Limit Theorems for a Storage Process with a Random Release Rule

We consider a discrete time Storage Process n X with a simple random walk input and a random release rule given by a family  of random variables whose probability laws n S  , 0 x U x    , 0  x x  form a convolution semigroup of measures, that is, . x y x y       The process n X obeys the equation: Under mild assumptions, we prove that the processes and 0 0 0, 0, X U X   , n n n S S U n   1.  n S U n X are simple random walks and derive a SLLN and a CLT for each of them.



form a convolution semigroup of measures, that is, .
The process n X obeys the equation: Under mild assumptions, we prove that the processes and

Introduction and Assumptions
The formal structure of a general storage process displays two main parts: the input process and the release rule.The input process, mostly a compound Poisson process

 
A t , describes the material entering in the system during the interval   . The release rule is usually given by a function representing the rate at which material flows out of the system when its content is x .So the state   X t of the system at time obeys the well known equation: Limit theorems and approximation results have been obtained for the process   X t by several authors, see [1][2][3][4][5] and the references therein.In this paper we study a discrete time new storage process with a simple random walk input n and a random release rule given by a family of random variables  where S  , 0 , x U x  x U has to be interpreted as the amount of material removed when the state of the system is .
x Hence the evolution of the system obeys the following equation: 0 X 0,  where 0 , for i.i.d.positive random variables 0 0, We will make the following assumptions: x y We will assume that for each x , x  is supported by Consequently, for x y  the distribution of y x U U  is the same as that of y x U  , (see 2.2   ii ).1.2.Also we will need some smoothness properties for the stochastic process These will be achieved if we impose the following continuity condition: where 0  is the unit mass at 0 and the limit is in the sense of the weak convergence of measures.

B
of the positive real line R and form the infinite Now, as usual define random variables n Z on by: Then the n Z are independent identically distributed with common distribution  We will assume that

 
iii .For every x the increments are independent.
is right continuous with left hand limit (cadlag). From is measurable on the product sp R ace 2 2.3.The basic probability space for the storage process n X will be the product Then we define is the simple random walk with: Since is a simple random walk, the random variables and have the same distribution for .

The Main Results
The main objective is to establish limit theorems for the processes n S and n X .Since the behavior of n is well understood, we will focus attention on the structure of the process n S U .The outstanding fact is that n S U itself is a simple random walk.First we need some preparation.S 3.1.Proposition: For every measurable bounded function , the function : Proof: Assume first continuous and bounded, then from (1.2) we have Now by (1.1) we have x y x y x y x by (1.2) and the bounded convergence theorem.Consequently the function right continuous for all hence it is measurable if 0, x  f is continuous and bounded.Next consider the class of functions: , such that the function d , measurable.

H
then H is a vector space satisfying the conditions of Theorem I,T20 in [6].
, ,P  F with probability distribution .


Then the function defined on   by: for every measurable positive function .In particular the probability distribution of is given by: : and its expectation is equal to (3.4) is a simple change of variable formula since .
1 2 , the random variables have the same probability distribution. , It is enough to show that for every positive measurable function , we have: Copyright © 2012 SciRes.AM Since 1 2 , E to both sides of this formula we get the first equality of (3.7).To get the second one, observe that the function We prove that for all integers and all positive measurable functions we have: Therefore, applying first 2 P E in the L.H.S of (3.8), we get the formula: .
1 2 1 2 P P P P To achieve the proof, write as follows: , where the are independent with the same distribution given by according to (3.5).■ 3.9.Proposition: For every positive measurable function , we have: :   being the n-fold convolution of the probability . In particular the distribution law of the process is given by: and its expectation is: Copyright © 2012 SciRes.AM Proof: We have: and, by Proposition 3.1, the function . So formula (3.9) is proved.To get the Now we turn to the structure of the process n X .We need the following technical lemm n a: 3.11.Lemma: For every Borel positive functio : ble rectangles, we deduce that .
t onot e is ex-actly the class of all Borel positive functions.■ 3.12.Theorem: The random variables  are independent ith the same distribution given by: for B  R B w Consequently the storage process , is a simple rando .12).h integ d ea m walk with the basic distribution (3 Proof: For eac er 0 r  , an ch So it is enough to prove that for all and all Borel positive functions From the construction of the process that for we know fixed, the random variables , we get: is the same as that of


, we have for each Borel positive function : nctions of the random variables r Z , thus they are independent under the probability erefo 1 .Th P re, applying 1 P E to both sides of (3.14) we ge 3).■ t (3.1 As for the process n X , the coun art of proposition 3.9 is the following: .15.Proposition: B , then we have: For the proof, use the formula . This ibution of process in this case: gives the distr the release e e quantity Likewi s st ositi 14, th Since we have n c B P cS B , we deduc that the release rule consists in removing from n S th of the random variables Z U and 1 X are finite.The conditions can respectively written as be where the second equality comes from the Taylor expansion of . It is well known that this limit is the characte function of the random variable   0,1 .N onents of The sam works for using the comp the process e proof n n R , X as given i eorem 3.12.■ In so ge systems, the changes due and release do not take place regularly in time.So it would be more realistic to consider the time parameter n Th me stora to supply n as random.We will do so in what follows and will consider the asymptotic distributions of the processes U , and

1 . 1 .
The probability distributions  of the random variables  form a convolution semigroup of measures:

By Proposition 3 . 1 , 1 PE
the R.H.S of these equalities are random variables of 1  , independent under 1 since they are measurable functions of the independent random variables P , to both sides of formula (*) we get the proof of (3.8):

1 
the characteristic function of some Borel set B. ■ 3.10.Remark: Let  be the distribution of 1

4 . 2 .
One can give more examples in this way by choosing the distribution  or/and the semigoup Proposition: Under the conditions:


Now the conclusion results from the fact that n S U and n X are simple random walks and the Lindberg Central Limit Theorem.To see this, we use the method of characteristic functions.Let us denote by f  the characteristic function of the random variable  .Since by The m

X
, when properly normalized and random-ized.First let us put for each , It is a simple adaptation of[7], VIII.4,Theorem 4process more tractable when c to those models studied elsewhere (see Introduction).In particular the conditions led to a simple structure of raneasily established for each of them.Moreover, a s tly more general limit theore has been obtained when tim s adequately randomized and both processes n S U and n ligh X properly normalized.
Then the normalized sequences of random variables: