Let (ξ1,η1), (ξ2,η2),… be independent identically distributed N2-valued random vectors with arbitrarily dependent components. The sequence (Θk)k∈N defined by Θk=Πk−1·ηk, where Π0=1 and Πk=ξ1·…·ξk for k∈N, is called a multiplicative perturbed random walk. Arithmetic properties of the random sets {Π1,Π2,…,Πk}⊂N and {Θ1,Θ2,…,Θk}⊂N, k∈N, are studied. In particular, distributional limit theorems for their prime counts and for the least common multiple are derived.

Least common multiplemultiplicative perturbed random walkprime counts11A0560F0511K65National Research Foundation of Ukraine2020.02/0014The research was supported by the National Research Foundation of Ukraine (project 2020.02/0014 “Asymptotic regimes of perturbed random walks: on the edge of modern and classical probability”). Introduction

Let (ξ1,η1), (ξ2,η2),… be independent copies of an N2-valued random vector (ξ,η) with arbitrarily dependent components. Denote by (Πk)k∈N0 (as usual, N0:=N∪{0}) the standard multiplicative random walk defined by
Π0:=1,Πk=ξ1·ξ2⋯ξk,k∈N.
A multiplicative perturbed random walk is the sequence (Θk)k∈N given by
Θk:=Πk−1·ηk,k∈N.
Note that if P{η=ξ}=1, then Πk=Θk for all k∈N. If P{ξ=1}=1, then (Θk)k∈N is just a sequence of independent copies of a random variable η. In this article we investigate some arithmetic properties of the random sets (Πk)k∈N and (Θk)k∈N.

To set the scene, we introduce first some necessary notation. Let P denote the set of prime numbers. For an integer n∈N and p∈P, let λp(n) denote the multiplicity of prime p in the prime decomposition of n, that is,
n=∏p∈Ppλp(n).
For every p∈P, the function λp:N↦N0 is totally additive in the sense that
λp(mn)=λp(m)+λp(n),p∈P,m,n∈N.
The set of functions (λp)p∈P is a basic brick from which many other arithmetic functions can be constructed. For example, with GCD(A) and LCM(A) denoting the greatest common divisor and the least common multiple of a set A⊂N, respectively, we have
GCD(A)=∏p∈Ppminn∈Aλp(n)andLCM(A)=∏p∈Ppmaxn∈Aλp(n).

The listed arithmetic functions applied either to A={Π1,…,Πn} or A={Θ1,…,Θn} are the main objects of investigation in the present paper. From the additivity of λp we infer
Sk(p):=λp(Πk)=∑j=1kλp(ξj),p∈P,k∈N0,
and
Tk(p):=λp(Θk)=∑j=1k−1λp(ξj)+λp(ηk),p∈P,k∈N.
Fix any p∈P. Formulae (1) and (2) demonstrate that S(p):=(Sk(p))k∈N0 is a standard additive random walk with the generic step λp(ξ), whereas the sequence T(p):=(Tk(p))k∈N is a particular instance of an additive perturbed random walk, see [6], generated by the pair (λp(ξ),λp(η)).

As is suggested by (1) and (2) the first step in the analysis of S(p) and T(p) should be the derivation of the joint distribution (λp(ξ),λp(η))p∈P. The next lemma confirms that the finite-dimensional distributions of the infinite vector (λp(ξ),λp(η))p∈P, are expressible via the probability mass function of (ξ,η). However, the obtained formulae are not easy to handle except some special cases. For i,j∈N, put
ui:=P{ξ=i},vj:=P{η=j},wi,j:=P{ξ=i,η=j}.

This follows from
P{λq(ξ)≥kq,λq(η)≥ℓq,q∈P,q≤p}=P{∏q≤p,q∈Pqkqdividesξ,∏q≤p,q∈Pqℓqdividesη}=∑i,j=1∞wKi,Lj.
Obviously, if ξ and η are independent, then
∑i,j=1∞wKi,Lj=(∑i=1∞uKi)(∑j=1∞vLj).
□

We proceed with the series of examples.

For α>1, let P{ξ=k}=(ζ(α))−1k−α, k∈N, where ζ is the Riemann zeta-function. For k∈N, p1,…,pk∈P and j1,…,jk∈N0 we have
P{λp1(ξ)≥j1,…,λpk(ξ)≥jk}=P{p1j1⋯pkjkdividesξ}=∑i=1∞P{ξ=(p1j1⋯pkjk)i}=(p1j1⋯pkjk)−α=p1−αj1⋯pk−αjk.
Thus, (λp(ξ))p∈P are mutually independent and λp(ξ) has a geometric distribution on N0 with parameter p−α, for every fixed p∈P.

For β∈(0,1), let P{ξ=k}=βk−1(1−β), k∈N. Then
P{λp(ξ)≥k}=1−ββ∑j=1∞βpkj=(1−β)(βpk−1)1−βpk,k∈N0.

Let Poi(λ) be a random variable with the Poisson distribution with parameter λ and put
P{ξ=k}=P{Poi(λ)=k|Poi(λ)≥1}=(eλ−1)−1λk/k!,k∈N.
Then
P{λp(ξ)≥k}=(eλ−1)−1∑j=1∞λpkj/(pkj)!=(0Fpk(;1pk,2pk,…,pk−1pk;(λpk)pk)−1),
where 0Fpk is the generalized hypergeometric function, see Chapter 16 in [10].

In all examples above, the distribution of λp(ξ) for every fixed p∈P is extremely light-tailed. It is not that difficult to construct ‘weird’ distributions where all λp(ξ) have infinite expectations.

Let (gp)p∈P be any probability distribution supported by P, gp>0, and (tk)k∈N0 any probability distribution on N such that ∑k=1∞ktk=∞ and tk>0. Define a probability distribution h on Q:=⋃p∈P{p,p2,…} by
h({pk})=gptk,p∈P,k∈N.
If ξ is a random variable with distribution h, then
P{λp(ξ)≥k}=gp∑j=k∞tj,k∈N,p∈P,
which implies E[λp(ξ)]=gp∑k=1∞ktk=∞, p∈P.

This example can be modified by taking g:=∑p∈Pgp<1 and charging all points of N∖Q (this set contains 1 and all integers having at least two different prime factors) with arbitrary positive masses of the total weight 1−g. The obtained probability distribution charges all points of N and still possesses the property that all λp’s have infinite expectations.

Let X be a random variable taking values in N. Since
logX=∑p∈Pλp(X)logp,
we conclude that E[(λp(X))k]<∞, for all p∈P, whenever E[logkX]<∞, k∈N. It is also clear that the converse implication is false in general. However, when k=1 the inequality E[logX]<∞ is in fact equivalent to ∑p∈PE[λp(X)]logp<∞. As we have seen in the above examples, checking that E[(λp(X))k]<∞ might be a much more difficult task than proving a stronger assumption E[logkX]<∞. Thus, we shall mostly work under moment conditions on logξ and logη.

Our standing assumption throughout the article is
μξ:=E[logξ]<∞,
which, by the above reasoning, implies E[λp(ξ)]<∞, p∈P.

From Donsker’s invariance principle we immediately obtain the following proposition. Let D:=D([0,∞),R) be the Skorokhod space endowed with the standard J1-topology.

Assume thatE[log2ξ]∈(0,∞). Then,((S⌊ut⌋(p)−utE[λp(ξ)]t)u≥0)p∈P⟹((Wp(u))u≥0)p∈P,t→∞,on the product spaceDN, where, for alln∈Nand allp1<p2<⋯<pn,pi∈P,i≤n,((Wp1(u))u≥0,…,(Wpn(u))u≥0)is an n-dimensional centered Wiener process with covariance matrixC=‖Ci,j‖1≤i,j≤ngiven byCi,j=Cj,i=Cov(λpi(ξ),λpj(ξ)).

According to the proof of Proposition 1.3.13 in [6], see pp. 28–29 therein, the following holds true for the perturbed random walks T(p), p∈P.

Assume thatE[log2ξ]∈(0,∞)andlimt→∞t2P{λp(η)≥t}=0,p∈P.Then,((T⌊ut⌋(p)−utE[λp(ξ)]t)u≥0)p∈P⟹((Wp(u))u≥0)p∈P,t→∞,on the product spaceDN.

Since P{λp(η)logp≥t}≤P{logη≥t}, the condition
limt→∞t2P{logη≥t}=0
is clearly sufficient for (5).

From the continuous mapping theorem under the assumptions of Proposition 2 we infer
((max1≤k≤⌊ut⌋(Tk(p)−kE[λp(ξ)])t)u≥0)p∈P⟹((sup0≤v≤uWp(v))u≥0)p∈P,t→∞,
see Proposition 1.3.13 in [6].

Formula (7), for a fixed p∈P, belongs to the realm of limit theorems for the maximum of a single additive perturbed random walk. This circle of problems is well-understood, see Section 1.3.3 in [6] and [7], in the situation when the underlying additive standard random walk is centered and attracted to a stable Lévy process. In our setting the perturbed random walks (Tk(p))k∈N and (Tk(q))k∈N are dependent whenever p,q∈P, p≠q, which make derivation of the joint limit theorems harder and leads to various asymptotic regimes.

Note that (5) implies E[λp(η)]<∞ and (6) implies E[logη]<∞. Theorem 5 below tells us that under such moment conditions and assuming also E[log2ξ]<∞ the maxima max1≤k≤nTk(p), p∈P, of noncentered perturbed random walks T(p) have the same behavior as Sn(p), p∈P as n→∞.

Assume thatE[log2ξ]<∞andE[λp(η)]<∞,p∈P. Suppose further thatP{ξis divisible byp}=P{λp(ξ)>0}>0,p∈P.Then, ast→∞,((max1≤k≤⌊tu⌋Tk(p)−E[λp(ξ)]tut1/2)u≥0)p∈P⟶f.d.d.((Wp(u))u≥0)p∈P.Moreover, if also (5) holds for allp∈P, then (9) holds on the product spaceDN.

If (8) holds only for some P0⊆P, then (9) holds with P0 instead of P.

In the next result we shall assume that η dominates ξ in a sense that the asymptotic behavior of max1≤k≤nTk(p) is regulated by the perturbations (λp(ηk))k≤n for all p∈P0, where P0 is a finite subset of prime numbers and those p’s dominate all other primes.

Assume (4). Suppose further that there exists a finite setP0⊆P,d:=|P0|, such that the distributional tail of(λp(η))p∈P0is regularly varying at infinity in the following sense. For some positive function(a(t))t>0and a measure ν satisfyingν({x∈Rd:‖x‖≥r})=c·r−α,c>0,α∈(0,1), it holdstP{(a(t))−1(λp(η))p∈P0∈·}⟶vν(·),t→∞,on the space of locally finite measures on(0,∞]dendowed with the vague topology. Then((max1≤k≤⌊tu⌋Tk(p)a(t))u≥0)p∈P0⟶f.d.d.((Mp(u))u≥0)p∈P0,t→∞,where((Mp(u))u≥0)p∈P0is a multivariate extreme process defined by(Mp(u))p∈P0=supk:tk≤uyk,u≥0.Here the pairs(tk,yk)are the atoms of a Poisson point process on[0,∞)×(0,∞]dwith the intensity measureLEB⊗νand the supremum is taken coordinatewise. Moreover, suppose thatE[λp(η)]<∞, forp∈P∖P0. Then((max1≤k≤⌊tu⌋Tk(p)a(t))u≥0)p∈P∖P0⟶f.d.d.0,t→∞.

We shall deduce Theorems 5 and 6 in Section 3 by proving general limit results for coupled perturbed random walks.

Limit theorems for the <inline-formula id="j_vmsta241_ineq_165"><alternatives><mml:math>
<mml:mi mathvariant="normal">LCM</mml:mi>
<mml:mspace width="0.1667em"/></mml:math><tex-math><![CDATA[$\mathrm{LCM}\hspace{0.1667em}$]]></tex-math></alternatives></inline-formula>

The results from the previous section will be applied below to the analysis of
Pn:=LCM({Π1,Π2,…,Πn})andTn:=LCM({Θ1,Θ2,…,Θn}).
A moment’s reflection shows that the analysis of Pn is trivial. Indeed, by definition, Πn−1 divides Πn and thereupon Pn=Πn for n∈N. Thus, assuming that σξ2:=Var(logξ)∈(0,∞), an application of the Donsker functional limit theorem yields
(logP⌊tu⌋−μξtut1/2)u≥0⟹(σξW(u))u≥0,t→∞,
on the Skorokhod space D, where (W(u))u≥0 is a standard Brownian motion and μξ=E[logξ] was defined in (4).

A simple structure of the sequence (Pn)n∈N breaks down completely upon introducing the perturbations (ηk), which makes the analysis of (Tn)n∈N a much harder problem. As an illustration, consider the case ξ=1 in which
Tn=LCM(η1,…,ηn).
Thus, the problem encompasses, as a particular case, the investigation of the LCM of an independent sample. This itself constitutes a highly nontrivial challenge. Note that
logTn=log∏p∈Ppmax1≤k≤n(λp(ξ1)+⋯+λp(ξk−1)+λp(ηk))=∑p∈Pmax1≤k≤nTk(p)logp,
which shows that the asymptotics of Tn is intimately connected with the behavior of max1≤k≤nTk(p), p∈P.

As one can guess from Theorem 5 in a ‘typical’ situation relation (14) holds with logT⌊tu⌋ replacing logP⌊tu⌋. The following heuristics suggest the right form of assumptions ensuring that perturbations (ηk)k∈N have an asymptotically negligible impact on logTn. Take a prime p∈P. Its contribution to logTn (up to a factor logp) is max1≤k≤nTk(p). According to Theorem 5, this maximum is asymptotically the same as Sn(p). However, as p gets large, the mean E[λp(ξ)] of the random walk Sn−1(p) becomes small because of the identity
∑p∈PE[λp(ξ)]logp=E[logξ]<∞.
Thus, for large p∈P, the remainder max1≤k≤nTk(p)−Sn−1(p) can, in principle, become larger than Sn−1(p) itself if the tail of λp(η) is sufficiently heavy. In order to rule out such a possibility, we introduce the deterministic sets
P1(n):={p∈P:P{λp(ξ)>0}≥n−1/2}andP2(n):=P∖P1(n),
and bound the rate of growth of max1≤k≤nλp(ηk) for all p∈P2(n). It is important to note that under the assumption (8) it holds
minP2(n)=min{p∈P:p∈P2(n)}=min{p∈P:P{λp(ξ)>0}<n−1/2}→∞,n→∞.
Therefore, if E[logξ]<∞ and (8) holds, then
limn→∞∑p∈P2(n)E[λp(ξ)]logp=0.

AssumeE[log2ξ]<∞,E[logη]<∞, (8) and the following two conditions:∑p∈PE[((λp(η)−λp(ξ))+)2]logp<∞and∑p∈P2(n)E[(λp(η)−λp(ξ))+]logp=o(n−1/2),n→∞.Then(logT⌊tu⌋−μξtut1/2)u≥0⟶f.d.d.(σξW(u))u≥0,t→∞,whereμξ=E[logξ]<∞,σξ2=Var[logξ]and(W(u))u≥0is a standard Brownian motion.

If E[log2η]<∞, then (17) holds true. Indeed, since we assume E[log2ξ]<∞,
E[∑p∈P((λp(η)−λp(ξ))+)2logp]≤E[∑p∈P(λp2(η)+λp2(ξ))logp]≤1log2E[(∑p∈Pλp(η)logp)2]+E[(∑p∈Pλp(ξ)logp)2]=1log2(E[log2η]+E[log2ξ])<∞.
The condition (18) can be replaced by a stronger one which only involves the distribution of η, namely
∑p∈P2(n)E[λp(η)]logp=o(n−1/2),n→∞.
Taking into account (16) and the fact that E[logη]<∞, the assumption (20) is nothing else but a condition of the speed of convergence of the series
∑p∈PE[λp(η)]logp=E[logη].

In the settings of Example 1, let ξ and η be arbitrarily dependent with
P{ξ=k}=1ζ(α)kα,P{η=k}=1ζ(β)kβ,k∈N,
for some α,β>1. Note that E[log2ξ]<∞ and E[log2η]<∞. Direct calculations show that
P1(n)={p∈P:p−α≥n−1/2}={p∈P:p≤n1/(2α)},P2(n)={p∈P:p>n1/(2α)}.
From the chain of relations
E[λp(η)]=∑j≥1P{λp(η)≥j}=∑j≥1p−βj=p−β1−p−β≤2p−β,
and using the notation π(x) for the number of primes smaller than x, we obtain
∑p∈P2(n)E[λp(η)]logp≤2∑p∈P,p>n1/(2α)logppβ=2∫(n1/(2α),∞)logxxβdπ(x)∼2∫n1/(2α)∞logxxβdxlogx=2n(1−β)/(2α)β−1,n→∞.
Here the asymptotic equivalence follows from the prime number theorem and integration by parts, see, for example Eq. (16) in [3]. Thus, (20) holds if
12+1−β2α<0⟺α+1<β.

In the settings of Theorem 6 the situation is much simpler in a sense that almost no extra assumptions are needed to derive a limit theorem for Tn.

Under the same assumptions as in Theorem6and assuming additionally that∑p∈P∖P0E[λp(η)]logp<∞,it holds(logT⌊tu⌋a(t))u≥0⟶f.d.d.(∑p∈P0Mp(u)logp)u≥0,t→∞.

Note that in Theorem 9 it is allowed to take ξ=1, which yields the following limit theorem for the LCM of an independent integer-valued random variables.

Under the same assumptions on η as in Theorem6, it holds(logLCM(η1,η2,…,η⌊tu⌋)a(t))u≥0⟶f.d.d.(∑p∈P0Mp(u)logp)u≥0,t→∞.

The results presented in Theorems 7 and 9 constitute a contribution to a popular topic in probabilistic number theory, namely, the asymptotic analysis of the LCM of various random sets. For random sets comprised of independent random variables uniformly distributed on {1,2,…,n} this problem has been addressed in [2–5, 9]. Some models with a more sophisticated dependence structure have been studied [1] and [8].

Limit theorems for coupled perturbed random walks

Theorems 5 and 6 will be derived from general limit theorems for the maxima of arbitrary additive perturbed random walks indexed by some parameters ranging in a countable set in the situation when the underlying additive standard random walks are positively divergent and attracted to a Brownian motion.

Let A be a countable or finite set of real numbers and
((X1(r),Y1(r)))r∈A,((X2(r),Y2(r)))r∈A,…
be independent copies of an R2×|A| random vector (X(r),Y(r))r∈A with arbitrarily dependent components. For each r∈A, the sequence (Sk∗(r))k∈N0 given by
S0∗(r):=0,Sk∗(r):=X1(r)+⋯+Xk(r),k∈N,
is an additive standard random walk. For each r∈A, the sequence (Tk∗(r))k∈N defined by
Tk∗(r):=Sk−1∗(r)+Yk(r),k∈N,
is an additive perturbed random walk. The sequence ((Tk∗(r))k∈N)r∈A is a collection of (generally) dependent additive perturbed random walks.

Assume that, for eachr∈A,μ(r):=E[X(r)]∈(0,∞),Var[X(r)]∈[0,∞)andE[Y(r)]<∞. Then((max1≤k≤⌊tu⌋Tk∗(r)−μ(r)tut1/2)u≥0)r∈A⟶f.d.d.((Wr(u))u≥0)r∈A,t→∞,where, for alln∈Nand arbitraryr1<r2<⋯<rnwithri∈A,i≤n,((Wr1(u))u≥0,…,(Wrn(u))u≥0)is an n-dimensional centered Wiener process with covariance matrixC=‖Ci,j‖1≤i,j≤nwith the entriesCi,j=Cj,i=Cov(X(ri),X(rj)).

We shall prove an equivalent statement that, as t→∞,
((max0≤k≤⌊tu⌋Tk+1∗(r)−μ(r)tut1/2)u≥0)r∈A⟶f.d.d.((Wr(u))u≥0)r∈A,
which differs from (23) by a shift of the subscript k. By the multidimensional Donsker theorem,
((S⌊tu⌋∗(r)−μ(r)tut1/2)u≥0)r∈A⟹((Wr(u))u≥0)r∈A,t→∞,
in the product topology of DN. Fix any r∈A and write
max0≤k≤⌊tu⌋Tk+1∗(r)−μ(r)tu=max0≤k≤⌊tu⌋(Sk∗(r)−S⌊tu⌋∗(r)+Yk+1(r))+S⌊tu⌋∗(r)−μ(r)tu.
In view of (24) the proof is complete once we can show that
n−1/2(max0≤k≤n(Sk∗(r)−Sn∗(r)+Yk+1(r)))→P0,n→∞.
Let (X0(r),Y0(r)) be a copy of (X(r),Y(r)) which is independent of the vector (Xk(r),Yk(r))k∈N. Since the collection
((X1(r),Y1(r)),…,(Xn+1(r),Yn+1(r)))
has the same distribution as
((Xn(r),Yn(r)),…,(X0(r),Y0(r))),
the variable
max0≤k≤n(Sk∗(r)−Sn∗(r)+Yk+1(r))
has the same distribution as
max(Y0(r),max0≤k≤n−1(−Sk∗(r)+Yk+1(r)−Xk+1(r))).

By assumption, E(−S1∗(r))∈(−∞,0) and E(Y(r)−X(r))+<∞. Hence, by Theorem 1.2.1 and Remark 1.2.3 in [6],
limk→∞(−Sk∗(r)+Yk+1(r)−Xk+1(r))=−∞a.s.
As a consequence, the a.s. limit
limn→∞max(Y0(r),max0≤k≤n−1(−Sk∗(r)+Yk+1(r)−Xk+1(r))=max(Y0(r),maxk≥0(−Sk∗(r)+Yk+1(r)−Xk+1(r))
is a.s. finite. This completes the proof of (26). □

Proposition 3 tells us that fluctuations of max1≤k≤⌊tu⌋Tk∗(r) on the level of finite-dimensional distributions are driven by the Brownian fluctuations of S⌊tu⌋∗(r). According to formula (25), a functional version of this statement would be true if we could check that, for every fixed T>0,
t−1/2supu∈[0,T]max0≤k≤⌊tu⌋(Sk∗(r)−S⌊tu⌋∗(r)+Yk+1(r))→P0,t→∞.
But the left-hand side is bounded from below by
t−1/2supu∈[0,T]Y⌊tu⌋+1(r)=t−1/2max0≤k≤⌊Tt⌋+1Yk(r).
Under the sole assumption E[Y(r)]<∞ this maximum does not converge to zero in probability, as t→∞. Thus, under the standing assumptions of Proposition 3 the functional convergence does not hold.

To deduce the finite-dimensional convergence (9) we apply Proposition 3 with A=P, X(p)=λp(ξ) and Y(p)=λp(η). The assumption (8) in conjunction with E[log2ξ]<∞ implies that E[λp(ξ)]∈(0,∞) and Var[λp(ξ)]∈[0,∞), for all p∈P.

Suppose that (5) holds true for all p∈P. Fix p∈P, t>0, and note that by the subadditivity of the supremum and the fact that (Sk(p))k∈N0 is nondecreasing we have
S⌊tu⌋−1(p)≤max1≤k≤⌊tu⌋Tk(p)≤S⌊tu⌋−1(p)+max1≤k≤⌊tu⌋λp(ηk),u≥0.
Assumption (5) implies that, for every fixed T>0,
t−1/2supu∈[0,T]max1≤k≤⌊tu⌋λp(ηk)=t−1/2max1≤k≤⌊tT⌋λp(ηk)→P0,t→∞.
By Proposition 1 and taking into account (27) this means that (9) holds true on the product space DN. □

AssumeE[X(r)]<∞,r∈A. Assume further that there exists a finite setA0⊆A,d:=|A0|, such that the distributional tail of(Y(r))r∈A0is regularly varying at infinity in the following sense. For some positive function(a(t))t>0and a measure ν satisfyingν({x∈Rd:‖x‖≥r})=c·r−α,c>0,α∈(0,1), it holdstP{(a(t))−1(Y(r))r∈A0∈·}⟶vν(·),t→∞,on the space of locally finite measures on(0,∞]dendowed with the vague topology. Then((max1≤k≤⌊tu⌋Tk∗(r)a(t))u≥0)r∈A0⟶f.d.d.((Mr(u))u≥0)r∈A0,t→∞,where((Mr(u))u≥0)r∈A0is defined as in (12). IfE[|Y(r)|]<∞, forr∈A∖A0, then also((max1≤k≤⌊tu⌋Tk∗(r)a(t))u≥0)r∈A∖A0⟶f.d.d.0,t→∞.

According to Corollary 5.18 in [11]
((max1≤k≤⌊tu⌋Yk(r)a(t))u≥0)r∈A0⟹((Mr(u))u≥0)r∈A0,t→∞,
in the product topology of DN. The function (a(t))t≥0 is regularly varying at infinity with index 1/α>1. Thus, by the law of large numbers, for all r∈A, (min1≤k≤⌊tu⌋Sk−1∗(r)a(t))u≥0⟶f.d.d.0,t→∞,(max1≤k≤⌊tu⌋Sk−1∗(r)a(t))u≥0⟶f.d.d.0,t→∞, and (29) follows from the inequalities
min1≤k≤⌊tu⌋Sk−1∗(r)+max1≤k≤⌊tu⌋Yk(r)≤max1≤k≤⌊tu⌋Tk∗(r)≤max1≤k≤⌊tu⌋Sk−1∗(r)+max1≤k≤⌊tu⌋Yk(r).
In view of (31) and (32), to prove (30) it suffices to check that
((max1≤k≤⌊tu⌋Yk(r)a(t))u≥0)⟶f.d.d.0,t→∞,
for every fixed r∈A∖A0. This, in turn, follows from
Yn(r)n⟶a.s.0,n→∞,r∈A∖A0,
which is a consequence of the assumption E[|Y(r)|]<∞, r∈A∖A0, and the Borel–Cantelli lemma. □

Follows immediately from Proposition 4 applied with A=P, X(p)=λp(ξ) and Y(p)=λp(η). □

Proof of Theorem <xref rid="j_vmsta241_stat_013">7</xref>

We aim at proving that
∑p∈P(max1≤k≤nTk(p)−Sn−1(p))logpn⟶P0,n→∞,
which together with the relation
∑p∈PSn(p)logp=logΠn=logPn,n∈N,
implies Theorem 7 by the Slutsky lemma and (14).

Let (ξ0,η0) be an independent copy of (ξ,η) which is also independent of (ξn,ηn)n∈N. By the same reasoning as we have used in the proof of (26) we obtain
(max1≤k≤nTk(p)−Sn−1(p))p∈P=d(max(λp(η0),max1≤k<n(λp(ηk)−λp(ξk)−Sk−1(p))))p∈P.
Taking into account
∑p∈Pλp(η0)logp=logη0,
we see that (33) is a consequence of
∑p∈Pmax1≤k<n(λp(ηk)−λp(ξk)−Sk−1(p))+logpn⟶P0,n→∞.
Since, for every fixed p∈P,
maxk≥1(λp(ηk)−λp(ξk)−Sk−1(p))+<∞a.s.
by assumption (8), it suffices to check that, for every fixed ε>0,
limM→∞lim supn→∞P{∑p∈P,p>Mmax1≤k<n(λp(ηk)−λp(ξk)−Sk−1(p))+logp>εn}.
In order to check (37), we divide the sum into two disjoint parts with summations over P1(n) and P2(n). For the first sum, by Markov’s inequality, we obtain
P{∑p∈P1(n),p>Mmax1≤k<n(λp(ηk)−λp(ξk)−Sk−1(p))+logp>εn/2}≤2εn∑p∈P1(n),p>ME(max1≤k<n(λp(ηk)−λp(ξk)−Sk−1(p))+)logp≤2εn∑p∈P1(n),p>Mlogp∑k≥1E(λp(ηk)−λp(ξk)−Sk−1(p))+=2εn∑p∈P1(n),p>Mlogp∑j≥1P{λp(η)−λp(ξ)=j}∑k≥1E(j−Sk−1(p))+≤2εn∑p∈P1(n),p>Mlogp∑j≥1jP{λp(η)−λp(ξ)=j}∑k≥0P{Sk(p)≤j}≤2εn∑p∈P1(n),p>Mlogp∑j≥1jP{λp(η)−λp(ξ)=j}2jE[(λp(ξ)∧j)],
where the last estimate is a consequence of Erickson’s inequality for renewal functions, see Eq. (6.5) in [6]. Further, since for p∈P1(n),
E[(λp(ξ)∧j)]≥P{λp(ξ)≥1}=P{λp(ξ)>0}≥n−1/2,
we obtain
P{∑p∈P1(n),p>Mmax1≤k<n(λp(ηk)−λp(ξk)−Sk−1(p))+logp>εn/2}≤4ε∑p∈P1(n),p>MlogpE[((λp(η)−λp(ξ))+)2]≤4ε∑p∈P,p>MlogpE[((λp(η)−λp(ξ))+)2].
The right-hand side converges to 0, as M→∞ by (17). For the sum over P2(n) the derivation is simpler. By Markov’s inequality
P{∑p∈P2(n),p>Mmax1≤k<n(λp(ηk)−λp(ξk)−Sk−1(p))+logp>εn/2}≤2εnE[∑p∈P2(n),p>Mmax1≤k<n(λp(ηk)−λp(ξk)−Sk−1(p))+logp]≤2nεnE[∑p∈P2(n),p>M(λp(ηk)−λp(ξk))+logp],
and the right-hand side tends to zero as n→∞ in view of (18). The proof is complete.

Proof of Theorem <xref rid="j_vmsta241_stat_016">9</xref>

From Theorem 6 with the aid of the continuous mapping theorem we conclude that
(∑p∈P0max1≤k≤⌊tu⌋Tk(p)logpa(t))u≥0⟶f.d.d.(∑p∈P0Mp(u)logp)u≥0,
as t→∞. It suffices to check
(∑p∈P∖P0max1≤k≤⌊tu⌋Tk(p)logpa(t))u≥0⟶f.d.d.0,t→∞.
Since (a(t)) is regularly varying at infinity, (38) follows from
∑p∈P∖P0E[max1≤k≤nTk(p)]logpa(n)→0,n→∞,
by Markov’s inequality. To check the latter, note that
∑p∈P∖P0E[max1≤k≤nTk(p)]logp≤∑p∈P∖P0E[Sn−1(p)+max1≤k≤nλp(ηk)]logp≤(n−1)∑p∈P∖P0E[λp(ξ)]logp+n∑p∈P∖P0E[λp(η)]logp≤(n−1)E[logξ]+n∑p∈P∖P0E[λp(η)]logp=O(n),n→∞,
where we have used the inequality E[logξ]<∞ and the assumption (21). Using that α∈(0,1) and (a(t)) is regularly varying at infinity with index 1/α, we obtain (39).

Acknowledgement

We thank the anonymous referee for useful comments on the first version of the manuscript.

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