1. Introduction
In this study, we consider a discrete random walk with three dimensions
, defined as follows: our walk starts from the origin at time (0, 0, 0). At each time step, the walk moves by one positive unit in the direction of the x-axis (1, 0, 0) with probability
; in the direction of the y-axis (0, 1, 0) with probability
; or in the direction of the z-axis (0, 0, 1) with probability
. Or, it resets to (0, 0, 0) with probability
, or shifts one unit in all three directions (1, 1, 1) with probability
such that
. Here are some examples illustrating the evolution of our model:
The final altitude of the random walks with a length 7 in the above examples are 0, 2, 6 and 2, respectively.
In this current paper, our goals are twofold: we will analyze the statistical properties of the final altitude denoted by by utilizing the moment-generating function (MGF) to derive its mean and variance. We will also examine the limit distribution, as well as the mean and variance, of each random walk , .
Numerous studies have explored the statistical properties of discrete random walks in one and higher dimensions using tools such as moment-generating functions, kernel methods, and singularity analysis (see [
1,
2,
3]). For example, in one dimension, if one concentrates on articles which play a role in our study, Abdelkader and Aguech determined the cumulative function of the height, the mean and the variance of the final altitude in the Moran random walk with reset and short memory in [
4]. Furthermore, Banderier and Nicodème studied the height for bounded discrete walks and proved that the height of bridges is converged to a Rayleigh limit law asymptotically in [
5]. Additionally, Aguech, Althagafi, and Banderier analyzed the properties of the final altitude for the Moran random walk with one dimension in [
6]. Finally, Banderier and Wallner investigated some statistics like the number of catastrophes for lattice paths with catastrophes in [
7].
In higher dimensions, Althagafi and Abdelkader demonstrated that the age of each component follows a shifted geometric distribution asymptotically, and they derived the mean and the variance of the final altitude using the moment-generating function in the two-dimensional Moran model [
8]. Additionally, Itoh, Mahmoud, and Takahashi showed that the stationary distribution is a convolution of geometric random variables and derived important results regarding normal distribution in soliton physics for wave propagation [
9]. Aguech, Althagafi, and Banderier investigated various Moran models and proved that the heights of these walks asymptotically follow a discrete Gumbel distribution [
6]. Other relevant articles discuss the Moran process [
10,
11].
The study of three-dimensional random walks is significant and new, as it investigates random walks with non-uniform initial probabilities
, partially generalizing the results of Itoh and Mahmoud in [
12] (which focused on random walks with uniform probabilities in any dimension). Furthermore, it extends the work of Althagafi and Abdelkader [
8], in which the authors studied the final altitude for the two-dimensional random walk. Additionally, the results concerning the mean and the variance of the final altitude for the three-dimensional random walk generalize Theorem 2 in [
6]. This study also provides insights into the general case of systems composed of
n components with non-uniform initial probabilities
.
The analysis of the final altitude of random walks has broad applications across various fields, as random walks are widely used to model complex problems. For instance, Moran walks model mutation transmission in genetics [
12,
13,
14], and they have applications in graph theory [
15]. Additionally, random walks contribute to physics, such as the soliton wave model [
9], which describes a stochastic system of particles representing a unidirectional wave. The final altitude is also relevant for estimating maximum electricity consumption and reservoir water levels.
The structure of this current paper is organized as follows. In
Section 2, we describe our Moran random walk with three dimensions. In
Section 3, we state our main result concerning the statistical properties such as the mean and the variance of the final altitude using the moment-generating function. In
Section 4, we simulate the final altitude and the height statistics of our random walk using the R-program. In
Section 5, we explore recursive equations relating the sequences of multivariate polynomials
and
at two consecutive times
and
n using the recursive equations between
and
. In
Section 6, we show that the three random walks
,
, and
converge to geometric distribution asymptotically. Also, we establish the explicit expressions of their means and variances via the moment-generating function tool. In
Section 7, we prove our main results. Finally, in
Section 8, we present the conclusions of this work and we put forth some ideas for the next work.
5. Recursive Equations
Our goal in this section is to find the recursive equation between
and
at times
and
n. For this, we find a closed form of the probability
at the time
n associated to the model (
2). Define the probability
by the following:
The next lemma introduce a recursive equation between and at two times and n when and .
Lemma 1. We have the following recursion of probabilities: for all and , Proof. We distinguish five cases.
Case 1: For
, denote by
and
the following events:
using the conditional probability, we have
since
.
Case 2: For
and
, denote by
and
the following events:
using the conditional probability, we have
The proofs of Case 3 and Case 4 is similar to the proof of Case 2.
Case 5: For
, denote by
and
the following events:
and using the conditional probability, we have
we finish the proof.
□
Now, the sequence of multivariate polynomials
(for
) is defined by:
Taking
and
and
in Equation (
21), we obtain the probability-generating functions
,
and
for the random walks
,
and
, respectively:
Remark 2. We have the following identities: for all Lemma 1 leads to find a recursive equation related between , , , and . It is used to find the distribution of , and .
Proposition 1. For all , the sequence of multivariate polynomials satisfies the recursive equation:where . Proof. Applying Equation (21),
and using Lemma 1 and Equations (23)–(25),
where, replacing
,
,
and
in Equation (
27), we obtain the following: for all
and the proof is finished. □
6. Distribution of ,
Our goal here is to establish the asymptotic law, the mean, and the variance of the walks , , and at time n days using the moment-generating function , , and .
The next theorem present the asymptotic limit of the walks , using the moment generating functions.
Theorem 3. The probability-generating functions of are given by The walk converges, in law, to some shifted geometric (Geo) random variables, asymptotically:where , . Proof. Putting
in Proposition 1, we obtain
Similarly
Passing to the limit of
, then we have the following: for
The proof is finished. □
From Theorem 3, we deduce the expressions of the mean and the variance of , .
Corollary 1. and are given by:where , defined in Theorem 3. Proof. We derive
defined in Theorem 3 with respect to
one time for
putting
in Equation (
29) and applying Equation (
1), then we obtain
We derive Equation (
29) with respect to
another time, for
and putting
, then we have
Combining Equations (
1), (
30) and (
31), we obtain
after calculation and simplification, we obtain
□
8. Conclusions and Perspectives
In this work, we study the joint law of the final altitude of the three components , and . Furthermore, we use the probability-generating function to establish that the limit distribution of the three walks converges to a shifted-geometric distribution with parameter . Also, we find the explicit expressions of the mean and the variance of each random walk . Finally, we find the moment-generating function of the maximum random walk, denoted by , , and giving the closed forms of their mean and variance.
In the next work, we will generalize the results in [
12] (i.e., (
). That means we will analyze the Moran random model for any finite dimension with non-uniform initial probabilities (
). We will study the mean and the variance of
,
, …,
using the moment-generating function tool. For this, we will refer to the paper [
8].