Limit Theorems for a Storage Process with a Random Release Rule

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.

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

1. 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, 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. So the state of the system at time obeys the well known equation:

.

Limit theorems and approximation results have been obtained for the process by several authors, see [1-5] and the references therein. In this paper we study a discrete time new storage process with a simple random walk input and a random release rule given by a family of random variables where has to be interpreted as the amount of material removed when the state of the system is Hence the evolution of the system obeys the following equation: where, for i.i.d. positive random variables with and

We will make the following assumptions:

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

We will assume that for each, is supported by the interval that is, Consequently, for the distribution of is the same as that of, (see 2.2).

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 is the unit mass at 0 and the limit is in the sense of the weak convergence of measures.

1.3. The two families of random variables and are independent.

2. Construction of the Processes and

2.1. Let be a probability measure on the Borel sets of the positive real line and form the infinite product space Now, as usual define random variables on by:

, if

Then the are independent identically distributed with common distribution We will assume that and

2.2. Let be a semigroup of convolution of probability measures on with and satisfying (1.2) then, it is well known, that there is a probability space and a family of positive random variables defined on this space such that the following properties hold:

. Under the distribution of is,

. For, the random variables and have under the same distribution

. For every the increments are independent.

. For almost all the function

is right continuous with left hand limit (cadlag).

From we deduce:

. The function is measurable on the product space

2.3. The basic probability space for the storage process will be the product Then we define by the following recipe:

if where is the simple random walk with:

2.4. Since is a simple random walk, the random variables and have the same distribution for.

3. The Main Results

The main objective is to establish limit theorems for the processes and. Since the behavior of is well understood, we will focus attention on the structure of the process. The outstanding fact is that itself is a simple random walk. First we need some preparation.

3.1. Proposition: For every measurable bounded function, the function

is measurable. Thus for any Borel set of the function is measurable.

Proof: Assume first continuous and bounded, then from (1.2) we have

Now by (1.1) we have

by (1.2) and the bounded convergence theorem. Consequently the function is right continuous for all hence it is measurable if is continuous and bounded. Next consider the class of functions:

then is a vector space satisfying the conditions of Theorem I,T20 in [6]. Moreover, by what just proved, contains the continuous bounded functions therefore contains every measurable bounded function ■

3.2. Remark: Let, be the expectation operators with respect to respectively. Since we have, by Fubini theorem. ■

3.3. Proposition: Let be a positive random variable on with probability distribution Then the function defined on by:

is a random variable such that

for every measurable positive function. In particular the probability distribution of is given by:

and its expectation is equal to

Proof: Define by

and by

It is clear that is measurable. Also is measurable by 2.2 so is measurable.

(3.4) is a simple change of variable formula since ■

3.7. Proposition: For all, the random variables have the same probability distribution.

Proof: It is enough to show that for every positive measurable function, we have:

Since we can write:

But for each fixed we get from 2.2

Applying to both sides of this formula we get the first equality of (3.7). To get the second one, observe that the function is measurable (Proposition 3.1) and use the fact that under, the random variables and have the same probability distribution by 2.4. ■

3.8. Theorem: The process is a simple random walk with:

and

Proof: We prove that for all integers and all positive measurable functions we have:

Let be fixed in. By 2.2 under the random variables

are independent. Therefore, applying first in the L.H.S of (3.8), we get the formula:

But have distributions , , respectively. Thus:

By Proposition 3.1, the R.H.S of these equalities are random variables of, independent under since they are measurable functions of the independent random variables Therefore, applying to both sides of formula (*) we get the proof of (3.8):

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:

Proof: We have:

and, by Proposition 3.1, the function

is a measurable function of

. Since is a simple random walk with the having distribution the random variable has the distribution. So, by a simple change of variable we get:

. So formula (3.9) is proved. To get the distribution law of the process, take equal to the characteristic function of some Borel set B. ■

3.10. Remark: Let be the distribution ofthat is and let

, then as a direct consequence of theorem 3.8,

    ■

Now we turn to the structure of the process. We need the following technical lemma:

3.11. Lemma: For every Borel positive function

, the function

is measurable.

Proof: Start with, the characteristic function of the measurable rectangle, in which case we have Since by proposition 3.1, the function is measurable we deduce that is measurable in this case. Next consider the family

It is easy to check that is a monotone class closed under finite disjoint unions. Since it contains the measurable rectangles, we deduce that Finally consider the following class of Borel positive functions

It is clear that is closed under addition and, by the step above, it contains the simple Borel positive functions. By the monotone convergence theorem, is exactly the class of all Borel positive functions. ■

3.12. Theorem: The random variables are independent with the same distribution given by: for

Consequently the storage process

, is a simple random walk with the basic distribution (3.12).

Proof: For each integer, and each put:

So it is enough to prove that for all and all Borel positive functions, we have:

From the construction of the process we know that for fixed, the random variables are independent under (see 2.2 (iii)). So, applying to, we get:

Now, since under, the distribution of

is the same as that of

, we have for each Borel positive function

From lemma 3.11, the functions

are Borel functions of the random variables, thus they are independent under the probability Therefore, applying to both sides of (3.14) we get (3.13). ■

As for the process, the counterpart of proposition 3.9 is the following:

3.15. Proposition: If is positive measurable and if, then we have:

For the proof, use the formula and routine integration.

3.16. Example: Let and let us take as measure the unit mass at the point, that is, the Dirac measure . It easy to check that for all in Then for every probability measure on

we have:. This gives the distribution of the release process in this case:

Since we have, we deduce that the release rule consists in removing from the quantity

Likewise it is straightforward, from Proposition 3.14, that

from which we deduce that the distribution of the storage process is

One can give more examples in this way by choosing the distribution or/and the semigoup. Consider the following simple example:

3.17. Example: Take the 0 - 1 Bernoulli distribution with probability of success In this case the semigroup is a sequence of probabilities with supported by for and is the Binomial distribution. So we get from proposition 3.9

Likewise we get the distribution of from proposition 3.15 as :

. ■

4. Limit Theorems

Due to the simple structure of the processes and (Theorems 3.8, 3.12), it is not difficult to establish a SLLN and a CLT for them.

4.1. Theorem: For the storage process and the release rule process, we have:

and

Proof: Since and are simple random walks with and we have:

and, by the classical S.L.L.N.

So we deduce:

and

4.2. Proposition: Under the conditions:

and

, the variances and

of the random variables and are finite. The conditions can respectively be written as

and

.

Proof: We have

, so the first condition gives. On the other hand we have

and

Since the variance of is finite we have

, so the conclusion follows. ■

Finally we get under the conditions of proposition 4.2:

4.3. Theorem: Assume the conditions of proposition 4.2. Then the normalized sequences of random variables:

and

both converge in distribution to the Normal law

Proof: The condition of the theorem insures the finiteness of the variances and Now the conclusion results from the fact that and are simple random walks and the Lindberg Central Limit Theorem. To see this, we use the method of characteristic functions. Let us denote by the characteristic function of the random variable. Since by Theorem 3.8 the components of have the same distribution as, we have

where the second equality comes from the Taylor expansion of. It is well known that this limit is the characteristic function of the random variable The same proof works for, using the components of the process as given in Theorem 3.12. ■

In some storage systems, the changes due to supply and release do not take place regularly in time. So it would be more realistic to consider the time parameter as random. We will do so in what follows and will consider the asymptotic distributions of the processes, and, when properly normalized and randomized. First let us put for each

, and

.

Then we have:

4.4. Theorem: Let be a sequence of integral valued random variables, independent of the and.

If converges in probability to 1, as, then the randomized processes:

and

both converge in distribution to the Normal law

Proof: It is a simple adaptation of [7], VIII.4, Theorem 4, p. 265. ■

5. Conclusion

In this paper, we presented a simple stochastic storage process with a random walk input and a natural release rule. Realistic conditions are prescribed which make this process more tractable when compared to those models studied elsewhere (see Introduction). In particular the conditions led to a simple structure of random walk for the processes and, which has given explicitly their distributions, and a rather good insight on their asymptotic behavior since a SLLN and a CLT has been easily established for each of them. Moreover, a slightly more general limit theorem has been obtained when time is adequately randomized and both processes and properly normalized.

6. Acknowledgements

I gratefully would like to thank the Referee for his appropriate comments which help to improve the paper.

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