Limit Theorems for a Storage Process with a Random Release Rule

Lakhdar Meziani

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

DOI: 10.4236/am.2012.311222                

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_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.

Keywords

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

Conflicts of Interest

The authors declare no conflicts of interest.

References

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