurrent Markov chains. J. Appl. Prob. 27, 4459.Google Scholar
Glynn, J. E., and Glynn, P. W. (1996). A diffusion approximation for a network of reservoirs with power law release rule. Stoch. Hydrol. Hydraul. 10, 1737.Google Scholar
Glynn, J. E., and Glynn, P. W. (1999). On the behaviour of a cascade of nonlinear reservoirs. In preparation.Google Scholar
Klemeš, V., and Boruvka, L. (1975). Output from a cascade of discrete linear reservoirs with stochastic input. J. Hydrology 27, 113.CrossRefGoogle Scholar
Klemeš, V., Klemeš, I., and Glynn, J. E. (1985). Discrete time linear cascade under time averaging. J. Hydrology 77, 107123.CrossRefGoogle Scholar
Nash, J. E. (1957). The form of the instantaneous unit hydrograph. I.A.S.H. Pub. 45, 14121.Google Scholar
Philipp, W., and Stout, W. (1975). Almost sure invariance principles for partial sums of weakly dependent random variables. Mem. Amer. Math. Soc. 161, 1140.Google Scholar
Vere-Jones, D. (1968). Some applications of probability generating functionals to the study of input/output streams. J. Roy. Statist. Soc. 30, 321333.Google Scholar