We consider a sequence
Let
In 1976 Dziubdziela-Kopocinski [
Taking account of these three limit laws we get those of the Type 1:
Type 2:
Type 3:
The authors have presented the expressions of the probability density function
In this work, we assume that
In this section we recall some relevant results for future use.
Let
Let
For form a Markov chain with equal probability of transition. That means
The assertions
Let
If
For
For
Let
For all
For all
Let
If
If
If
The proof is obvious.
If
Let
Throughout the following, we consider
Let
The number of coincidences, which means the number of times
Let us study the nature of the following series
As
An
Using Theorems
Choosing
Indeed
Hence we get
Since
Then
Let us consider and give some properties of the following sequences:
Consider the following
Consider the following
Consider
Consider the following
Now, let us consider the two following sequences:
Consider the following
Since the number of coincidences is finite, there exists
Between
By the same method, we have
Let
According to
The sequence of
Considering the sub-
But
Consequently
By the same method
In this section we will prove the asymptotic independence of
Let
Since there exists an
Let
Thus,
We conclude that for all for all
This sequential reasoning will be taken each time we have either a record for the
Thus, we prove that
Indeed, if we denote, for the sake of simplicity,
In this paper we proved the asymptotic independence of the
The author declares that there is no conflict of interests regarding the publication of this paper.