A standard random walk formed by partial sums of independent identically distributed random vectors is one of the most simple, classical and well-studied discrete-time random processes. Regarded as a Markov chain this process is time-homogeneous, meaning that its transition probabilities do not depend on time, and its increments are space-homogeneous meaning that their distributions do not depend on the current state of the process. There are many various generalizations of this basic model leading to interesting and far-reaching theories. A whole class of such extensions is provided by a notion of locally perturbed random walks. In a wide sense, a locally perturbed random walk is a Markov chain whose transition probabilities coincide with the transition probabilities of some standard random walk only outside a given subset of a state space. The first model of this flavor is due to Harrison and Shepp [5] who investigated simple random walk on $\mathbb{Z}$ with a perturbation at state 0. That is, if the process is at state $x\in \mathbb{Z}\setminus \{0\}$ it jumps to $x\pm 1$ with probabilities $1/2$, whereas from the state 0 it jumps to $\pm 1$ with probability $p\in [0,1]\setminus \{1/2\}$. Their main result says that the scaling limit of the locally perturbed random walk is a skew Brownian motion, rather than a standard Brownian motion occurring in the case of a simple symmetric random walk when $p=1/2$. Recent results and generalizations of this model can be found in [7, 9] and [10].

Another representative of the class of locally perturbed random walks is a planar random walk in a semi-infinite strip studied in [2]. This model deals with a two-dimensional random walk on ${\mathbb{Z}^{2}}$ which behaves as a standard random walk before hitting a horizontal line $\{(x,y)\in {\mathbb{Z}^{2}}:y=c\}$ and afterwards the walk ‘sticks’ to that line and behaves as a one-dimensional random walk on that line. Thus, the perturbation in that random walk is due to the presence of an impenetrable barrier that changes the behavior of the walk when it is reached. In [2] such a process appeared in the context of statistical analysis of the low inventory problem and, therefore, is of purely applied nature. The main focus in [2] was put on the analysis of that model when $c\to \infty $. Motivated by the aforementioned study we introduce and analyze in this note a new class of multidimensional locally perturbed random walks, which we call random walks with sticky barriers (RWSB) and which generalizes the model treated in [2].

Let $d\in \mathbb{N}$ be an integer which will remain fixed throughout the paper and denotes the dimension of an underlying space. For a sequence ${\mathbf{b}_{n}}:={({b_{1}^{(n)}},{b_{2}^{(n)}},\dots ,{b_{d}^{(n)}})_{n\in \mathbb{N}}}$ of ${\mathbb{N}^{d}}$-valued vectors,2 introduce a sequence of lattice sets where ${\mathbb{N}_{0}}:=\mathbb{N}\cup \{0\}$. The random walk with sticky barriers ${({\mathbf{S}_{k}}(n))_{k\in {\mathbb{N}_{0}}}}:={({\mathbf{S}_{k}})_{k\in {\mathbb{N}_{0}}}}$ is a discrete-time ${\mathbb{N}_{0}^{d}}$-valued random process whose evolution can be described by the following informal rules (a formal description will be given below):

Note that the notion of RWSB is genuinely multidimensional and becomes almost degenerate when $d=1$. Thus, the simplest nontrivial situation occurs in dimension two and from now on we always assume that $d\ge 2$. As has already been mentioned, in case $d=2$ the concept of RWSB (in a slightly simplified variant with ${b_{2}^{(n)}}=\infty $) was proposed in [2] for the statistical analysis of a shared inventory problem with two types of goods. This model, on the one hand, is rich enough to take into account peculiarities of a finite inventory, but at the same time is simple enough to allow for a closed form of many important statistical quantities related to the model, see [2] for the details. Let us also stress that the model studied here, as well as its particular case treated in [2], can easily be generalized to the nonlattice settings. That is, one can easily drop the assumption that the increments are ${\mathbb{N}_{0}^{d}}$-valued and define a RWSB with state space ${\mathbb{R}^{d}}$. Most of the results obtained in this paper hold also in this more general settings upon necessary amendments.

The rest of the paper is organized as follows. In Section 2 we give a formal specification of the model by defining the process ${({\mathbf{S}_{k}})_{k\in {\mathbb{N}_{0}}}}$ and various related quantities. The main results are given in Section 3 and divided into three parts: strong laws of large numbers, asymptotic expansions for expectations and functional limit theorems. The proofs are given in Section 4. In the Appendix we collected some basic facts from renewal theory which is the main toolbox for our analysis.

We shall also use the following notational conventions. For a vector $\mathbf{x}=({x_{1}},{x_{2}},\dots ,{x_{d}})\in {\mathbb{N}_{0}^{d}}$ we denote by $|\mathbf{x}|=|{x_{1}}|+|{x_{2}}|+\cdots +|{x_{d}}|$ its Manhattan norm. We shall also use the notation $\mathbf{x}\cdot \mathbf{y}$ for the coordinatewise product of vectors, that is, if $\mathbf{y}=({y_{1}},{y_{2}},\dots ,{y_{d}})\in {\mathbb{N}_{0}^{d}}$, then $\mathbf{x}\cdot \mathbf{y}=({x_{1}}{y_{1}},{x_{2}}{y_{2}},\dots ,{x_{d}}{y_{d}})$.

Let $n\in \mathbb{N}$ be a fixed integer. Denote by ${\mathbf{B}_{d}}:={\{0,1\}^{d}}$ the set of binary strings of length d. The vector $\mathbf{m}=({m_{1}},{m_{2}},\dots ,{m_{d}})\in {\mathbf{B}_{d}}$ represents the already reached barriers, that is, ${m_{i}}=0$, for some $i=1,\dots ,d$, if and only if a RWSB, to be constructed below, did not yet reach the barrier ${b_{i}^{(n)}}$. Put ${\mathbf{B}^{\prime }_{d}}:={\{0,1\}^{d}}\setminus \{1,1,\dots ,1\}$. Let $(\Omega ,\mathcal{F},\mathbb{P})$ be a fixed probability space, which we assume to be rich enough to accommodate an array ${\mathbf{X}_{k}^{(\mathbf{m})}}=({X_{k,1}^{(\mathbf{m})}},{X_{k,2}^{(\mathbf{m})}},\dots ,{X_{k,d}^{(\mathbf{m})}})$, $\mathbf{m}\in {\mathbf{B}^{\prime }_{d}}$, $k\in \mathbb{N}$, of mutually independent ${\mathbb{N}_{0}^{d}}$-valued random variables. For every fixed $\mathbf{m}\in {\mathbf{B}^{\prime }_{d}}$, the variables ${({\mathbf{X}_{k}^{(\mathbf{m})}})_{k\in \mathbb{N}}}$ are assumed to be identically distributed and represent the increments of the RWSB when it has hit barriers determined by m. A random walk with sticky barriers is a Markov chain ${\mathbf{S}_{k}}:=({S_{k,1}},{S_{k,2}},\dots ,{S_{k,d}})$, $k\in {\mathbb{N}_{0}}$, that depends on the parameter n, and which is defined formally by the recursive formula where the minimum is taken coordinatewise, the binary vector represents the barriers already reached before k-th step, and are the moments at which $({\mathbf{S}_{k}})$ hits a new barrier.3 Note that, for every fixed $n\in \mathbb{N}$, $({\mathbf{S}_{k}})$ is indeed a Markov chain with the discrete state space ${\mathbf{R}_{n}}$ and the inhomogeneous (in space) transition kernel and, for $\mathbf{m}\in {\mathbf{B}_{d}}$, ${\mathbf{X}^{(\mathbf{m})}}=({X_{1}^{(\mathbf{m})}},{X_{2}^{(\mathbf{m})}},\dots ,{X_{d}^{(\mathbf{m})}})$ is a generic copy of the i.i.d. variables ${\mathbf{X}_{k}^{(\mathbf{m})}}$, $k\in \mathbb{N}$. By definition, ${\tau _{d}}(n)$ is the absorption time when the RWSB hits ${\mathbf{b}_{n}}$ and stays there forever. Furthermore, without loss of generality we can and do assume in what follows that, for all $k\in \mathbb{N}$ and $\mathbf{m}\in {\mathbf{B}_{d}}$,

Despite a formidable form, formula (1) has a simple interpretation that agrees with the informal description given in the introduction. Namely, the evolution of ${({\mathbf{S}_{k}})_{k\in {\mathbb{N}_{0}}}}$ before the absorption time ${\tau _{d}}(n)$ is divided into disjoint parts. For every integer $k<{\tau _{d}}(n)$ there is a unique index $i=0,\dots ,d-1$, such that $k\in [{\tau _{i}}(n),{\tau _{i+1}}(n))$. Thus, using that ${\mathbf{I}_{k}}={\mathbf{I}_{{\tau _{i}}(n)}}$ and $|{\mathbf{I}_{k}}|=i$ if $k\in [{\tau _{i}}(n),{\tau _{i+1}}(n))$, and denoting $l:=k-{\tau _{i}}(n)$, we can write Moreover, by definition of ${\tau _{i+1}}(n)$ and in view of assumption (4), the minimum above can be dropped if $0\le l\le {\tau _{i+1}}(n)-{\tau _{i}}(n)-2$. Thus, which means that, for every $i=0,\dots ,d-1$, the process $({\mathbf{S}_{{\tau _{i}}(n)+l}}-{\mathbf{S}_{{\tau _{i}}(n)}})$ is the standard zero-delayed random walk on the time interval $0\le l<{\tau _{i+1}}(n)-{\tau _{i}}(n)-1$. We also emphasize that in our definition of $({\mathbf{S}_{k}})$ the numeration of increments starts afresh from 1 on each time interval $[{\tau _{i}}(n),{\tau _{i+1}}(n))$.

Another useful representation of $({\mathbf{S}_{k}})$ is Indeed, (1) yields, for $k\le {\tau _{d}}(n)$, which is the right-hand side of (6).

Throughout the paper we shall make the following assumptions on the distribution of ${\mathbf{X}^{(\mathbf{m})}}$ and use the following notation. By $\mathbb{E}$ we denote expectation with respect to the probability measure $\mathbb{P}$. We suppose that, for every $\mathbf{m}\in {\mathbf{B}^{\prime }_{d}}$, it holds and

We now state the assumptions on the sequence of barriers $({\mathbf{b}_{n}})$. We assume that there exist pairwise distinct positive numbers ${\rho _{1}},\dots ,{\rho _{d}}$ such that, for every fixed $\lambda >0$, that is, the sequence ${({b_{i}^{(n)}})_{n\in \mathbb{N}}}$ is regularly varying with index ${\rho _{i}}$, for every $i=1,\dots ,d$. See [1] for the comprehensive information on regularly varying functions and sequences. Note that without loss of generality we may assume that $0<{\rho _{1}}<{\rho _{2}}<\cdots <{\rho _{d}}$, since we can always permute the coordinates of all involved variables to achieve the desired order of indices ${\rho _{i}}$’s. Thus, from now on we suppose additionally that We shall also occasionally allow the sequence ${({\mathbf{b}_{n}})_{n\in \mathbb{N}}}$ to be random. In this case we shall assume that the barriers and the steps of the corresponding RWSB are independent and that the limit relation (9) holds almost surely for some deterministic ${\rho _{1}},\dots ,{\rho _{d}}$, which, therefore, can again be arranged in the increasing order.

Let $\{{\mathbf{e}_{1}},{\mathbf{e}_{2}},\dots ,{\mathbf{e}_{d}}\}$ denote the standard unit basis of ${\mathbb{R}^{d}}$ and put

Let us introduce some additional notation. For $\mathbf{m}\in {\mathbf{B}^{\prime }_{d}}$, let be a standard zero-delayed random walk. Further, for $\mathbf{A}\subset {\mathbb{N}_{0}^{d}}$, set For every fixed $\mathbf{m}\in {\mathbf{B}_{d}}$, let ${\pi _{\mathbf{m}}}:{\mathbb{R}^{d}}\mapsto {\mathbb{R}^{d-|\mathbf{m}|}}$ be the projection mapping which removes coordinates of $\mathbf{x}\in {\mathbb{R}^{d}}$ with indices $i=1,\dots ,d$ such that ${m_{i}}=1$. For example, and for $j=0,\dots ,d$.

The following simple observation lies in the core of all subsequent proofs.

We shall now briefly describe our proof strategy. In accordance with (28), for every $n\in \mathbb{N}$ and $j=1,\dots ,d$, and, further, The most important properties of the RWSB ${({\mathbf{S}_{k}^{\to j}})_{k\in {\mathbb{N}_{0}}}}$ are: In conjunction with the fact that the event $\{{\mathbf{I}_{{\tau _{j-1}}(n)}}=\mathbf{y}\}$ for a particular deterministic $\mathbf{y}={\mathbf{s}_{j-1}}$ has probability tending to one, see (32) and (33) below, equations (30) and (31) demonstrate how to reduce the analysis of ${\tau _{j}}(n)-{\tau _{j-1}}(n)$ and ${\mathbf{S}_{{\tau _{j}}(n)}}-{\mathbf{S}_{{\tau _{j-1}}(n)}}$, for arbitrary $j=1,\dots ,d$, to the analysis of the hitting time of the first barrier by the RWSB ${({\mathbf{S}_{k}^{\to j}})_{k\in {\mathbb{N}_{0}}}}$. In order to quantify the above statement, for every fixed $a>0$, $j=0,\dots ,d-1$ and $n\in \mathbb{N}$, define events We shall see below that