Let (Xk,ξk)k∈N be a sequence of independent copies of a pair (X,ξ) where X is a random process with paths in the Skorokhod space D[0,∞) and ξ is a positive random variable. The random process with immigration (Y(u))u∈R is defined as the a.s. finite sum Y(u)=∑k≥0Xk+1(u−ξ1−⋯−ξk)1{ξ1+⋯+ξk≤u}. We obtain a functional limit theorem for the process (Y(ut))u≥0, as t→∞, when the law of ξ belongs to the domain of attraction of an α-stable law with α∈(0,1), and the process X oscillates moderately around its mean E[X(t)]. In this situation the process (Y(ut))u≥0, when scaled appropriately, converges weakly in the Skorokhod space D(0,∞) to a fractionally integrated inverse stable subordinator.

Fractionally integrated inverse stable subordinatorsrandom process with immigrationshot noise process60F0560K05Introduction and main result

Let (Xk,ξk)k∈N be a sequence of independent copies of a pair (X,ξ) where X is a random process with paths in D[0,∞) and ξ is a positive random variable. We impose no conditions on the dependence structure of (X,ξ). Hereafter N0 denotes the set of non-negative integers {0,1,2,…}.

Let (Sn)n∈N0 be a standard zero-delayed random walk:
S0:=0,Sn:=ξ1+⋯+ξn,n∈N,
and let (ν(t))t∈R be the corresponding first-passage time process for (Sn)n∈N0:
ν(t):=inf{k∈N0:Sk>t},t∈R.t\},\hspace{1em}t\in \mathbb{R}.\]]]>
The random process with immigrationY=(Y(u))u∈R is defined as a finite sum
Y(u):=∑k≥0Xk+1(u−Sk)1{Sk≤u}=∑k=0ν(u)−1Xk+1(u−Sk),u∈R.
This family of random processes was introduced in [11] as a generalization of several known objects in applied probability including branching processes with immigration (in case of X being a branching process) and renewal shot noise processes (in case of X(t)=h(t) a.s. for some h∈D[0,∞)). The process X is usually called a response process, or a response function if X(t)=h(t) a.s. for some deterministic function h.

The problem of weak convergence of random processes with immigration was addressed in [11, 12, 16] where the authors give a more or less complete picture of the weak convergence of finite-dimensional distributions of (Y(ut))u≥0 or (Y(u+t))u∈R, as t→∞. The case of renewal shot noise process has received much attention in the past years, see [6, 9, 10, 14]. A comprehensive survey of the subject is given in Chapter 3 of the recent book [7].

A much more delicate question of weak convergence of Y in functional spaces, to the best of our knowledge, was only investigated either for particular response processes, or in the simple case when ξ is exponentially distributed. In the latter situation Y is called a Poisson shot noise process. In the list below η is a random variable which satisfies certain assumptions specified in the corresponding papers:

if ξ has exponential distribution and either X(t)=1{η>t}t\}}$]]> or X(t)=t∧η, functional limit theorems for Y were derived in [18];

if X(t)=1{η>t}t\}}$]]> and Eξ<∞, a functional limit theorem for Y was established in [8];

if X(t)=1{η≤t}, functional limit theorems for Y are given in [1];

if ξ has exponential distribution and X(t)=ηf(t) for some deterministic function f, limit theorems for Y were obtained in [15];

in [12, 16] sufficient conditions for weak convergence of (Y(u+t))u∈R to a stationary process with immigration were found.

In this paper we treat the case where ξ is heavy-tailed, more precisely we assume that
P{ξ>t}∼t−αℓξ(t),t→∞,t\}\sim {t}^{-\alpha }\ell _{\xi }(t),\hspace{1em}t\to \infty ,\]]]>
for some ℓξ slowly varying at infinity, and α∈(0,1). Assuming (2), we obtain a functional limit theorem for a quite general class of response processes. The class of such processes can be described by a common property: they do not “oscillate to much” around the mean E[X(t)], which itself varies regularly with parameter ρ>−α-\alpha $]]>. Let us briefly outline our approach based on ideas borrowed from [11]. Put h(t):=E[X(t)] and write1

In what follows we always assume that h exists and is a càdlàg function.

Y(t)=∑k≥0(Xk+1(t−Sk)−h(t−Sk))1{Sk≤t}+∑k≥0h(t−Sk)1{Sk≤t}.
We investigate the two summands in the right-hand side separately. The second summand is a standard renewal shot noise process with response function h. Under condition (2) and assuming that
h(t)=E[X(t)]∼tρℓh(t),t→∞,
for some ρ∈R and a slowly varying function ℓh, it was proved in [10, Theorem 2.9] and [14, Theorem 2.1] that
(P{ξ>t}h(t)∑k≥0h(ut−Sk)1{Sk≤ut})u>0⟹f.d.(Jα,ρ(u))u>0,t→∞,t\}}{h(t)}\sum \limits_{k\ge 0}h(ut-S_{k})\mathbb{1}_{\{S_{k}\le ut\}}\bigg)_{u>0}\stackrel{\mathrm{f}.\mathrm{d}.}{\Longrightarrow }\big(J_{\alpha ,\rho }(u)\big)_{u>0},\hspace{1em}t\to \infty ,\]]]>
where Jα,ρ=(Jα,ρ(u))u≥0 is a so-called fractionally integrated inverse α-stable subordinator. The process Jα,ρ is defined as a pathwise Lebesgue–Stieltjes integral
Jα,ρ(u)=∫[0,u](u−y)ρdWα←(y),u≥0.
In this formula Wα←(y):=inf{t≥0:Wα(t)>y}y\}$]]>, y≥0, is a generalized inverse of an α-stable subordinator (Wα(t))t≥0 with the Laplace exponent
−logEe−sWα(1)=Γ(1−α)sα,s≥0.
It is also known that convergence of finite-dimensional distributions (5) can be strengthened to convergence in the Skorokhod space D(0,∞) endowed with the J1-topology if ρ>−α-\alpha $]]>, see Theorem 2.1 in [14]. If ρ≤−α the process (Jα,ρ(u))u≥0, being a.s. finite for every fixed u≥0, has a.s. locally unbounded trajectories, see Proposition 2.5 in [14].

Turning to the first summand in (3) we note that it is the a.s. limit of a martingale (R(j,t),Fj)j∈N, where Fj:=σ((Xk,ξk):1≤k≤j) and
R(j,t):=∑k=0j−1(Xk+1(t−Sk)−h(t−Sk))1{Sk≤t},j∈N.
Applying the martingale central limit theory it is possible to show that under appropriate assumptions (which are of no importance for this paper)
(P{ξ>t}v(t)∑k≥0(Xk+1(ut−Sk)−h(ut−Sk))1{Sk≤ut})u>0⟹f.d.(Z(u))u>0,t\}}{v(t)}}\sum \limits_{k\ge 0}\big(X_{k+1}(ut-S_{k})-h(ut-S_{k})\big)\mathbb{1}_{\{S_{k}\le ut\}}\bigg)_{u>0}\stackrel{\mathrm{f}.\mathrm{d}.}{\Longrightarrow }\big(Z(u)\big)_{u>0},\]]]>
as t→∞, for a non-trivial process Z, where v(t):=E[(X(t)−h(t))2] is the variance of X, see Proposition 2.2 in [11].

We are interested in situations when the second summand in (3) asymptotically dominates, more precisely we are looking for conditions ensuring
P{ξ>t}h(t)supu∈[0,T]|∑k≥0(Xk+1(ut−Sk)−h(ut−Sk))1{Sk≤ut}|→P0,t→∞,t\}}{h(t)}\underset{u\in [0,\hspace{0.1667em}T]}{\sup }\bigg|\sum \limits_{k\ge 0}\big(X_{k+1}(ut-S_{k})-h(ut-S_{k})\big)\mathbb{1}_{\{S_{k}\le ut\}}\bigg|\stackrel{\mathbb{P}}{\to }0,\hspace{1em}t\to \infty ,\]]]>
for every fixed T>00$]]>. From what has been mentioned above it is clear that this can happen only if
limt→∞P{ξ>t}v(t)h2(t)=0.t\}v(t)}{{h}^{2}(t)}=0.\]]]>
Restricting our attention to the case where v is regularly varying with index β∈R, i.e.
v(t)∼tβℓv(t),t→∞,
we see that (8) holds if β<α+2ρ and fails if β>α+2ρ\alpha +2\rho $]]>. As long as we do not make any assumptions on distributional or path-wise properties of X such as e.g., monotonicity, self-similarity or independence of increments, it can be hardly expected that condition (8) alone is sufficient for (7). Nevertheless, we will show that (7) holds true under additional assumptions on the asymptotic behavior of higher centered moments E[(X(t)−h(t))2l], l=1,2,…, and an additional technical assumption. Our first main result treats the case where the moments of the normalized process ([X(t)−h(t)]/v(t))t≥0 are bounded uniformly in t≥0. Denote by (Xˆ(t))t≥0 the centered process (X(t)−h(t))t≥0.

Assume that for allt≥0andl∈Nwe haveE[|X(t)|l]<∞. Further, assume that the following conditions are fulfilled:

relation (2) holds for someα∈(0,1);

relation (4) holds for someρ>−α-\alpha $]]>;

relation (9) holds for someβ∈(−α,α+2ρ);

there existsδ>00$]]>such that for everyl∈Nthe following two conditions hold:E[Xˆ(t)2l]≤Clvl(t),t≥0,andE[supy∈[0,δ)|Xˆ(t)−Xˆ(t−y)1{y≤t}|l]≤Cltl(ρ−ε),t≥0,for someCl∈(0,∞)andε>00$]]>.

Our second main result is mainly applicable when the process X is almost surely bounded by some (deterministic) constant. We have the following theorem.

Assume that for allt≥0andl∈Nwe haveE|X(t)|l<∞and conditions (A1), (A2) of Theorem1are valid. Further, suppose that for everyl∈Nthere exists a constantCl>00$]]>such thatE[Xˆ(t)2l]=E[(X(t)−h(t))2l]≤Clh(t),t≥0,and for someδ>00$]]>the functiont↦E[supy∈[0,δ)|Xˆ(t)−Xˆ(t−y)1{y≤t}|l]is either directly Riemann integrable or locally bounded andE[supy∈[0,δ)|Xˆ(t)−Xˆ(t−y)1{y≤t}|l]=O(P{ξ>t}),t→∞.t\}\big),\hspace{1em}t\to \infty .\]]]>Then (12) holds.

Obviously, our results are far from being optimal and leave a lot of space for improvements, yet they are applicable to several models given in the next section.

ApplicationsThe number of busy servers in a <inline-formula id="j_vmsta76_ineq_087"><alternatives>
<mml:math><mml:mi mathvariant="italic">G</mml:mi><mml:mo mathvariant="normal" stretchy="false">/</mml:mo><mml:mi mathvariant="italic">G</mml:mi><mml:mo mathvariant="normal" stretchy="false">/</mml:mo><mml:mi>∞</mml:mi></mml:math>
<tex-math><![CDATA[$G/G/\infty $]]></tex-math></alternatives></inline-formula> queue

Consider a G/G/∞ queue with customers arriving at 0=S0<S1<S2<⋯. Upon arrival each customer is served immediately by one of infinitely many idle servers and let the service time of the kth customer be ηk, a copy of a positive random variable η. Put X(t):=1{η>t}t\}}$]]>, then the random process with immigration
Y(u)=∑k≥01{Sk≤u<Sk+ηk+1},u≥0,
represents the number of busy servers at time u≥0. The process (Y(u))u≥0 may also be interpreted as the difference between the number of visits to [0,t] of the standard random walk (Sk)k≥0 and the perturbed random walk (Sk+ηk+1)k≥1, see [2], or as the number of active sources in a communication network, see [17, 18]. An introduction to renewal theory for perturbed random walks can be found in [7].

Assume that (2) holds and
P{η>t}∼tρℓη(t),t→∞,t\}\sim {t}^{\rho }\ell _{\eta }(t),\hspace{1em}t\to \infty ,\]]]>
for some ρ∈(−α,0] and ℓη slowly varying at infinity. Note that
h(t)=P{η>t}∼tρℓη(t),t→∞.t\}\sim {t}^{\rho }\ell _{\eta }(t),\hspace{1em}t\to \infty .\]]]>
Moreover, for every l∈N and every δ>00$]]>,
E[Xˆ(t)2l]=P{η>t}P{η≤t}(P2l−1{η>t}+P2l−1{η≤t})≤h(t)t\}\mathbb{P}\{\eta \le t\}\big({\mathbb{P}}^{2l-1}\{\eta >t\}+{\mathbb{P}}^{2l-1}\{\eta \le t\}\big)\le h(t)\]]]>
and
E[supy∈[0,δ)|Xˆ(t)−Xˆ(t−y)1{y≤t}|l]≤2l−1E[supy∈[0,δ)|Xˆ(t)−Xˆ(t−y)1{y≤t}|]≤2lP{η>t}1{t≤δ}+2l(P{η>t−δ}−P{η>t})1{t>δ}.t\}\mathbb{1}_{\{t\le \delta \}}+{2}^{l}\big(\mathbb{P}\{\eta >t-\delta \}-\mathbb{P}\{\eta >t\}\big)\mathbb{1}_{\{t>\delta \}}.\end{array}\]]]>
The function on the right-hand side is directly Riemann integrable. Indeed, we have
∑n≥1supδn≤y≤δ(n+1)(P{η>y−δ}−P{η>y})≤∑n≥1(P{η>(n−1)δ}−P{η>(n+1)δ})=P{η>0}+P{η>δ}≤2,y-\delta \}-\mathbb{P}\{\eta >y\}\big)\\{} & \displaystyle \le \sum \limits_{n\ge 1}\big(\mathbb{P}\big\{\eta >(n-1)\delta \big\}-\mathbb{P}\big\{\eta >(n+1)\delta \big\}\big)=\mathbb{P}\{\eta >0\}+\mathbb{P}\{\eta >\delta \}\le 2,\end{array}\]]]>
and the claim follows from the remark after the definition of direct Riemann integrability given on p. 362 in [5].

From Theorem 2 we obtain the following result, complementing Theorem 1.2 in [8] that treats the case Eξ<∞.

Assume that(ξ,η)is a random vector with positive components such that (2) and (15) hold forα∈(0,1)andρ∈(−α,0], respectively. Let(ξk,ηk)k∈Nbe a sequence of independent copies of(ξ,η)and(Sk)k∈N0be a random walk defined by (1). Then(P{ξ>t}P{η>t}∑k≥01{Sk≤ut<Sk+ηk+1})u>0⇒(Jα,ρ(u))u>0,t→∞,t\}}{\mathbb{P}\{\eta >t\}}\sum \limits_{k\ge 0}\mathbb{1}_{\{S_{k}\le ut0}\Rightarrow \big(J_{\alpha ,\rho }(u)\big)_{u>0},\hspace{1em}t\to \infty ,\]]]>weakly onD(0,∞)endowed with theJ1-topology.

We do not assume independence of ξ and η.

Shot noise processes with a random amplitude

Assume that X(t)=ηf(t), where η is a non-degenerate random variable and f:[0,∞)→R is a fixed càdlàg function. The corresponding random process with immigration
Y(t)=∑k≥0ηk+1f(t−Sk)1{Sk≤t},t≥0,
where (ηk)k∈N is a sequence of independent copies of η, may be interpreted as a renewal shot noise process in which the common response function f is scaled at a shot Sk by a random factor ηk+1. In case where (ξk)k∈N have exponential distribution and are independent of (ηk)k∈N such processes were used in mathematical finance as a model of stock prices with long-range dependence in asset returns, see [15].

Note that if E|η|l<∞ for all l∈N, then
h(t)=(Eη)f(t),v(t)=Var(η)f2(t),E[(X(t)−h(t))2l]=E[(η−Eη)2l]f2l(t)≤Clvl(t),l∈N,
for some Cl>00$]]>. Assume now that f varies regularly with index ρ>−α-\alpha $]]> and additionally satisfies
supy∈[0,δ)|f(t)−f(t−y)|=O(tρ−ε),t→∞,
for some δ>00$]]> and ε>00$]]>. Then
E[supy∈[0,δ)|Xˆ(t)−Xˆ(t−y)1{y≤t}|l]=E|η−Eη|lsupy∈[0,δ)|f(t)−f(t−y)1{y≤t}|l=O(tl(ρ−ε)),t→∞.
Hence, all assumptions of Theorem 1 hold (if Eη<0, Theorem 1 is applicable to the process −X) and we have the following result.

Assume thatE|η|l<∞for alln∈N,Eη≠0and (2) holds. Iff:[0,∞)→Rsatisfiesf(t)∼tρℓf(t),t→∞,for someρ>−α-\alpha $]]>andℓfslowly varying at infinity, and (16) holds, then(P{ξ>t}f(t)Eη∑k≥0ηk+1f(ut−Sk)1{Sk≤ut})u>0⇒(Jα,ρ(u))u>0,t→∞,t\}}{f(t)\mathbb{E}\eta }\sum \limits_{k\ge 0}\eta _{k+1}f(ut-S_{k})\mathbb{1}_{\{S_{k}\le ut\}}\bigg)_{u>0}\Rightarrow \big(J_{\alpha ,\rho }(u)\big)_{u>0},\hspace{1em}t\to \infty ,\]]]>weakly onD(0,∞)endowed with theJ1-topology.

This result complements the convergence of finite-dimensional distributions provided by Example 3.3 in [11].

In general, condition (16) might not hold for a function f which is regularly varying with index ρ∈R. Take, for example,
f(t)=1+(−1)[t]log[t]1{t>1}.1\}}.\]]]>
Then, f is regularly varying with index ρ=0, but for every δ>00$]]> and large n∈N we have
supy∈[0,δ)|f(2n)−f(2n−y)|≥supy∈[0,δ∧1)|f(2n)−f(2n−y)|≥2log(2n).
Hence, (16) does not hold. On the other hand, if f is differentiable with an eventually monotone derivative f′, then (16) holds by the mean value theorem for differentiable functions and Theorem 1.7.2 in [3].

Proof of Theorems <xref rid="j_vmsta76_stat_001">1</xref> and <xref rid="j_vmsta76_stat_002">2</xref>

The proofs of Theorems 1 and 2 rely on the same ideas, so we will prove them simultaneously. Pick δ>00$]]> such that all assumptions of Theorem 1 or Theorem 2 hold. This δ>00$]]> remains fixed throughout the proof.

In view of assumptions (A1) and (A2) and the fact that h is càdlàg we infer from Theorem 2.1 in [14] that
(P{ξ>t}h(t)∑k≥0h(ut−Sk)1{Sk≤ut})u>0⇒(Jα,ρ(u))u>0t→∞,t\}}{h(t)}\sum \limits_{k\ge 0}h(ut-S_{k})\mathbb{1}_{\{S_{k}\le ut\}}\bigg)_{u>0}\Rightarrow \big(J_{\alpha ,\rho }(u)\big)_{u>0}\hspace{1em}t\to \infty ,\]]]>
weakly on D(0,∞) endowed with the J1-topology. Note that in Theorem 2.1 of [14] h is assumed monotone (or eventually monotone). However, this assumption is redundant. The only places which have to be adjusted in the proofs are two displays on p. 90, where h(0) should be replaced by supy∈[0,c]h(y).

Hence, from (3) we see that it is enough to check, for every fixed T>00$]]>, that
P{ξ>t}h(t)supu∈[0,T]|Y˜(ut)|→P0,t→∞,t\}}{h(t)}\underset{u\in [0,T]}{\sup }\big|\widetilde{Y}(ut)\big|\stackrel{\mathbb{P}}{\to }0,\hspace{1em}t\to \infty ,\]]]>
where Y˜(t):=∑k≥0(Xk+1(t−Sk)−h(t−Sk))1{Sk≤t} for t≥0. Moreover, it suffices to show that
P{ξ>t}h(t)|Y˜(t)|→a.s.0,t→∞.t\}}{h(t)}\big|\widetilde{Y}(t)\big|\stackrel{a.s.}{\to }0,\hspace{1em}t\to \infty .\]]]>
Indeed, for every fixed s>00$]]>,
P{ξ>t}h(t)supu∈[0,T]|Y˜(ut)|≤P{ξ>t}h(t)supu∈[0,s]|Y˜(u)|+P{ξ>t}h(t)supu∈[s,Tt]|Y˜(u)|≤P{ξ>t}h(t)supu∈[0,s]|Y˜(u)|+P{ξ>t}h(t)supu∈[s,Tt]h(u)P{ξ>u}supu∈[s,Tt]|P{ξ>u}h(u)Y˜(u)|.t\}}{h(t)}\underset{u\in [0,T]}{\sup }\big|\widetilde{Y}(ut)\big|\\{} & \displaystyle \hspace{1em}\le \frac{\mathbb{P}\{\xi >t\}}{h(t)}\underset{u\in [0,s]}{\sup }\big|\widetilde{Y}(u)\big|+\frac{\mathbb{P}\{\xi >t\}}{h(t)}\underset{u\in [s,Tt]}{\sup }\big|\widetilde{Y}(u)\big|\\{} & \displaystyle \hspace{1em}\le \frac{\mathbb{P}\{\xi >t\}}{h(t)}\underset{u\in [0,s]}{\sup }\big|\widetilde{Y}(u)\big|+\frac{\mathbb{P}\{\xi >t\}}{h(t)}\underset{u\in [s,Tt]}{\sup }\frac{h(u)}{\mathbb{P}\{\xi >u\}}\underset{u\in [s,Tt]}{\sup }\bigg|\frac{\mathbb{P}\{\xi >u\}}{h(u)}\widetilde{Y}(u)\bigg|.\end{array}\]]]>
Since t↦h(t)/P{ξ>t}t\}$]]> is regularly varying with positive index ρ+α,
supu∈[s,Tt]h(u)P{ξ>u}∼h(Tt)P{ξ>Tt}∼Tρ+αh(t)P{ξ>t},t→∞.u\}}\sim \frac{h(Tt)}{\mathbb{P}\{\xi >Tt\}}\sim {T}^{\rho +\alpha }\frac{h(t)}{\mathbb{P}\{\xi >t\}},\hspace{1em}t\to \infty .\]]]>
Sending t→∞ we obtain, for every fixed s>00$]]>,
lim supt→∞P{ξ>t}h(t)supu∈[0,T]|Y˜(ut)|≤Tρ+αsupu∈[s,∞)|P{ξ>u}h(u)Y˜(u)|.t\}}{h(t)}\underset{u\in [0,T]}{\sup }\big|\widetilde{Y}(ut)\big|\le {T}^{\rho +\alpha }\underset{u\in [s,\infty )}{\sup }\bigg|\frac{\mathbb{P}\{\xi >u\}}{h(u)}\widetilde{Y}(u)\bigg|.\]]]>
Sending now s→∞ shows that (19) implies (18). Let us first check that (19) holds along the arithmetic sequence (nδ)n∈N. According to the Borel–Cantelli lemma and Markov’s inequality it suffices to check that for some l∈N∑n=1∞(P{ξ>nδ}h(δn))2lE[Y˜(δn)2l]<∞.n\delta \}}{h(\delta n)}\bigg)}^{2l}\mathbb{E}\big[\widetilde{Y}{(\delta n)}^{2l}\big]<\infty .\]]]>
To check (20) we apply the Burkholder–Davis–Gundy inequality in the form given in Theorem 11.3.2 of [4], to obtain
E[Y˜(t)2l]≤KlE[(∑k≥0E(Xˆk+12(t−Sk)1{Sk≤t}|Fk))l]+KlE[supk≥0(Xˆk+12l(t−Sk)1{Sk≤t})],
for some constant Kl>00$]]>, where we recall the notation Fk=σ((Xj,ξj):1≤j≤k).

Using assumption (A4) we infer from (21):
E[Y˜(t)2l]≤KlE[(∑k≥0v(t−Sk)1{Sk≤t})l]+KlE[∑k≥0Xˆk+12l(t−Sk)1{Sk≤t}]≤KlE[(∑k≥0v(t−Sk)1{Sk≤t})l]+KlClE[∑k≥0vl(t−Sk)1{Sk≤t}].
If β≥0, then t↦vl(t) varies regularly with non-negative index lβ. Therefore, Lemma 1(i) yields
E(∑k≥0vl(t−Sk)1{Sk≤t})=O(vl(t)P{ξ>t}),t→∞.t\}}\bigg),\hspace{1em}t\to \infty .\]]]>
If β∈(−α,0), pick l∈N such that lβ<−α. Then vl(t)=O(P{ξ>t})t\})$]]>, as t→∞, and Lemma 1(iii) yields
E[∑k≥0vl(t−Sk)1{Sk≤t}]=O(1),t→∞.
Hence, in any case
E[∑k≥0vl(t−Sk)1{Sk≤t}]=O(vl(t)P{ξ>t})+O(1),t→∞.t\}}\bigg)+O(1),\hspace{1em}t\to \infty .\]]]>
To bound the first summand in (22) apply Lemma 1(i) to obtain
E[(∑k≥0v(t−Sk)1{Sk≤t})l]=O((v(t)P{ξ>t})l),t→∞.t\}}\bigg)}^{l}\bigg),\hspace{1em}t\to \infty .\]]]>
Combining this estimate with (23), we see that (20) holds if we pick l>(2ρ+α−β)−1{(2\rho +\alpha -\beta )}^{-1}$]]>. This proves (20) under assumptions of Theorem 1. □

From (21) and using (13) we have
E[Y˜(t)2l]≤KlC1lE[(∑k≥0h(t−Sk)1{Sk≤t})l]+KlClE[∑k≥0h(t−Sk)1{Sk≤t}].
Lemma 1(i) gives us the estimate
E[Y˜(t)2l]=O((h(t)P{ξ>t})l),t→∞.t\}}\bigg)}^{l}\bigg),\hspace{1em}t\to \infty .\]]]>
Therefore, (20) holds if we choose l∈N such that l(α+ρ)>11$]]>. This proves (20) under the assumptions of Theorem 2.

It remains to show that
P{ξ>nδ}h(nδ)supt∈[nδ,(n+1)δ)|∑k≥0(Xˆk+1((n+1)δ−Sk)1{Sk≤(n+1)δ}−Xˆk+1(t−Sk)1{Sk≤t})|→a.s.0,n\delta \}}{h(n\delta )}\underset{t\in [n\delta ,(n+1)\delta )}{\sup }\bigg|& \displaystyle \sum \limits_{k\ge 0}\big(\widehat{X}_{k+1}\big((n+1)\delta -S_{k}\big)\mathbb{1}_{\{S_{k}\le (n+1)\delta \}}\\{} & \displaystyle -\widehat{X}_{k+1}(t-S_{k})\mathbb{1}_{\{S_{k}\le t\}}\big)\bigg|\stackrel{a.s.}{\to }0,\end{array}\]]]>
as n→∞, which in turn is an obvious consequence of regular variation of t↦P{ξ>t}/h(t)t\}/h(t)$]]> and
P{ξ>n}h(n)∑k≥0Vk+1(nδ−Sk)1{Sk≤nδ}→a.s.0,n→∞,n\}}{h(n)}\sum \limits_{k\ge 0}V_{k+1}(n\delta -S_{k})\mathbb{1}_{\{S_{k}\le n\delta \}}\stackrel{a.s.}{\to }0,\hspace{1em}n\to \infty ,\]]]>
where Vk+1(t):=supy∈[0,δ)|Xˆk+1(t)−Xˆk+1(t−y)1{y≤t}|. □

Applying Lemma 2(i) with b(t)=tρ−ε and appropriate ε>00$]]> we obtain from (A5) that
E[(∑k≥0Vk+1(t−Sk)1{Sk≤t})l]=O((tρ−εP{ξ>t})l),t→∞.t\}}\bigg)}^{l}\bigg),\hspace{1em}t\to \infty .\]]]>
Hence (24) holds in view of the Borel–Cantelli lemma and Markov’s inequality, since
∑n=1∞P{P{ξ>n}h(n)∑k≥0Vk+1(nδ−Sk)1{Sk≤nδ}>ε}≤Cˆ∑n=1∞(nρ−εh(n))l<∞,n\}}{h(n)}\sum \limits_{k\ge 0}V_{k+1}(n\delta -S_{k})\mathbb{1}_{\{S_{k}\le n\delta \}}>\varepsilon \bigg\}\le \widehat{C}\sum \limits_{n=1}^{\infty }{\big({n}^{\rho -\varepsilon }h(n)\big)}^{l}<\infty ,\]]]>
for all l∈N such that εl>11$]]> and some Cˆ=Cˆl>00$]]>. □

If the function
t↦E[(supy∈[0,δ)|Xˆk+1(t)−Xˆk+1(t−y)1{y≤t}|)l]
is directly Riemann integrable, then
E[(∑k≥0Vk+1(t−Sk)1{Sk≤t})l]=o(1),t→∞
by Lemma 2(ii). Hence (24) holds by the same reasoning as above after applying the Borel–Cantelli lemma. If (14) holds, then the last centered formula also holds with O(1) in the right-hand side by Lemma 2(iii), whence (24). This finishes the proofs of Theorems 1 and 2. □

Acknowledgments

The work of A. Marynych was supported by the Alexander von Humboldt Foundation. We thank two anonymous referees for careful reading, valuable comments and corrections of our numerous blunders.

AppendixMoment convergence for renewal shot noise process

Letf:[0,∞)→Rbe a locally bounded measurable function and suppose that relation (2) holds for someα∈(0,1).

Assume thatf(t)∼tρℓf(t),t→∞,for someρ>−α-\alpha $]]>andℓfslowly varying at infinity. Let(Jα,ρ(u))u≥0be a fractionally integrated inverse stable subordinator defined in (6) (and below). Then, for everyl∈N,limt→∞E[(P{ξ>t}f(t)∑k≥0f(t−Sk)1{Sk≤t})l]=E(Jα,ρ(u))l=l!(Γ(1−α))l∏j=1lΓ(1+ρ+(j−1)(α+ρ))Γ(j(α+ρ)+1).t\}}{f(t)}\sum \limits_{k\ge 0}f(t-S_{k})\mathbb{1}_{\{S_{k}\le t\}}\bigg)}^{l}\bigg]\\{} & \displaystyle \hspace{1em}=\mathbb{E}{\big(J_{\alpha ,\rho }(u)\big)}^{l}\\{} & \displaystyle \hspace{1em}=\frac{l!}{{(\varGamma (1-\alpha ))}^{l}}\prod \limits_{j=1}^{l}\frac{\varGamma (1+\rho +(j-1)(\alpha +\rho ))}{\varGamma (j(\alpha +\rho )+1)}.\end{array}\]]]>

If f is directly Riemann integrable, then, for everyl∈N,E[(∑k≥0f(t−Sk)1{Sk≤t})l]=o(1),t→∞.

Iff(t)=O(P{ξ>t})t\})$]]>, ast→∞, then, for everyl∈N,E[(∑k≥0f(t−Sk)1{Sk≤t})l]=O(1),t→∞.

The formula for the moments of fractionally integrated inverse stable subordinator (the second equality in (25)) is known, see for example (3.65) in [7] or (2.17) in [10].

In case ρ∈(−α,0] this result is just Lemma 5.3 in [10]. A perusal of the proof of the aforementioned lemma shows that without any modifications the constraint ρ∈(−α,0] can be replaced by ρ>−α-\alpha $]]>. □

If l=1 and the distribution of S1 is non-lattice the claim follows from the classical key renewal theorem. If l=1 and the distribution of S1 is lattice, the claim still holds, see the penultimate centered formula on p. 94 in [13]. In particular, this means
0≤m1(t):=E[∑k≥0|f(t−Sk)|1{Sk≤t}]≤M1,t≥0,
for some constant M1>00$]]>. Applying formula (5.19) in [10] we obtain
ml(t):=E[(∑k≥0|f(t−Sk)|1{Sk≤t})l]=∫0trl(t−y)dU(y),
where U(y)=∑k≥0P{Sk≤y}, y≥0 is the renewal function and
rl(t)=∑j=0l−1vj|f(t)|l−j(t)mj(t),
for some real constants vj. We proceed by induction. Assume that we know
mj(t)→0,t→∞,j=1,…,l−1,
in particular,
0≤mj(t)≤Mj,t≥0,j=1,…,l−1.
Then
|rl(t)|≤∑j=0l−1Mj|vj||f(t)|l−j,t≥0,
and the right-hand side is directly Riemann integrable. By the same reasoning as in case l=1 we obtain
ml(t)→0,t→∞,
by the key renewal theorem. □

Again, let us consider the case l=1 first. Put Z(t):=t−Sν(t)−1 and note that
E[∑k≥0f(t−Sk)1{Sk≤t}]=Eg(Z(t)),
where g(t):=f(t)/P{ξ>t}t\}$]]>. Since g is bounded, we have Eg(Z(t))=O(1), as t→∞. For arbitrary l∈N the result follows from (26) and (27) by induction in the same vein as in the proof of part (ii). □

In the next lemma we give an upper bound on the moments of random process with immigration under assumption (2). Recall the notation Y(t)=∑k≥0Xk+1(t−Sk)1{Sk≤t}.

Assume that (2) holds for someα∈(0,1).

Suppose there exists a locally bounded measurable functionb:[0,∞)→[0,∞)such thatb(t)∼tβℓb(t),t→∞,for someβ>−α-\alpha $]]>andℓbslowly varying at infinity. If for everyl∈NE[|X(t)|l]≤bl(t),t≥0,then for everyl∈Nwe haveE[|Y(t)|l]=O((b(t)P{ξ>t})l),t→∞.t\}}\bigg)}^{l}\bigg),\hspace{1em}t\to \infty .\]]]>

Suppose that for everyl∈Nthere exists a directly Riemann integrable functionbl:[0,∞)→[0,∞)such thatE[|X(t)|l]≤bl(t),t≥0.Then, for everyl∈NE[|Y(t)|l]=o(1),t→∞.

Suppose that for every fixedl∈Nwe haveE[|X(t)|l]=O(P{ξ>t}),t→∞.t\}\big),\hspace{1em}t\to \infty .\]]]>Then, for everyl∈NE[|Y(t)|l]=O(1),t→∞.

Put al(t):=E[|X(t)|l] for l∈N and
Z(t):=∑k≥0|Xk+1(t−Sk)|1{Sk≤t},t≥0.
Clearly, E[|Y(t)|l]≤E[[Z(t)]l] for all t≥0 and l∈N. We prove (28), (29) and (30) with E[Z(t)l] replacing E[|Y(t)|l] in the left-hand sides. From the definition of random process with immigration it follows that
Z(t)=d|X(t)|+Zˆ(t−ξ)1{ξ≤t},t≥0,
where Zˆ(t)=dZ(t) for every fixed t≥0 and Zˆ(t) is independent of (X,ξ) in the right-hand side. Taking expectations we obtain
E[Z(t)]=a1(t)+E[Z(t−ξ1)]1{ξ≤t},t≥0,
whilst, for l≥2, we have
E[Z(t)l]=al(t)+∑j=1l−1ljE[|X(t)|l−j(Zˆ(t−ξ))j1{ξ≤t}]+E[Z(t−ξ)l1{ξ≤t}]=al(t)+∑j=1l−1lj∫0∞∫0tzl−jE[Z(t−y)j]P{|X(t)|∈dz,ξ∈dy}+E[Z(t−ξ)l1{ξ≤t}]≤al(t)+∑j=1l−1ljal−j(t)sup0≤y≤tE[Z(y)j]+E[Z(t−ξ)l1{ξ≤t}].

From Lemma 1(i) and formula (31) using the inequality a1(t)≤b(t), t≥0, we obtain
E[Z(t)]=O(b(t)P{ξ>t}),t→∞.t\}}\bigg),\hspace{1em}t\to \infty .\]]]>
Thus, (28) holds for l=1. We proceed by induction. Assume that for every j=1,…,l−1 there exists Cj>00$]]> such that
E[Z(t)j]≤Cj(b(t)P{ξ>t})j,t≥0.t\}}\bigg)}^{j},\hspace{1em}t\ge 0.\]]]>
This implies
sup0≤y≤tE[Z(y)j]≤Cjsup0≤y≤t(b(y)P{ξ>y})j∼Cj(b(t)P{ξ>t})j,y\}}\bigg)}^{j}\sim C_{j}{\bigg(\frac{b(t)}{\mathbb{P}\{\xi >t\}}\bigg)}^{j},\]]]>
where the last relation follows from the regular variation of t↦b(t)/P{ξ>t}t\}$]]> with positive index β+α. Hence, from equation (32) and the inequalities aj(t)≤bj(t), t≥0, j=1,…,l−1, we deduce
E[Z(t)l]≤C′bl(t)(P{ξ>t})l−1+E[Z(t−ξ)l1{ξ≤t}],t≥0,t\})}^{l-1}}+\mathbb{E}\big[Z{(t-\xi )}^{l}\mathbb{1}_{\{\xi \le t\}}\big],\hspace{1em}t\ge 0,\]]]>
for some C′=Cl′>00$]]>. Since t↦C′bl(t)/(P{ξ>t})l−1t\})}^{l-1}$]]> is regularly varying with index l(β+α)−α>−α-\alpha $]]>, Lemma 1(i) yields
E[Z(t)l]=O((b(t)P{ξ>t})l),t→∞.t\}}\bigg)}^{l}\bigg),\hspace{1em}t\to \infty .\]]]>

Arguing by induction as in the proof of case (i) we see from formulae (31) and (32) that
E[Z(t)l]≤bˆl′(t)+E[Z(t−ξ1)l1{ξ≤t}],t≥0,
for a directly Riemann integrable function bˆl′. The claim follows from the key renewal theorem.

For l=1 the claim follows from Lemma 1(iii) and formula (31). Using inductive argument once again we obtain from (32) that
E[Z(t)l]≤C″P{ξ>t}+E[Z(t−ξ1)l1{ξ≤t}],t≥0,t\}+\mathbb{E}\big[Z{(t-\xi _{1})}^{l}\mathbb{1}_{\{\xi \le t\}}\big],\hspace{1em}t\ge 0,\]]]>
for some C″=Cl″>00$]]> and the claim follows from Lemma 1(iii).

□
ReferencesAlsmeyer, G., Iksanov, A., Marynych, A.: Functional limit theorems for the number of occupied boxes in the Bernoulli sieve. Alsmeyer, G., Iksanov, A., Meiners, M.: Power and exponential moments of the number of visits and related quantities for perturbed random walks. Bingham, N.H., Goldie, C.M., Teugels, J.L.: Chow, Y.S., Teicher, H.: Feller, W.: Iksanov, A.: Functional limit theorems for renewal shot noise processes with increasing response functions. Iksanov, A.: Iksanov, A., Jedidi, W., Bouzeffour, F.: Functional limit theorems for the number of busy servers in a G/G/∞ queue. Preprint available at https://arxiv.org/pdf/1610.08662.pdfIksanov, A., Kabluchko, Z., Marynych, A.: Weak convergence of renewal shot noise processes in the case of slowly varying normalization. Iksanov, A., Marynych, A., Meiners, M.: Limit theorems for renewal shot noise processes with eventually decreasing response functions. Iksanov, A., Marynych, A., Meiners, M.: Asymptotics of random processes with immigration I: Scaling limits. Iksanov, A., Marynych, A., Meiners, M.: Asymptotics of random processes with immigration II: Convergence to stationarity. Iksanov, A.M., Marynych, A.V., Vatutin, V.A.: Weak convergence of finite-dimensional distributions of the number of empty boxes in the Bernoulli sieve. Iksanov, A., Kabluchko, Z., Marynych, A., Shevchenko, G.: Fractionally integrated inverse stable subordinators. Klüppelberg, C., Kühn, C.: Fractional Brownian motion as a weak limit of Poisson shot noise processes—with applications to finance. Marynych, A.: A note on convergence to stationarity of random processes with immigration. Mikosch, T., Resnick, S.: Activity rates with very heavy tails. Resnick, S., Rootzén, H.: Self-similar communication models and very heavy tails.