## Abstract

We introduce a continuous-time random walk process with correlated temporal structure. The dependence between consecutive waiting times is generated by weighted sums of independent random variables combined with a reflecting boundary condition. The weights are determined by the memory kernel, which belongs to the broad class of regularly varying functions. We derive the corresponding diffusion limit and prove its subdiffusive character. Analysing the set of corresponding coupled Langevin equations, we verify the speed of relaxation, Einstein relations, equilibrium distributions, ageing and ergodicity breaking.

## 1. Introduction

The standard mathematical description of diffusion processes is in terms of partial differential equations (Fokker–Planck, Kolmogorov equations) and, equivalently, in terms of Itô or Stratonovich stochastic differential equations. Studies of the diffusion processes have a glorious and exciting history. They began in 1827. In this year, Scottish botanist Robert Brown observed and investigated, for the first time, the random motion of pollen grains suspended in a liquid. It took almost 80 years to describe this chaotic (Brownian) motion of particles using mathematical equations. Einstein [1], in one of his four ‘annus mirabilis’ articles, derived an equation for Brownian motion from microscopic principles. Today, it is called the diffusion equation and has the well-known form ∂*p*/∂*t*=*K*(∂^{2}*p*/∂*x*^{2}). This equation was first described by Fick [2].

Einstein's explanation of diffusion was based on two fundamental assumptions: (i) the consecutive steps of the particle are independent with finite second moment and (ii) the mean time taken to perform a step is finite. However, in the 1970s researchers started to investigate situations in which the assumptions made by Einstein do not hold. Surprisingly, the way that photocopier machines work was the trigger for these developments [3]. At that time, the theory of ‘anomalous diffusion’ was born, but it has only been in the last decade that one has been able to observe its vivid development [4]. Today, an increasing number of experimentally observed processes can be described as anomalous. Starting from the signalling of biological cells to the foraging behaviour of animals, it seems that in many cases the overall motion of a particle is better described by steps and waiting times that are not independent and that can broaden power-law distributions [4].

The most popular models of anomalous dynamics are continuous-time random walks (CTRWs), which were introduced to physics by Montroll & Weiss [5]. A CTRW is a process in which the motion of a random walker is described by consecutive jumps and waiting times between them. Each pair of jumps and waiting times is drawn from some associated probability distributions. If both Einstein's assumptions hold, the CTRW converges in distribution to the Brownian motion (Wiener process). However, if we relax one of these assumptions, then we arrive at the class of fractional dynamics and anomalous diffusions [6–8].

In this paper, we analyse CTRWs for which the second of Einstein's assumptions does not hold. More precisely, we assume that waiting times have infinite mean. Moreover, we make an additional assumption that they are dependent. The dependence between consecutive waiting times is generated by weighted sums of independent random variables with diverging mean, combined with a reflecting boundary condition. The weights are determined by the memory kernel, which belongs to the broad class of regularly varying functions. Such correlated CTRWs are good candidates for modelling human mobility [9], financial market dynamics [10], or chaotic and turbulent flows [11]. The introduced model generalizes those studied in Chechkin *et al.* [12] and Tejedor & Metzler [13]. Recent progress in the field of CTRWs with correlated temporal and spatial structure can be found in Meerschaert *et al.* [14] and Magdziarz *et al.* [15,16]. The Langevin description of some classes of anomalous diffusions has been recently studied in Magdziarz [17], Magdziarz *et al.* [18] and Teuerle *et al.* [19]. We would like to emphasize that the word ‘correlated’ is used here in the broad sense and should be understood as ‘dependent’. As the waiting times considered in this paper are heavy tailed, the usual correlation coefficient between them is not defined.

In §2, we define the correlated model and derive its diffusion limit. We verify that it is subdiffusive and analyse its speed of relaxation. We also check the asymptotic behaviour of a correlated CTRW for the case of an exponential kernel. In §3, we introduce the Langevin description of the limit process and extend it to the non-zero external forces. Taking advantage of the Langevin picture we analyse the speed of relaxation, Einstein relations, equilibrium distributions, ageing and ergodicity breaking. The last section is the summary of the obtained results.

## 2. Limiting behaviour

### (a) The correlated model and its scaling limit

Let us begin with recalling the general definition of the CTRW process. We denote by *T*_{i}, *i*=1,2,…, the sequence of positive random variables, which represent the waiting times between consecutive jumps of the walker. Then, the process counts the number of jumps of the walker up to time *t*. Next, let *J*_{i}, *i*=1,2,…, be the sequence of random variables representing the consecutive jumps of the walker. Consequently, the CTRW process defined as
2.1describes the position of the walker at time *t*.

In this paper, we assume that the jumps *J*_{i} are independent, identically distributed (iid) *α*-stable random variables with the Fourier transform given in the well-known stretched exponential form [20]
2.2Moreover, we introduce the dependent sequence of waiting times *T*_{i} in the following manner:
2.3where *M* is the memory function and *ξ*_{j}, *j*=1,2,…, is the sequence of iid *γ*-stable random variables with the Fourier transform [20]
2.4Note that, for *β*=1, random variables *ξ*_{i} are positive. However, for other values of the skewness parameter *β*, *ξ*_{i} are supported also on the negative half-line. Therefore, the absolute value (reflecting boundary condition) must appear in (2.3).

We emphasize that taking the absolute value of sum (2.3) is one of a few different ways of reflecting the process at the origin. Another, common in the literature, way of reflecting the jump process is by subtracting its running infimum (see Bertoin [21], ch. 6).

Observe that, in definition (2.3), we use only values of function *M* at points that are positive integers. Hence, the sequence of waiting times *T*_{i} is in fact uniquely determined by the sequence of random variables *ξ*_{j} and the values of *M* at integer points. Thus, with no restrictions of generality, we will further assume for technical reasons that *M* is continuous and bounded on interval (0,1).

In order to investigate the limit behaviour of the CTRW (2.1), we need some assumptions on the scaling properties of function *M*, which are outlined in appendix C. In this paper, however, we will concentrate mainly on the wide class of regularly varying functions [22]. Recall that function *M*(*t*) mapping into is regularly varying, if for any
The number is the index of regular variation of *M*(*t*). A function *L*(*t*) satisfying for any is called slowly varying.

Regularly varying functions play a key role in calculus, differential equations, probability and number theory [22]; in addition, they have found widespread applications in other scientific fields, such as statistical physics, insurance mathematics or renewal theory (see [23] and references therein). The prominent examples of regularly varying functions are: power-law functions *M*(*t*)=*t*^{ρ}, logarithms or constants *M*(*t*)=const.

We emphasize that, by choosing *β*=1 in (2.3) and *M*(*t*)=*t*^{ρ} in (2.4), we obtain the correlated model introduced in Chechkin *et al.* [12]. On the other hand, choosing *β*=0 and *M*(*t*)=const., we arrive at the correlated CTRW derived in Tejedor & Metzler [13]. Thus, the CTRW (2.1) analysed in our paper generalizes these two models, by combining the underlying correlation mechanisms for waiting times *T*_{i} (weighted sum of *ξ*_{i}'s combined with a reflecting boundary condition). Moreover, as compared with Chechkin *et al.* [12], we extend the choice of the kernel *M*(*t*) to the wide class of regularly varying functions.

In the last part of this section, we separately analyse the case of the exponential kernel function, which does not belong to the family of regularly varying functions.

In the next theorem, which is the main result of this section, we derive the asymptotic diffusion limit of the correlated CTRW (2.1).

### Theorem 2.1

*Let J*_{i}*,* *be the sequence of iid α-stable random variables with Fourier transform (*2.2*). Let T*_{i}*,* *be the correlated sequence of waiting times defined by (*2.3*) with ξ*_{i} *given by (*2.4*) and M(t) regularly varying with index ρ>0. Assume that the jumps J*_{i}*,* *and the waiting times T*_{i}*,* *are independent. Then, the corresponding CTRW process R(t) (*2.1*) satisfies
*2.5*Here, the deterministic scaling sequence {B*_{n}*} is such that* *as* *. L*_{α}*(t) is the α-stable Lévy motion with Fourier transform* *. Moreover,
*2.6*where T*_{γ}*(y) is the γ-stable Levy motion with Fourier transform* *independent of L*_{α}*(t), whereas S*^{−1}*(t) is the inverse of S(τ), i.e.* *.*

### Proof.

See appendix E.

The scaling sequence *B*_{n} can be written in the form *B*_{n}=*L*(*n*)*n*^{1/(1+ρ+1/γ)}, where *L*(*n*) is some slowly varying function.

A more general result extending theorem 2.1 is formulated as theorem D.1 in appendix D.

The notation ‘→*d*’ in the above theorem means convergence in distribution. However, in the appendices, we prove even stronger results—functional convergence in the Skorohod topology (see Billingsley [24]). In particular, this convergence implies convergence of all finite-dimensional distributions.

Now, let us explain the structure of the limit process *X*(*t*). The *α*-stable Lévy motion *L*_{α}(*t*) appears here as the limit of rescaled cumulated jumps . Moreover, the cumulated waiting times converge (after proper rescaling) to the process *S*(*t*). It follows that the inverse process *S*^{−1}(*t*) is the scaling limit of the counting process *N*(*t*). Finally, the limit of the correlated CTRW has the form of subordination *X*(*t*)=*L*_{α}(*S*^{−1}(*t*)).

It appears that the limit process *X*(*t*) has nice scaling properties. Note first that for any *a*>0
Thus, *S*(*τ*) is self-similar with index 1+*ρ*+1/*γ*. It follows from the property that *S*^{−1}(*t*) is self-similar with index 1/(1+*ρ*+1/*γ*). Finally, applying the well-known fact that *L*_{α}(*t*) is 1/*α*-self-similar, we get that *X*(*t*)=*L*_{α}(*S*^{−1}(*t*)) is self-similar with index *H*=1/*α*(1+*ρ*+1/*γ*). Note that the parameter *H* can take any value from the infinite interval . Thus, depending on the choice of the parameters, the process *X*(*t*) can display scaling, which is characteristic for sub-, super- or normal diffusion [25]. However, the second moment is finite only for *α*=2. Then, the mean square displacement yields , where is the generalized diffusion constant. As the parameter 1/(1+*ρ*+1/*γ*) takes values in the interval (0,1/2], the mean square displacement of *X*(*t*) displays subdiffusive behaviour.

### (b) Exponential kernel

Now, we investigate in detail the diffusion limit of the correlated CTRW with exponential kernel *M*(*t*). This model was introduced in Chechkin *et al.* [12]. We show that, in the limit, the exponential kernel kills the dependence between waiting times. As a result, the scaling limit is the same as for the uncorrelated temporal structure.

### Theorem 2.2

*Let J*_{i}*,* *be the sequence of iid α-stable random variables with Fourier transform (*2.2*). Let T*_{i}*,* *be the correlated sequence of waiting times given by
*2.7*where ξ*_{i} *satisfy (*2.4*) with β=1. Assume that the jumps J*_{i}*,* *, and the waiting times T*_{i}*,* *are independent. Then, the corresponding CTRW process R(t) (*2.1*) satisfies
*2.8*Here, L*_{α}*(t) is the α-stable Lévy motion with Fourier transform* *. Moreover,* *is the γ-stable subordinator with Fourier transform
*2.9*which is independent of L*_{α}*(t), whereas* *is the inverse of* *i.e.* *.*

### Proof.

See appendix F.

The above result confirms that the limit process *Y* (*t*) of the correlated CTRW with an exponential kernel is equal (up to a constant) to the limit of the CTRW with uncorrelated waiting times *T*_{i} [4,8,26]. This shows that the exponential kernel kills the dependence between waiting times, and the correlation between successive rests of the walker can be observed only in the primary stage of the motion.

It should be noted that the probability density function of the scaling limit *Y* (*t*) satisfies the following fractional diffusion equation [4]:
Here, the operator stands for the fractional derivative of the Riemann–Liouville type, and ∇^{α} is the Riesz fractional derivative with the Fourier transform . *K*>0 is the generalized diffusion coefficient.

For *α*=2, the second moment of *Y* (*t*) is finite and given by , where . This result agrees with the asymptotic subdiffusive behaviour of the mean square displacement derived in Chechkin *et al.* [12] for large times.

## 3. Langevin framework

### (a) Coupled Langevin equations

The great advantage of the obtained diffusion limit (2.5) is that it can be used to describe the corresponding correlated dynamics in the framework of coupled Langevin equations. Applying the method of Fogedby [27], we get the following set of coupled Langevin equations for position *x* and time *t* of the considered process:
3.1Here, *Γ*_{α}(*s*)=d*L*_{α}(*s*)/d*s* and *Γ*_{γ}(*s*)=d*T*_{γ}(*s*)/d*s* are two independent noises. The first equation in (3.1) is the usual Langevin equation in the operational time *s*. The next two equations describe the relationship between the operational time *s* and the physical time *t*. Solving the above system of equations is rather straightforward. First, one solves the first equation to get the driving process *x*(*s*)=*L*_{α}(*s*) acting in the operational time *s*. In the next step, one solves the second and third equations to get *z*(*s*)=*T*_{γ}(*s*) and
3.2Finally, we obtain the solution of (3.1) in the subordination form *X*(*t*)=*x*(*s*(*t*))=*L*_{α}(*s*(*t*)), where *s*(*t*) is the inverse of *t*(*s*), i.e. . So, we arrive at exactly the same process as in (2.5).

To include external force *F*(*x*) in the description of correlated dynamics, one needs to add the drift term in the first equation in (3.1). Then, we obtain the following set of equations:
3.3which describes the correlated dynamics in the presence of external force *F*(*x*). Here, *m* is the particle mass and *η* denotes friction. The solution to the above set of equations is obtained in the same way as (3.1). We get that the solution is equal to *X*(*t*)=*x*(*s*(*t*)), where *s*(*t*) is the inverse of *t*(*s*) given in (3.2) and *x*(*s*) satisfies the following stochastic differential equation:

Based on (3.3), one can efficiently approximate numerically trajectories of *X*(*t*). It is enough to simulate the noises *Γ*_{α}(*s*) and *Γ*_{γ}(*s*), which can be done by the well-known method of simulating stable processes [20] and plugging the obtained results into (3.3).

### (b) Asymptotic properties

Now, let us apply previously obtained representations (3.1) and (3.3) to analyse the asymptotic properties of the correlated model. We start with the speed of single-mode relaxation. Assume that *F*≡0 and denote by the characteristic function (Fourier transform) of *X*(*t*). We have . Recall that *s*(*t*) is the inverse of *t*(*s*) given in (3.2). Denote
Then we have *a*(*s*)≤*t*(*s*)≤*b*(*s*) or, equivalently,
Here, *a*^{−1}(*t*) and *b*^{−1}(*t*) are the inverses of *a*(*s*) and *b*(*s*), respectively. Now, as both processes *a*(*s*) and *b*(*s*) are heavy-tailed with index *γ* and self-similar with index 1+*ρ*+1/*γ*, applying the above inequalities and Tauberian theorems, we get that *w*(*k*,*t*)≃1/*t*^{γ} for large *t*. The notation *w*(*k*,*t*)≃1/*t*^{γ} means that there exist positive constants *c*_{1} and *c*_{2}, such that *c*_{1}*t*^{−γ}<*w*(*k*,*t*)<*c*_{2}*t*^{−γ} for large *t*. Thus, the correlated model displays power-law single-mode relaxation.

In what follows, we will analyse properties which require the existence of moments of the process. Therefore, from now until the end of the paper, we assume that *α*=2. This means that the process *L*_{α=2}(*t*) is just the standard Brownian motion. We will denote it by *B*(*t*). Thus, its moments of any order exist. In particular, , where *D*>0 is the diffusion constant. Consequently, the second moment of the correlated model yields
3.4which is typical for subdiffusion. Here, depends on the distribution of the correlated waiting times.

It is natural to ask about the stationary solution of (3.3). As the subordinator *s*(*t*) tends to infinity as , we obtain that the stationary solution of (3.3) has the well-known form . Here, denotes the external potential. Comparing the above expression with the classical Gibbs–Boltzmann equilibrium distribution , we obtain that the correlated model satisfies the generalized Einstein relation
3.5which connects the diffusion parameter *D* with the dissipative parameter *η* and the energy *k*_{B}*T*.

Further on, if we assume that the force is constant *F*≡*F*_{0}, we obtain the following expression for the first moment of the correlated process *X*(*t*):
Comparing the above expression with the second moment in the force-free case (3.4), we arrive at the second Einstein relation .

Similar to other models of subdiffusion based on CTRWs, the studied process displays weak ergodicity breaking [28–33]. To see this, let us analyse the time-averaged mean square displacement
3.6Here, *Δ* is the lag time and *t* represents the total length of the observation. For standard diffusion processes, scales as 2*DΔ* for large *t*. However, after some standard calculations, we get the following expression for the correlated process:
for *Δ*≪*t*. Thus, the correlated process displays weak ergodicity breaking.

Another feature of anomalous dynamics displayed by CTRW models is ageing [34,35], manifested by the temporal decay of the response of the process to a sinusoidal, time-dependent force. Applying the subordination method to the correlated process under the influence of time-dependent force (see [34,17] for details), we get that the corresponding first moment of *X*(*t*) has the form
Because of the factor *u*^{1/(1+ρ+1/γ)−1} inside the integral, the first moment decays as time proceeds and we observe the so-called ‘death of linear response’ [34].

## 4. Summary

In this paper, we have introduced the CTRW process with correlated temporal structure. Consecutive waiting times were defined as weighted sums of independent random variables combined with a reflecting boundary condition. The weights, in turn, were determined by the regularly varying memory function. Our model generalizes two different correlated models introduced in Chechkin *et al.* [12] and Tejedor & Metzler [13].

We have analysed various asymptotic properties of the studied process. We have derived the corresponding diffusion limit and proved its subdiffusive character. It should be emphasized that the functional limit theorems proved in the appendices apply to the class of correlated processes, which is much wider than the CTRWs studied in the main part of the paper. Applying the subordination scheme we have extended the correlated model to include external forces. The derived coupled Langevin equations have been further used to analyse Einstein relations, speed of relaxation, equilibrium distribution, ergodicity breaking and ageing properties.

We have separately analysed the case of exponential memory function. We have shown that in such a setting the exponential kernel kills the dependence between waiting times and the correlation between successive rests of the walker can be observed only in the primary stage of the motion. However, the diffusion limit is the same as for the renewal CTRW with independent waiting times.

## Funding statement

The research of M.M. was partially supported by an NCN Maestro grant.

## Appendix

The structure of this appendix is as follows. Necessary definitions and notations are gathered in appendix A. In appendix B, we define two useful mappings and show when they are continuous. Next, in appendix C, we formulate two general assumptions on the sequence of jumps *J*_{k} and the sequence of innovations *ξ*_{k} and the function *M* defining the correlated time structure of the CTRW process. We also discuss in what situations these general assumptions are satisfied. Results from appendices B and C will be used in appendix D, in which we prove the general result dealing with the weak convergence of the sequence of scaled CTRW processes. In appendix E, we give the proof of theorem 2.1, which is the conclusion from appendix D. In appendix F, we prove theorem 2.2.

## Appendix A. Definitions and notations

Let denote the space of functions mapping into that are right-continuous having limits from the left (*cádlág* functions). We equip this space with the Skorohod topology. We define also two subspaces of ; namely, being the space of all *cádlág* functions starting from 0 that are non-negative, non-decreasing and unbounded from above, and the space containing all *cádlág* functions of bounded variation on any interval [0,*r*], *r*>0.

Let and denote the space of Radon measures on and the space of signed Radon measures on , respectively. For any function there exists measure such that *μ*_{y}((*a*,*b*])=*y*(*b*)−*y*(*a*) for any interval (*a*,*b*], 0≤*a*<*b*. Any function may be written in the form *z*=*z*^{+}−*z*^{−}, where . Hence, to any corresponds the measure defined as *μ*_{z}=*μ*_{z+}−*μ*_{z−}.

In the following appendices, we write to denote the Lebesgue integral of *f* with respect to the Radon measure *μ*_{y}, , while is to be understood in the Riemann–Stieltjes sense with *f* and *y* being the integrand and the integrator, respectively. For , we write . Recall that exists if *f* is continuous on [0,*r*], and then

Given functions *y*_{n}, , we say that sequence {*μ*_{yn}} of measures from converges vaguely to some measure , which we denote as , if for any function , which is continuous and has compact support. It is equivalent to convergence for all *t* being continuity points of *y*. Similarly, given functions *z*_{n}, , we say that measures converge vaguely to the measure if and .

## Appendix B. Continuity of two useful mappings

For any and , we define mapping as

### Lemma B.1

*Assume that the sequence of functions* *n*≥1, *converges to a continuous function* *x* *uniformly on any interval* [0,*r*], *r*>0, *and let the sequence* *n*≥1, *converge to* *in the Skorohod* *topology. Then for any* *r*>0

### Proof.

See the electronic supplementary material.

For any , and we define mapping

### Lemma B.2

*Let* *x*_{n}, *x* *be as in lemma* *B*.1 *and assume that sequence* *converges to* *in the Skorohod* *topology, where* *and* *z*=*z*^{+}−*z*^{−}. *Moreover, let* , *such that* *κ*_{n}→*vκ*. *Then for any* *r*>0

### Proof.

See the electronic supplementary material.

## Appendix C. General assumptions on a continuous-time random walk with correlated temporal structure

The construction of a CTRW with correlated temporal structure is based on the sequence of random vectors {(*J*_{i},*ξ*_{i})}_{i≥1} defined on the common probability space and the function *M*. Consecutive jumps of the walker are given by the sequence {*J*_{i}}, while waiting times between the jumps are of the form . We define the counting process and the CTRW .

Below we state assumptions (A1) and (A2) on the sequence {(*J*_{i},*ξ*_{i})} and the function *M*, respectively. Under these assumptions, we will be able to obtain asymptotic behaviour of the sequence of scaled CTRWs with correlated temporal structure.

(A1) There exist sequences {

*a*_{n}} and {*b*_{n}} such that the sequence of processes of scaled partial sums converges weakly in the Skorohod topology to the process such that*Ξ*=*Ξ*^{+}−*Ξ*^{−}and the trajectories of processes*Ξ*^{+}and*Ξ*^{−}are the elements of , almost surely.(A2) There exists a sequence {

*c*_{n}} and a continuous function*m*mapping into , such that where ,*t*≥0.

Before proceeding to investigate the asymptotic of the CTRW with correlated temporal structure (see appendix D), we discuss the situations in which the above assumptions hold true.

The following lemma, which is an extension of [36], theorem 7.1, is a useful tool for checking whether (A1) holds.

### Lemma C.1

*Let* {**X**_{n,j}≡(*X*_{1,n,j},*X*_{2,n,j})}_{n,j≥1} *be the array of random vectors such that* {**X**_{n,j}}_{j≥1} *is the iid sequence for any* *n*≥1. *Assume that*

(i)

*where**ν*_{0}*is a Lévy measure on*;(ii)

(iii)

*Let* {**Y**_{n,j}≡(*Y* _{1,n,j},*Y* _{2,n,j},*Y* _{3,n,j},*Y* _{4,n,j})}_{n,j≥1} *be the array of random vectors, where*
*Define the sequence of partial sum processes* **Y**_{n}≡(*Y* _{1,n},*Y* _{2,n},*Y* _{3,n},*Y* _{4,n})
*where* . *Then*

(

*a*)**Y**_{n}⇒**Y***in the Skorohod**topology, where***Y**≡(*Y*_{1},*Y*_{2},*Y*_{3},*Y*_{4})*is the Lévy process with the Fourier transform**where**Lévy measure**ν**is defined as*(

*b*)*processes**Y*_{3}and*Y*_{4}*are independent and**Y*_{2}=*Y*_{3}−*Y*_{4}.

### Proof.

See the electronic supplementary material.

### Remark C.2

One can prove results similar to [36], theorem 7.1 and lemma C.1 in situations where {**X**_{n,k}} is the array of dependent random vectors and the first coordinate of the limit process ** Y** has a gaussian component. For more details, see [37,38].

Assumption (A2) is also quite weak. In particular, it is satisfied for functions from a broad family of regularly varying functions with index *ρ*>0 (see appendix E). Assumption (A2) also holds when *M*_{n}(*t*)→*m*(*t*) for every *t*≥0 and functions *M*_{n} are non-decreasing.

## Appendix D. Asymptotic of a continuous-time random walk with correlated temporal structure

### Theorem D.1

*Let the assumptions (A1) and (A2) be satisfied with sequences {b*_{n}*} and {c*_{n}*} such that* *as* *. Define the sequence of processes* *. Then R*_{n}*⇒L°S*^{−1} *in the Skorohod* *topology, where* *.*

### Proof.

See the electronic supplementary material.

### Remark D.2

Let the assumptions of theorem D.1 be satisfied. If we additionally assume that *ξ*_{i}>0 almost surely for all *i*≥1 and that function *M* is non-negative, then
Furthermore, one can easily show that
and then .

## Appendix E. Proof of theorem 2.1

Let sequences {*J*_{k}} and {*ξ*_{k}} and function *M* be as in §2*a*. We will obtain theorem 2.1 as a conclusion from theorem D.1.

### Proof of theorem 2.1.

In order to use theorem D.1, we need to check if its assumptions are satisfied, i.e. if sequence {(*J*_{k},*ξ*_{k})} fulfils condition (A1) and the regularly varying function *M* satisfies condition (A2).

We begin with checking condition (A1). The main tool we use to verify that this condition holds will be lemma C.1; thus, we need to check its assumptions first. Before doing so, recall that {*J*_{k}} is the iid sequence with Fourier transform given by (2.2). As shown in [39], §1.2.6 we have that *J*_{k}∼*S*_{α}(1,0,0), *n*^{−1/α}(*J*_{1}+⋯+ *J*_{n})=*DJ*, where *J*∼*S*_{α}(1,0,0) is independent of {*J*_{k}} and the distribution of *J*_{k} is infinitely divisible with Lévy measure
Then, we also have that as (e.g. [40], theorem 3.2.2). Similarly, the assumption that *ξ*_{k} have Fourier transform (2.4) implies that *ξ*_{k}∼*S*_{γ}(1,*β*,0), *n*^{−1/α}(*ξ*_{1}+⋯+*ξ*_{n})=*Dξ*, where *ξ*∼*S*_{γ}(1,*β*,0) is independent of {*ξ*_{k}}, the distribution of *ξ*_{k} is infinitely divisible with Lévy measure
and that as .

Now, we define the array of random vectors {**X**_{n,k}}≡{(*n*^{−1/α}*J*_{k},*n*^{−1/γ}*ξ*_{k})} and check that it fulfils the assumptions of lemma C.1.

Take arbitrary such that {(0,0)}∉*A*×*B*. Then, as sequences {*J*_{k}} and {*ξ*_{k}} are assumed to be independent, we have that
Hence condition (i) holds with *ν*_{0}(d*x*×d*y*)≡*ν*_{α}(d*x*)*δ*_{0}(d*y*)+*δ*_{0}(d*x*)*ν*_{γ}(d*y*).

Assumption (ii) is obviously satisfied with and .

Next observe that for arbitrary small *ε*>0
and
Then, if we let *ε*↘0, it is easy to see that assumption (iii) is satisfied as well. Hence, assertions (a) and (b) of lemma C.1 hold true.

Now, we identify limit processes in condition (A1).

Note that distribution *S*_{α}(1,0,0) is symmetric; thus, for any *t*≥0
Then,
where *Y* _{1}(*t*) has the Fourier transform
which we identify as the Fourier transform of *α*-stable process *L*_{α} such that *L*_{α}(1)∼*S*_{α}(1,0,0) (e.g. [39], §1.2.6). Hence, *L*_{n}⇒*L*_{α} in the Skorohod topology, where *L*_{α} has the Fourier transform *ϕ*_{Lα(t)}(*k*)=*E*(*e*^{ikLα(t)})=*e*^{−t|k|α}.

Next, observe that . Then, the Fourier transform of *Y* _{2}(*t*) can be written in the form
where *T*_{γ} is the *γ*-stable process with *T*_{γ}(1)∼*S*_{γ}(1,*β*,0) and the Fourier transform
Moreover, for any *t*≥0, it follows that
Hence
Thus, we have shown that sequence {(*J*_{k},*ξ*_{k})} satisfies condition (A2) with sequences *a*_{n}≡*n*^{−1/α}, *b*_{n}≡*n*^{−1/γ} and processes *L*≡*L*_{α} and *Ξ*≡*T*_{γ}.

Next, we check condition (A2), assuming that *M* is a regularly varying function of index *ρ*>0. As mentioned in §2, without loss of generality we may assume that *M* is bounded on (0,1). Below we show that for any fixed *r*>0 we have that
E1Observe that for *t*∈[0,*r*] and for every *n*≥1 we have that 0<1/*n*≤*v*=*t*+1/*n*≤*r*+1/*n*≤*r*+1. Then
By the well-known uniform convergence theorem for regularly varying functions (e.g. [36], proposition 2.4), it follows that
and one can immediately see that also
This proves (E.1), and hence condition (A2) holds with *c*_{n}=*M*(*n*) and *m*(*t*)=*t*^{ρ}.

We have shown that under the assumptions made in §2*a* conditions (A1) and (A2) are satisfied. Then from theorem D.1, we get convergence of the sequence of scaled CTRW processes *R*_{n}(⋅)≡*n*^{−1/α}*R*(*n*^{1+1/γ}*M*(*n*)⋅)⇒*L*_{α}(*S*^{−1}(⋅)) in the Skorohod topology, where *L*_{α} and *S* are as in the statement of theorem 2.1.

Finally, let {*B*_{n}} be the sequence of scaling constants such that behaves asymptotically as *n*, i.e. , as . Note that as . Then the sequence of scaled processes has the same limit as the sequence *R*_{n}(⋅), and the convergence (2.5) also follows. This completes the proof of theorem 2.1.

## Appendix F. Proof of theorem 2.2

### Proof of theorem 2.2.

Let {*J*_{k}} and {*ξ*_{k}} be as assumed in §2*b*. Define the sequence of processes
F1Then from the definition of *T*_{i} (2.7) we get
F2where processes *Ξ*_{n} and *Z*_{n} are defined as
Using [36, corollary 7.1], one can easily check that *Ξ*_{n} converges weakly in the Skorohod topology to the *γ*-stable subordinator *T*_{γ} with the Fourier transform of the form , . We show that *Z*_{n}⇒*z*≡0 in the Skorohod topology. As *ξ*_{1} is *γ*-stable, the moments are finite for all 0≤*δ*≤*γ*. We fix some *δ*<*γ*. Then, by *γ*<1 we have that *δ*<1. Next, observe that for any 0<*δ*<1 and *x*_{i}≥0, *i*=1,2,…,*n*, the following inequality holds: Using this inequality and fact that *ξ*_{i} are iid, we get

Hence, as and by Chebyshev's inequality, for any *ε*>0, we get
Thus, (convergence in probability) for any *t*≥0 and because trajectories of *Z*_{n} are non-decreasing it follows also that , which finally implies the weak convergence *Z*_{n}⇒*z*≡0 in the topology. This together with (F.2) and convergence *Ξ*_{n}⇒*T*_{γ} gives
F3in the Skorohod topology. As {*J*_{i}} and {*ξ*_{i}} are mutually independent, we immediately get that is independent of *L*_{α}. Moreover from (F.3), it follows that *L*_{α} has the Fourier transform given by (2.9).

Now we show convergence (2.8). Observe that for any *t*≥0
We define the auxiliary sequence of CTRW processes as
By [36], corollary 7.1, it follows that *L*_{n}⇒*L*_{α} and that sequences of processes *L*_{n}, *S*_{n} are independent. This, together with (F.3), implies joint convergence in the Skorohod topology. Then from [41], theorem 3.6, it follows that in the Skorohod topology.

We conclude the proof of theorem 2.2 with the observation that *R*(*nt*)/*n*^{γ/α}=*R*_{nγ}(*t*). As with , *n*^{−γ/α}*R*(*nt*) has the same limit as sequence *R*_{n} and convergence (2.8) follows.

- Received June 26, 2013.
- Accepted July 30, 2013.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.