Limit Theorems for a Storage Process with a Random Release Rule

Affiliation(s)

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

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: *X*_{0} = 0, *U*_{0} = 0, *X*_{n} = *S*_{n} － *U*_{Sn}, *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.

We consider a discrete time Storage Process

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.

L. Meziani, "Limit Theorems for a Storage Process with a Random Release Rule,"

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.

[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.