Limit Theorems for a Storage Process with a Random Release Rule ()
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:
, (1.1)
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:
(1.2)
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:
(2.3)
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:
(3.3)
is a random variable such that
for every measurable positive function
. In particular the probability distribution of 
is given by:
(3.5)
and its expectation is equal to
(3.6)
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:
(3.4)
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:
(3.8)
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:
(3.9)
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 of
that 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 
(3.12)
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:
(3.13)
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:
(3.14)
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.