Limit Theorems for a Storage Process with a Random Release Rule

Lakhdar Meziani

doi:10.4236/am.2012.311222

Home > Journals > Physics & Mathematics > AM

AM> Vol.3 No.11, November 2012

Share This Article:

Open Access

Limit Theorems for a Storage Process with a Random Release Rule

Abstract Full-Text HTML XML

Download as PDF (Size:254KB) PP. 1607-1613

DOI: 10.4236/am.2012.311222 2,694 Downloads 4,203 Views

Author(s) Leave a comment

Lakhdar Meziani

Affiliation(s)

Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia.

ABSTRACT

We consider a discrete time Storage Process X_n with a simple random walk input S_n and a random release rule given by a family {U_x, x ≥ 0} of random variables whose probability laws {U_x, x ≥ 0} form a convolution semigroup of measures, that is, μ_x × μ_y = μ_{x + y} The process X_n obeys the equation: X₀ = 0, U₀ = 0, X_n = S_n － U_{S_n}, n ≥ 1. Under mild assumptions, we prove that the processes and are simple random walks and derive a SLLN and a CLT for each of them.

KEYWORDS

Storage Process; Random Walk; Strong Law of Large Numbers; Central Limit Theorem

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

L. Meziani, "Limit Theorems for a Storage Process with a Random Release Rule," Applied Mathematics, Vol. 3 No. 11, 2012, pp. 1607-1613. doi: 10.4236/am.2012.311222.

References

[1]	E. Cinlar and M. Pinsky, “On Dams with Additive Inputs and a General Release Rule,” Journal of Applied Probability, Vol. 9, No. 2, 1972, pp. 422-429. doi:10.2307/3212811

[2]	E. Cinlar and M. Pinsky, “A Stochastic Integral in Storage Theory,” Probability Theory and Related Fields, Vol. 17, No. 3, 1971, pp. 227-240. doi:10.1007/BF00536759

[3]	J. M. Harrison and S. I. Resnick, “The Stationary Distribution and First Exit Probabilities of a Storage Process with General Release Rule,” Mathematics of Operations Research, Vol. 1, No. 4, 1976, pp. 347-358. doi:10.1287/moor.1.4.347

[4]	J. M. Harrison and S. I. Resnick, “The Recurrence Classification of Risk and Storage Processes,” Mathematics of Operations Research, Vol. 3, No. 1, 1978, pp. 57-66. doi:10.1287/moor.3.1.57

[5]	K. Yamada, “Diffusion Approximations for Storage Processes with General Release Rules,” Mathematics of Operations Research, Vol. 9, No. 3, 1984, pp. 459-470. doi:10.1287/moor.9.3.459

[6]	P. A. Meyer, “Probability and Potential,” Hermann, Paris, 1975.

[7]	W. Feller, “An Introduction to Probability Theory and Its Applications,” 2nd Edition, Wiley, Hoboken, 1970.