EQUATIONS UNDER KESTEN’S CONDITION

D. BURACZEWSKI, E. DAMEK, T. MIKOSCH AND J. ZIENKIEWICZ

Abstract. In this paper we prove large deviations results for partial sums constructed from the solution to a stochastic recurrence equation. We assume Kesten’s condition [17] under which the solution of the stochastic recurrence equation has a marginal distribution with power law tails, while the noise sequence of the equations can have light tails. The results of the paper are analogs of those obtained by A.V. and S.V. Nagaev [20, 21] in the case of partial sums of iid random variables. In the latter case, the large deviation probabilities of the partial sums are essentially determined by the largest step size of the partial sum. For the solution to a stochastic recurrence equation, the magnitude of the large deviation probabilities is again given by the tail of the maximum summand, but the exact asymptotic tail behavior is also influenced by clusters of extreme values, due to dependencies in the sequence. We apply the large deviation results to study the asymptotic behavior of the ruin probabilities in the model.

1. Introduction Through the last 40 years, the stochastic recurrence equation

*Y**n**= A**n**Y**n−1**+ B**n**,* *n ∈ Z ,*
(1.1)

*and its stationary solution have attracted much attention. Here ((A**i**, B**i**)) is an iid sequence, A**i**> 0*
*a.s. and B**i* *assumes real values. (In what follows, we write A, B, Y, . . . , for generic elements of the*
*strictly stationary sequences (A**i**), (B**i**), (Y**i**), . . ., and we also write c for any positive constant whose*
value is not of interest.)

*It is well known that if E log A < 0 and E log*^{+}*|B| < ∞, there exists a unique strictly stationary*
*ergodic causal solution (Y**i*) to the stochastic recurrence equation (1.1) with representation

*Y**n*=
X*n*
*i=−∞*

*A**i+1**· · · A**n**B**i**,* *n ∈ Z ,*
*where, as usual, we interpret the summand for i = n as B**n*.

*One of the most interesting results for the stationary solution (Y**i*) to the stochastic recurrence
equation (1.1) was discovered by Kesten [17]. He proved under general conditions that the marginal
*distributions of (Y**i*) have power law tails. For later use, we formulate a version of this result due to
Goldie [10].

Theorem 1.1. *(Kesten [17], Goldie [10]) Assume that the following conditions hold:*

*2000 Mathematics Subject Classification. Primary 60F10; secondary 91B30, 60G70.*

*Key words and phrases. Stochastic recurrence equation, large deviations, ruin probability.*

D. Buraczewski and E. Damek were partially supported by MNiSW grant N N201 393937. Thomas Mikosch’s research is partly supported by the Danish Natural Science Research Council (FNU) Grants 09-072331 ”Point process modelling and statistical inference” and 10-084172 “Heavy tail phenomena: Modeling and estimation”. J. Zienkiewicz was supported by MNiSW grant N N201 397137. D. Buraczewski was also supported by European Commission via IEF Project (contract number PIEF-GA-2009-252318 - SCHREC).

1

*•* *There exists α > 0 such that*

*EA*^{α}*= 1 .*
(1.2)

*•* *ρ = E(A*^{α}*log A) and E|B|*^{α}*are both finite.*

*•* *The law of log A is nonarithmetic.*

*•* *For every x, P{Ax + B = x} < 1.*

*Then Y is regularly varying with index α > 0. In particular, there exist constants c*^{+}_{∞}*, c*^{−}_{∞}*≥ 0 such*
*that c*^{+}_{∞}*+ c*^{−}_{∞}*> 0 and*

*P{Y > x} ∼ c*^{+}_{∞}*x*^{−α}*,* *and P{Y ≤ −x} ∼ c*^{−}_{∞}*x*^{−α}*as x → ∞ .*
(1.3)

*Moreover, if B ≡ 1 a.s. then the constant c*^{+}_{∞}*takes on the form*
*c**∞**:= E[(1 + Y )*^{α}*− Y*^{α}*]/(αρ) ,*

Goldie [10] also showed that similar results remain valid for the stationary solution to stochastic
*recurrence equations of the type Y**n* *= f (Y**n−1**, A**n**, B**n**) for suitable functions f satisfying some*
contractivity condition.

*The power law tails (1.3) stimulated research on the extremes of the sequence (Y**i*) . Indeed, if
*(Y**i**) were iid with tail (1.3) and c*^{+}_{∞}*> 0, then the maximum sequence M**n**= max(Y*1*, . . . , Y**n*) would
satisfy the limit relation

*n→∞*lim *P{(c*^{+}_{∞}*n)*^{−1/α}*M**n**≤ x} = e*^{−x}* ^{−α}*= Φ

*α*

*(x) ,*

*x > 0 ,*(1.4)

where Φ*α* denotes the Fr´echet distribution, i.e. one of the classical extreme value distributions; see
*Gnedenko [11]; cf. Embrechts et al. [6], Chapter 3. However, the stationary solution (Y**i*) to (1.1)
is not iid and therefore one needs to modify (1.4) as follows: the limit has to be replaced by Φ^{θ}* _{α}*for

*some constant θ ∈ (0, 1), the so-called extremal index of the sequence (Y*

*i*); see de Haan et al. [12];

cf. [6], Section 8.4.

The main objective of this paper is to derive another result which is a consequence of the power
*law tails of the marginal distribution of the sequence (Y**i*): we will prove large deviation results for
the partial sum sequence

*S**n* *= Y*1*+ · · · + Y**n**,* *n ≥ 1 ,* *S*0*= 0 .*

*This means we will derive exact asymptotic results for the left and right tails of the partial sums S**n*.
Since we want to compare these results with those for an iid sequence we recall the corresponding
classical results due to A.V. and S.V. Nagaev [20, 21] and Cline and Hsing [3].

Theorem 1.2. *Assume that (Y**i**) is an iid sequence with a regularly varying distribution , i.e. there*
*exists an α > 0, constants p, q ≥ 0 with p + q = 1 and a slowly varying function L such that*

*P{Y > x} ∼ pL(x)*

*x*^{α}*and P{Y ≤ −x} ∼ qL(x)*

*x*^{α}*as x → ∞.*

(1.5)

*Then the following relations hold for α > 1 and suitable sequences b*_{n}*↑ ∞:*

*n→∞*lim sup

*x≥b**n*

¯¯

¯¯*P{S**n**− ES**n* *> x}*

*n P{|Y | > x}* *− p*

¯¯

¯¯ = 0 (1.6)

*and*

*n→∞*lim sup

*x≥b**n*

¯¯

¯¯*P{S**n**− ES**n* *≤ −x}*

*n P{|Y | > x}* *− q*

¯¯

¯*¯ = 0 .*
(1.7)

*If α > 2 one can choose b**n* =*√*

*an log n, where a > α − 2, and for α ∈ (1, 2], b**n* *= n*^{δ+1/α}*for any*
*δ > 0.*

*For α ∈ (1, 2] one can choose a smaller normalization b**n*if one knows the slowly varying function
*L appearing in (1.5). Moreover, if α ∈ (0, 1) or α = 1 and E|Y | = ∞, a result similar to equations*
*(1.6) and (1.7) can be obtained with the centering constants ES** _{n}* replaced by zero. A functional
version of Theorem 1.2 with multivariate regularly varying summands was proved in Hult et al.

[13] and the results were used to prove asymptotic results about multivariate ruin probabilities.

Large deviation results for iid heavy-tailed summands are also known when the distribution of the
summands is subexponential, including the case of regularly varying tails; see the recent paper
Denisov et al. [5] and the references therein. In this case, the regions where the large deviations
hold very much depend on the decay rate of the tails of the summands. For semi-exponential tails
(such as for the log-normal and the heavy-tailed Weibull distributions) the large deviation regions
*(b**n**, ∞) are much smaller than those for summands with regularly varying tails. In particular, x = n*
*is not necessarily contained in (b**n**, ∞).*

The aim of this paper is to study large deviation probabilities for a particular dependent sequence
*(Y**n**) as described in Kesten’s Theorem 1.1. For dependent sequences (Y**n*) much less is known about
*the large deviation probabilities for the partial sum process (S**n*). Gantert [8] proved large deviation
results of logarithmic type for mixing subexponential random variables. Davis and Hsing [4] and
Jakubowski [14, 15] proved large deviation results of the following type: there exist sequences
*s**n**→ ∞ such that*

*P{S**n* *> a**n**s**n**}*
*n P{Y > a**n**s**n**}* *→ c**α*

*for suitable positive constants c**α* *under the assumptions that Y is regularly varying with index*
*α ∈ (0, 2), n P (|Y | > a**n**) → 1 and (Y**n*) satisfies some mixing conditions. Both Davis and Hsing
*[4] and Jakubowski [14, 15] could not specify the rate at which the sequence (s**n*) grows to infinity,
*and an extension to α > 2 was not possible. These facts limit the applicability of these results,*
*for example for deriving the asymptotics of ruin probabilities for the random walk (S**n*). Large
*deviations results for particular stationary sequences (Y**n*) with regularly varying finite-dimensional
distributions were proved in Mikosch and Samorodnitsky [19] in the case of linear processes with iid
*regularly varying noise and in Konstantinides and Mikosch [18] for solutions (Y**n*) to the stochastic
*recurrence equation (1.1), where B is regularly varying with index α > 1 and EA*^{α}*< 1. This means*
*that Kesten’s condition (1.2) is not satisfied in this case and the regular variation of (Y**n*) is due to
*the regular variation of B. For these processes, large deviation results and ruin bounds are easier*
*to derive by applying the “heavy-tail large deviation heuristics”: a large value of S** _{n}* happens in the
most likely way, namely it is due to one very large value in the underlying regularly varying noise

*sequence, and the particular dependence structure of the sequence (Y*

*n*) determines the clustering

*behavior of the large values of S*

*n*

*. This intuition fails when one deals with the partial sums S*

*n*

*under the conditions of Kesten’s Theorem 1.1: here a large value of S**n* is not due to a single large
*value of the B**n**’s or A**n*’s but one needs to consider an increasing number of these quantities for
building up a very large value of the partial sums.

The paper is organized as follows. In Section 2 we prove an analog to Theorem 1.2 for the partial
*sum sequence (S**n*) constructed from the solution to the stochastic recurrence equation (1.1) under
the conditions of Kesten’s Theorem 1.1. The proof of this result is rather technical: it is given in
Section 3 where we split the proof into a series of auxiliary results. There we treat the different cases
*α ≤ 1, α ∈ (1, 2] and α > 2 by different tools and methods. In particular, we will use exponential*
tail inequalities which are suited for the three distinct situations. In contrast to the iid situation
*described in Theorem 1.2, we will show that the x-region where the large deviations hold cannot*
*be chosen as an infinite interval (b**n**, ∞) for a suitable lower bound b**n* *→ ∞, but one also needs*
*upper bounds c**n* *≥ b**n*. In Section 4 we apply the large deviation results to get precise asymptotic
*bounds for the ruin probability related to the random walk (S**n*). This ruin bound is an analog of

the celebrated result by Embrechts and Veraverbeke [7] in the case of a random walk with iid step sizes.

2. Main result

The following is the main result of this paper. It is an analog of the well known large deviation result of Theorem 1.2.

Theorem 2.1. *Assume that the conditions of Theorem 1.1 are satisfied and additionally there exists*
*ε > 0 such that EA*^{α+ε}*and E|B|*^{α+ε}*are finite. Then the following relations hold:*

(1) *For α ∈ (0, 2], M > 2,*

(2.1) sup

*n* sup

*n*^{1/α}*(log n)*^{M}*≤x*

*P{S**n**− d**n* *> x}*

*n P{|Y | > x}* *< ∞ ,*
*If additionally e*^{s}^{n}*≥ n*^{1/α}*(log n)*^{M}*and lim**n→∞**s**n**/n = 0 then*

(2.2) lim

*n→∞* sup

*n*^{1/α}*(log n)*^{M}*≤x≤e*^{sn}

¯¯

¯¯*P{S**n**− d**n* *> x}*

*n P{|Y | > x}* *−* *c*^{+}_{∞}*c**∞*

*c*^{+}*∞**+ c*^{−}*∞*

¯¯

¯*¯ = 0 ,*

*where d**n**= 0 or d**n**= ES**n* *according as α ∈ (0, 1] or α ∈ (1, 2].*

(2) *For α > 2 and any c**n* *→ ∞,*

(2.3) sup

*n* sup

*c**n**n*^{0.5}*log n≤x*

*P{S**n**− ES**n* *> x}*

*n P{|Y | > x}* *< ∞ .*
*If additionally c**n**n*^{0.5}*log n ≤ e*^{s}^{n}*and lim**n→∞**s**n**/n = 0 then*

(2.4) lim

*n→∞* sup

*c**n**n*^{0.5}*log n≤x≤e*^{sn}

¯¯

¯¯*P{S**n**− ES**n**> x}*

*n P{|Y | > x}* *−* *c*^{+}_{∞}*c**∞*

*c*^{+}*∞**+ c*^{−}*∞*

¯¯

¯*¯ = 0 .*

*Clearly, if we change from the variables B**n* *to −B**n* in the above results we obtain the corre-
*sponding asymptotics for the left tail of S**n**. For example, for α > 1 the following relation holds*
*uniformly for the x-regions indicated above:*

*n→∞*lim

*P{S**n**− nEY ≤ −x}*

*n P{|Y | > x}* = *c*^{−}_{∞}*c**∞*

*c*^{+}*∞**+ c*^{−}*∞*

*.*

Remark 2.2. The deviations of Theorem 2.1 from the iid case (see Theorem 1.2) are two-fold. First,
*the extremal clustering in the sequence (Y**n*) manifests in the presence of the additional constants
*c**∞* *and c*^{±}_{∞}*. Second, the precise large deviation bounds (2.2) and (2.4) are proved for x-regions*
*bounded from above by a sequence e*^{s}^{n}*for some s*_{n}*→ ∞ with s*_{n}*/n → 0. It is shown in the course*
*of the proof (see Section 3.3) that (2.2) and (2.4) cannot be extended to unbounded x-regions, i.e.*

in the latter case the upper bounds (2.1) and (2.3) are the best one can achieve.

3. Proof of the main result

3.1. Preliminaries. In what follows, it will be convenient to use the following notation
Π*ij* =

½ *A**i**· · · A**j* *i ≤ j*

1 otherwise and Π*j*= Π*1j**,*
and

*Y*e*i*= Π*2i**B*1+ Π*3i**B*2*+ · · · + Π**ii**B**i−1**+ B**i**,* *i ≥ 1 .*

*Since Y**i*= Π*i**Y*0+ e*Y**i* the following decomposition is straightforward:

*S**n* *= Y*0

X*n*
*i=1*

Π*i*+
X*n*

*i=1*

*Y*e*i**=: Y*0*η**n*+ e*S**n**,*
where

(3.1) *S*e* _{n}*= e

*Y*

_{1}

*+ · · · + eY*

_{n}*and η*

*= Π*

_{n}_{1}

*+ · · · + Π*

_{n}*,*

*n ≥ 1 .*

*We start with some rough bound on the tail of |Y*0

*|η*

*n*.

Lemma 3.1. *Let (s**n**) be a sequence such that s**n**/n → 0. Then for any sequence (b**n**) with b**n**→ ∞*
*the following relations hold:*

*n→∞*lim sup

*b*_{n}*≤x≤e*^{sn}

*P{|Y*0*| η**n* *> x}*

*n P{|Y | > x}* *= 0 and* lim sup

*n→∞* sup

*b*_{n}*≤x*

*P{|Y*0*| η**n**> x}*

*n P{|Y | > x}* *< ∞,*
*Proof.* *The infinite series η =* P_{∞}

*i=0*Π*i* has the distribution of the stationary solution to the sto-
*chastic recurrence equation (1.1) with B ≡ 1 a.s. and therefore, by Theorem 1.1, P (η > x) ∼*
*c**∞**x*^{−α}*,* *x → ∞ . It follows from a slight modification of Jessen and Mikosch [16], Lemma 4.1(4),*
*and the independence of Y*0 *and η that*

(3.2) *P{|Y*0*| η > x} ∼ c x*^{−α}*log x ,* *x → ∞ .*
*Since s**n**/n → 0 as n → ∞ we have*

sup

*b**n**≤x≤e*^{sn}

*P{|Y*0*| η**n* *> x}*

*n P{|Y | > x}* *≤* sup

*b**n**≤x≤e*^{sn}

*P{|Y*0*| η > x}*

*n P{|Y | > x}* *→ 0 .*
*There exist c*0*, x*0*> 0 such that P {|Y*0*| > y} ≤ c*0*y*^{−α}*for y > x*0. Therefore

*P{|Y*0*| η**n**> x} ≤ P{x/η**n**≤ x*0*} + c*0*x*^{−α}*Eη*^{α}* _{n}*1

_{{x/η}

_{n}

_{>x}_{0}

_{}}*≤ cx*

^{−α}*Eη*

^{α}

_{n}*.*

*By Bartkiewicz et al. [1], Eη*

_{n}

^{α}*≤ cn. Hence*

*I**n*= sup

*b**n**≤x*

*P{|Y*0*| η**n**> x}*

*n P{|Y | > x}* *≤ sup*

*b**n**≤x*

*cx*^{−α}*Eη*^{α}_{n}

*n P{|Y | > x}* *< ∞.*

This concludes the proof. ¤

Remark 3.2. The arguments presented in Section 3.3 (see in particular (3.22)) will show that
*we cannot expect the limit of I**n* *to exist without further restrictions on the x-region; see also*
*Lemma 3.10 for precise bounds of the ratio P{Y*0*η**n**> x}/n P{|Y | > x}.*

*In view of Lemma 3.1 it suffices to bound the ratios P{ eS*_{n}*− d*_{n}*> x}/(n P{|Y | > x}) uniformly*
*for the considered x-regions. Here and in what follows, slightly abusing notation, d**n* denotes E e*S**n*

*for α > 1 and zero for α ≤ 1.*

*For any x in the considered large deviation regions, we define various quantities to be used*
throughout the proof.

*•* *m = [(log x)*^{0.5+σ}*] for some positive number σ < 1/4, where [·] denotes the integer part.*

*•* *n*0*= [ρ*^{−1}*log x], where ρ = E(A*^{α}*log A).*

*•* *n*1*= n*0*− m and n*2*= n*0*+ m*

*•* *For α > 1, let D be the smallest integer such that −D log EA > α − 1. Notice that the*
*latter inequality makes sense since EA < 1 due to (1.2) and the convexity of the function*
*ψ(h) = EA*^{h}*, h > 0.*

*•* *For α ≤ 1, fix some β < α and let D be the smallest integer such that −D log EA*^{β}*> α − β*
*where, by the same remark as above, EA*^{β}*< 1.*

*•* *Let n*3be the smallest integer satisfying

*D log x ≤ n*3*,* *x > 1 .*
(3.3)

*Notice that since the function Ψ(h) = log ψ(h) is convex, putting β = 1 if α > 1, by the*
*choice of D we have* _{D}^{1} *<* ^{Ψ(α)−Ψ(β)}_{α−β}*< Ψ*^{0}*(α) = ρ, therefore n*2*< n*3*if x is sufficiently large.*

*Now, for fixed n we make the change of indices i → j = n − i + 1 and, abusing notation and*
*suppressing the dependence on n, we again write*

*Y*e*j**= B**j*+ Π*jj**B**j+1**+ · · · + Π**j,n−1**B**n**.*
*Let n*4*= min(j + n*3*, n). We use the decomposition*

*Y*e*j* = e*U**j*+ f*W**j**,*
(3.4)

with *U*e*j**= B**j*+ Π*jj**B**j+1**+ · · · + Π**j,n*4*−1**B**n*4*.*
Clearly, f*W**j* *vanishes if j ≥ n − n*3.

Lemma 3.3. *For any small δ > 0, there exists a constant c > 0 such that*
Pn¯¯

¯
X*n*
*j=1*

(f*W**j**− c**j*)

¯¯

*¯ > x*
o

*≤ c n x*^{−α−δ}*,* *x > 1 ,*
(3.5)

*where c**j**= 0 or c**j* = Ef*W**j* *according as α ≤ 1 or α > 1.*

*Note that the statement of Lemma 3.3 is nontrivial for n > n*3.

*Proof.* *Assume first that α > 1. Since EfW**j* *is finite, −D log EA > α − 1 and D log x ≤ n*3, we have
*for some positive δ*

(3.6) *E|fW**j**| ≤* *(EA)*^{n}^{3}

*1 − EAE|B| ≤ c e**D log x log EA**≤ c x*^{−(α−1)−δ}*,*
and hence by Markov’s inequality

Pn¯¯

¯
X*n*
*j=1*

(f*W*_{j}*− EfW** _{j}*)

¯¯

*¯ > x*
o

*≤ 2 x** ^{−1}*
X

*n*

*j=1*

*E|fW*_{j}*| ≤ c n x*^{−α−δ}*.*

*If β < α ≤ 1 an application of Markov’s inequality yields for some positive δ,*
P

nX^{n}

*j=1*

*W*f*j* *> x*
o

*≤ x** ^{−β}*
X

*n*

*j=1*

*E|fW**j**|*^{β}*≤ x*^{−β}*nE|B|*^{β}*(EA** ^{β}*)

^{n}^{3}

*(1 − EA*

*)*

^{β}*≤ cx*^{−β}*ne**D log x log EA*^{β}*≤ c n x*^{−α−δ}*.*

*In the last step we used the fact that −D log EA*^{β}*> α − β.* ¤
*By virtue of (3.5) and the decomposition (3.4) is suffices to study the probabilities P{*P_{n}

*j=1*( e*U**j**−*
*a**j**) > x}, where a**j* *= 0 for α ≤ 1 and a**j*= E e*U**j* *for α > 1. We further decompose eU**i*into

*U*e*i*= e*X**i*+ e*S**i*+ e*Z**i**,*
*where for i ≤ n − n*3,

*X*e*i* *= B**i*+ Π*ii**B**i+1**+ · · · + Π**i,i+n*1*−2**B**i+n*1*−1**,*
*S*e*i* = Π*i,i+n*1*−1**B**i+n*1*+ · · · + Π**i,i+n*2*−1**B**i+n*2*,*
(3.7)

*Z*e*i* = Π*i,i+n*2*B**i+n*2+1*+ · · · + Π**i,i+n*3*−1**B**i+n*3*.*

*For i > n − n*3, define e*X**i**, eS**i**, eZ**i* *as follows: For n*2*< n − i < n*3 choose e*X**i**, eS**i* as above and
*Z*e*i*= Π*i,i+n*2*B**i+n*2+1*+ · · · + Π**i,n−1**B**n**.*

*For n*1*≤ n − i ≤ n*2, choose e*Z**i*= 0, e*X**i* as before and

*S*e*i*= Π*i,i+n*1*−1**B**i+n*1*+ · · · + Π**i,n−1**B**n**.*
*Finally, for n − i < n*1, define e*S**i**= 0, eZ**i*= 0 and

*X*e_{i}*= B** _{i}*+ Π

_{ii}*B*

_{i+1}*+ · · · + Π*

_{i,n−1}*B*

_{n}*.*

*Let p*1*, p, p*3 *be the largest integers such that p*1*n*1 *≤ n − n*1*+ 1, pn*1*≤ n − n*2 *and p*3*n*1 *≤ n − n*3,
respectively. We study the asymptotic tail behavior of the corresponding block sums given by

*X**j* =

*jn*1

X

*i=(j−1)n*1+1

*X*e*i**,* *S**j* =

*jn*1

X

*i=(j−1)n*1+1

*S*e*i**,* *Z**j* =

*jn*1

X

*i=(j−1)n*1+1

*Z*e*i**,*
(3.8)

*where j is less or equal p*1*, p, p*3 respectively.

*3.2. Block sums of length log x. In a series of auxiliary results we will now study the tail behavior*
*of the single block sums X*1*, S*1*, Z*1defined in (3.8).

Lemma 3.4. *Assume ψ(α+²) = EA*^{α+²}*< ∞ for some ² > 0. Then there is a constant C = C(²) >*

*0 such that ψ(α + γ) ≤ C e*^{ργ}*for |γ| ≤ ²/2, where ρ = E(A*^{α}*log A).*

*Proof.* *By a Taylor expansion and since ψ(α) = 1, ψ*^{0}*(α) = ρ, we have for some θ ∈ (0, 1),*
*ψ(α + γ) = 1 + ργ + 0.5ψ*^{00}*(α + θγ)γ*^{2}*.*

(3.9)

*If |θγ| < ²/2 then, by assumption, ψ*^{00}*(α + θγ) = EA*^{α+θγ}*(log A)*^{2} *is bounded by a constant c > 0.*

Therefore,

*ψ(α + γ) ≤ 1 + ργ + cγ*^{2}*= e**log(1+ργ+c γ*^{2})*≤ C e*^{ργ}*.*

¤ Next we study the tails of

*X*_{1} =

*n*1

X

*i=1*

*(|B**i**| + Π**ii**|B**i+1**| + · · · + Π**i,i+n*1*−2**|B**i+n*1*−1**|) ,*

*Z*_{1} =

*n*1

X

*i=1*

(Π*i,i+n*2*|B**i+n*2+1*| + · · · + Π**i,i+n*3*−1**|B**i+n*3*|) .*

Remark 3.5. *We notice that | eX*1*| and | eZ*1*| are stochastically dominated by X*_{1}*and Z*_{1}, respectively.

*Therefore the bounds in Lemmas 3.6 and 3.7 also apply to the tails of | eX*1*| and | eZ*1*|, respectively.*

Lemma 3.6. *There exist positive constants C*1*, C*2*, C*3 *such that*
*P{X*_{1}*> x} ≤ C*1*x*^{−α}*e*^{−C}^{2}^{(log x)}^{C3}*,* *x > 1.*

*Proof.* *We have X*_{1}=P_{n}_{0}

*k=m+1**R**k**, where for m < k ≤ n*0,

*R**k* = Π*1,n*0*−k**|B**n*0*−k+1**| + · · · + Π**i,i+n*0*−k−1**|B**i+n*0*−k**| + · · · + Π**n*1*,n*1*+n*0*−k−1**|B**n*1*+n*0*−k**| .*
*Notice that for x sufficiently large,*

n X^{n}^{0}

*k=m+1*

*R**k**> x*
o

*⊂*

*n*0

[

*k=m+1*

*{R**k**> x/k*^{3}*}.*

*Indeed, on the set {R**k* *≤ x/k*^{3}*, m < k ≤ n*0*} we have for some c > 0 and sufficiently large x, by*
*the definition of m = [(log x)** ^{0.5+σ}*],

*n*0

X

*k=m+1*

*R**k* *≤* *x*
*m + 1*

X*∞*
*k=1*

1

*k*^{2} *≤ c* *x*

*(log x)*^{0.5+σ}*< x .*
*We conclude that, with I**k* *= P{R**k**> x/k*^{3}*},*

P
n X^{n}^{0}

*k=m+1*

*R**k* *> x*
o

*≤*

*n*0

X

*k=m+1*

*I**k**.*

*Next we study the probabilities I**k**. Let δ = (log x)** ^{−0.5}*. By Markov’s inequality,

*I*

*k*

*≤ (x/k*

^{3})

^{−(α+δ)}*ER*

_{k}

^{α+δ}*≤ (x/k*

^{3})

^{−(α+δ)}*n*

^{α+δ}_{0}

*(EA*

*)*

^{α+δ}

^{n}^{0}

^{−k}*E|B|*

^{α+δ}*.*

*By Lemma 3.4 and the definition of n*0

*= [ρ*

^{−1}*log x],*

*I**k**≤ c (x/k*^{3})^{−(α+δ)}*n*^{α+δ}_{0} *e*^{(n}^{0}^{−k)ρδ}*≤ c x*^{−α}*k*^{3(α+δ)}*n*^{α+δ}_{0} *e*^{−kρδ}*.*

*Since k ≥ (log x)*^{0.5+σ}*≥ m there are positive constants ζ*1*, ζ*2 *such that kδ ≥ k*^{ζ}^{1}*(log x)*^{ζ}^{2} and
*therefore for sufficiently large x and appropriate positive constants C*1*, C*2*, C*3,

*n*0

X

*k=m+1*

*I**k**≤ c x*^{−α}*n*^{α+δ}_{0}

*n*1

X

*k=m+1*

*e*^{−ρ k}^{ζ1}^{(log x)}^{ζ2}*k*^{3(α+δ)}*≤ C*1*x*^{−α}*e*^{−C}^{2}^{(log x)}^{C3}*.*

This finishes the proof. ¤

Lemma 3.7. *There exist positive constants C*4*, C*5*, C*6 *such that*
*P{Z*_{1}*> x} ≤ C*4*x*^{−α}*e*^{−C}^{5}^{(log x)}^{C6}*,* *x > 1.*

*Proof.* *We have Z*_{1}=P_{n}_{3}_{−n}_{2}

*k=1* *R*e*k**, where*

*R*e*k*= Π*1,n*2*+k**|B**n*2*+k+1**| + · · · + Π**i,i+n*2*+k−1**|B**i+n*2*+k**| + · · · + Π**n*1*,n*1*+n*2*+k−1**|B**n*1*+n*2*+k**|.*

*As in the proof of Lemma 3.6 we notice that, with J**k* *= P{ eR**k* *> x/(n*2*+ k)*^{3}*}, for x sufficiently*
large

*P{*

*n*3X*−n*2

*k=1*

*R*e*k* *> x} ≤*

*n*3X*−n*2

*k=1*

*J**k**.*

*Next we study the probabilities J**k**. Choose δ = (n*2*+ k)*^{−0.5}*< ²/2 with ² as in Lemma 3.4. By*
Markov’s inequality,

*J**k* *≤ ((n*2*+ k)*^{3}*/x)** ^{α−δ}*E e

*R*

^{α−δ}

_{k}*≤ ((n*2

*+ k)*

^{3}

*/x)*

^{α−δ}*n*

^{α−δ}_{1}

*(EA*

*)*

^{α−δ}

^{n}^{2}

^{+k}*E|B|*

^{α−δ}*.*

*By Lemma 3.4 and since n*2

*+ k = n*0

*+ m + k,*

*(EA** ^{α−δ}*)

^{n}^{2}

^{+k}*≤ c e*

^{−δρ(n}^{2}

^{+k)}*≤ c x*

^{−δ}*e*

^{−δρ(m+k)}*.*

*There is ζ*3*> 0 such that δ(m + k) ≥ (log x + k)*^{ζ}^{3}*. Hence, for appropriate constants C*4*, C*5*, C*6*> 0,*

*n*3X*−n*2

*k=1*

*J**k* *≤ c x*^{−α}*n*^{α−δ}_{1}

*n*3X*−n*2

*k=1*

*(n*2*+ k)*^{3(α−δ)}*e**−ρ(log x+k)*^{ζ3}*≤ C*4*x*^{−α}*e*^{−C}^{5}^{(log x)}^{C6}*.*

This finishes the proof. ¤

The next lemma is a first major step towards the proof of the main result. For the formulation
*of the result and its proof, recall the definitions of S*1and e*S**i*defined in (3.8) and (3.7), respectively.

Lemma 3.8. *Assume that c*^{+}_{∞}*> 0 and let (b**n**) be any sequence such that b**n* *→ ∞. Then the*
*following relation holds:*

*n→∞*lim sup

*x≥b**n*

¯¯

¯ *P{S*1*> x}*

*n*1*P{Y > x}− c**∞*

¯¯

*¯ = 0 .*
(3.10)

*If c*^{+}_{∞}*= 0 then*

*n→∞*lim sup

*x≥b**n*

*P{S*1*> x}*

*n*1*P{|Y | > x}* *= 0 .*
(3.11)

*Proof.* *For i ≤ n*1, consider
*S*e*i**+ S*^{0}_{i}

= Π*i,n*1*B**n*1+1*+ · · · + Π**i,i+n*1*−2**B**i+n*1*−1*+ e*S**i*+ Π*i,i+n*2*B**i+n*2+1*+ · · · + Π**i,n*2*+n*1*−1**B**n*2*+n*1

= Π*i,n*1*(B**n*1+1*+ A**n*1+1*B**n*1+2*+ · · · + Π**n*1*+1,n*2*+n*1*−1**B**n*2*+n*1*) .*
Notice that

*P{|S*_{1}^{0}*+ · · · + S*_{n}^{0}_{1}*| > x} ≤ n*1*P{|S*_{1}^{0}*| > x/n*1*}.*

Therefore and by virtue of Lemmas 3.6 and 3.7 (cf. Remark 3.5) there exist positive constants
*C*7*, C*8*, C*9such that

*P{|S*_{1}^{0}*+ · · · + S*^{0}_{n}_{1}*| > x} ≤ C*7*x*^{−α}*e*^{−C}^{8}^{(log x)}^{C9}*,* *x ≥ 1 .*
*Therefore and since S*1=P_{n}_{1}

*i=1**S*e*i* it suffices for (3.10) to show that

*n→∞*lim sup

*x≥b**n*

¯¯

¯*P{S*1+P_{n}_{1}

*i=1**S*_{i}^{0}*> x}*

*n*1*P{Y > x}* *− c**∞*

¯¯

*¯ = 0 .*
We observe that

*S*1+

*n*1

X

*i=1*

*S*_{i}^{0}*=: U T*1 *and T*1*+ T*2 *d*

*= Y ,*
where

*U* = Π*1,n*1+ Π*2,n*1*+ · · · + Π**n*1*,n*1*,*

*T*1 *= B**n*1+1+ Π*n*1*+1,n*1+1*B**n*1+2*+ · · · + Π**n*1*+1,n*2*+n*1*−1**B**n*2*+n*1*,*
*T*2 = Π*n*1*+1,n*2*+n*1*B**n*2*+n*1+1+ Π*n*1*+1,n*2*+n*1+1*B**n*2*+n*1+2*+ · · · .*
*We will prove the following two relations: for some positive constants C*10*, C*11*, C*12,

*P{|U T*2*| > x} = C*10*x*^{−α}*e*^{−C}^{11}^{(log x)}^{C12}*,* *x > 1 ,*
(3.12)

and

*n→∞*lim sup

*x≥b**n*

¯¯

¯*P{U (T*_{1}*+ T*_{2}*) > x}*

*n*1*P{Y > x}* *− c**∞*

¯¯

*¯ = 0 ,*
(3.13)

*provided c*^{+}_{∞}*> 0 or*

*n→∞*lim sup

*x≥b**n*

*P{U (T*1*+ T*2*) > x}*

*n*1*P{|Y | > x}* *= 0 ,*
(3.14)

*if c*^{+}* _{∞}* = 0. A combination of (3.12) and (3.13) yields (3.10). The proof of (3.12) is given in
Lemma 3.9 and an argument for the proofs of (3.13) and (3.14) is indicated in Remark 3.11. This

proves the lemma. ¤

Lemma 3.9. *Relation (3.12) holds.*

*Proof.* The same argument as in the proof of Lemma 3.6 yields
*P{|U T*2*| > x} ≤*

X*∞*
*k=0*

*P{U Π**n*1*+1,n*1*+n*2*+k**|B**n*1*+n*2*+k+1**| > x/(log x + k)*^{3}*}.*

*Write δ = (log x + k)*^{−0.5}*. Then by Lemma 3.4, Markov’s inequality and since n*2*= n*0*+ m,*
*P{U Π**n*1*+1,n*1*+n*2*+k**|B**n*1*+n*2*+k+1**| > x/(log x + k)*^{3}*}*

*≤ (log x + k)*^{3(α−δ)}*x*^{−(α−δ)}*EU*^{α−δ}*(EA** ^{α−δ}*)

^{n}^{2}

^{+k}*E|B|*

^{α−δ}*≤ c (log x + k)*^{3(α−δ)}*x*^{−(α−δ)}*e*^{−(n}^{2}^{+k)ρδ}

*≤ c e*^{−(m+k)ρδ}*(log x + k)*^{3(α−δ)}*x*^{−α}*.*
*There is ζ > 0 such that (m + k)δ ≥ (log x + k)** ^{ζ}* and therefore,

*P{|U T*2*| > x} ≤ c x** ^{−α}*
X

*∞*

*k=0*

*e*^{−(log x+k)}^{ζ}^{ρ}*(log x + k)*^{3(α−δ)}

*≤ c x*^{−α}*e*^{−(log x)}^{ζ}^{ρ/2}*.*

This proves the lemma. ¤

Lemma 3.10. *Assume that Y and η**k**(defined in (3.1)) are independent and ψ(α+²) = EA*^{α+²}*< ∞*
*for some ² > 0. Then for n*1 *= n*0*− m = [ρ*^{−1}*log x] − [(log x)*^{0.5+σ}*] for some σ < 1/4 and any*
*sequences b**n* *→ ∞ and r**n**→ ∞ the following relation holds:*

*n→∞*lim sup

*r**n**≤k≤n*1*,b**n**≤x*

¯¯

¯*P{η**k**Y > x}*

*k P{Y > x}* *− c**∞*

¯¯

*¯ = 0 ,*
*provided c*^{+}_{∞}*> 0. If c*^{+}_{∞}*= 0 then*

*n→∞*lim sup

*r**n**≤k≤n*1*,b**n**≤x*

*P{η**k**Y > x}*

*k P{|Y | > x}* *= 0.*

Remark 3.11. *The proofs of relations (3.13) and (3.14) follow by observing that U* *= η*^{d}_{n}_{1} and
*Y* *= T** ^{d}* 1

*+ T*2.

*Proof.* *Assume first c*^{+}_{∞}*> 0. We have by independence of Y and η**k**, for any k ≥ 1, x > 0 and r > 0,*
*P{η*_{k}*Y > x}*

*k P{Y > x}* =

³ Z

*(0,x/r]*

+ Z

*[x/r,∞)*

*´ P{Y > x/z}*

*k P{Y > x}* *dP(η**k* *≤ z) = I*1*+ I*2*.*
*For every ε ∈ (0, 1) there is r > 0 such that for x ≥ r and z ≤ x/r,*

*P{Y > x/z}*

*P{Y > x}* *∈ z*^{α}*[1 − ε, 1 + ε] and P{Y > x}x*^{α}*≥ c*^{+}_{∞}*− ε .*
*Hence for sufficiently large x,*

*I*_{1}*∈ k*^{−1}*Eη*_{k}* ^{α}*1

_{{η}

_{k}

_{≤x/r}}*[1 − ε, 1 + ε] and I*

_{2}

*≤ c k*

^{−1}*x*

^{α}*P{η*

_{k}*> x/r} ≤ c k*

^{−1}*Eη*

^{α}*1*

_{k}

_{{η}

_{k}

_{>x/r}}*.*We have

*I*1*∈ (k*^{−1}*Eη*_{k}^{α}*− k*^{−1}*Eη*_{k}* ^{α}*1

_{{η}

_{k}

_{>x/r}}*)[1 − ε, 1 + ε]*

and by virtue of Bartkiewicz et al. [1], lim*k→∞**k*^{−1}*Eη*_{k}^{α}*= c**∞**. Therefore it is enough to prove that*

*n→∞*lim sup

*r**n**≤k≤n*1*,b**n**≤x*

*k*^{−1}*Eη*_{k}* ^{α}*1

_{{η}

_{k}

_{>x}}*= 0.*

(3.15)

*By the H¨older and Markov inequalities we have for ² > 0,*
*Eη*_{k}* ^{α}*1

*{η*

*k*

*>x}*

*≤ (Eη*

^{α+²}*)*

_{k}*¡*

^{α/(α+²)}*P{η**k**> x}*¢_{²/(α+²)}

*≤ x*^{−²}*Eη*_{k}^{α+²}*.*
(3.16)

*Next we study the order of magnitude of Eη*^{α+²}_{k}*. By definition of η**k*,
*Eη*^{α+²}_{k}*= EA*^{α+²}*E(1 + η**k−1*)^{α+²}

*= EA** ^{α+²}*¡

*E(1 + η**k−1*)^{α+²}*− E(η*_{k−1}* ^{α+²}*)¢

*+ EA*^{α+²}*Eη*_{k−1}^{α+²}*.*
Thus we get the recursive relation

*Eη*^{α+²}* _{k}* =
X

*k*

*i=1*

*(EA** ^{α+²}*)

*¡*

^{k−i+1}*E(1 + η**i−1*)^{α+²}*− E(η*^{α+²}* _{i−1}*)¢

*≤ c*
X*k*
*i=1*

*(EA** ^{α+²}*)

^{k−i+1}*≤ c*

*(EA*

*)*

^{α+²}

^{k}*EA*

^{α+²}*− 1.*(3.17)

*Indeed, we will prove that if ² < 1 then there is a constant c such that for i ≥ 1,*
*E(1 + η**i*)^{α+²}*− Eη*^{α+²}_{i}*≤ c .*

*If α + ² ≤ 1 then this follows from the concavity of the function f (x) = x*^{α+²}*, x > 0. If α + ² > 1*
we use the mean value theorem to obtain

*E(1 + η**i*)^{α+²}*− Eη*_{i}^{α+²}*≤ (α + ²) E(1 + η**i*)^{α+²−1}*≤ (α + ²)Eη*_{∞}^{α+²−1}*< ∞.*

*Now we choose ² = k** ^{−0.5}*. Then by (3.16), (3.17) and Lemma 3.4,

*Eη*^{α}* _{k}*1

_{{η}

_{k}

_{>x}}*≤ c*

*(EA*

*)*

^{α+²}

^{k}*EA*^{α+²}*− 1x*^{−²}*≤ ce*^{ρn}^{1}^{/}^{√}^{k−log x/}^{√}^{k}

*EA*^{α+²}*− 1* *≤ c* *e*^{−ρm/}^{√}^{k}*EA*^{α+²}*− 1.*

*In the last step we used that k ≤ n*1 *= n*0*− m, where n*0 *= [ρ*^{−1}*log x]. Moreover, since m =*
*[(log x)*^{0.5+σ}*], m/√*

*k ≥ 2 c*1*(log x)*^{σ}*for some c*1 *> 0. On the other hand, setting γ = ² = k** ^{−0.5}* in

*(3.9), we obtain EA*

^{α+²}*− 1 ≥ ρk*

^{−0.5}*/2. Combining the bounds above, we finally arrive at*

sup

*r**n**≤k≤n*1*,b**n**≤x*

*k*^{−1}*Eη*_{k}* ^{α}*1

_{{η}

_{k}

_{>x}}*≤ c e*

^{−c}^{1}

^{(log x)}

^{σ}*for constants c, c*1*> 0. This estimate yields the desired relation (3.15) and thus completes the proof*
*of the first part of the lemma when c*^{+}_{∞}*> 0.*

*If c*^{+}_{∞}*= 0 we proceed in the same way, observing that for any δ, z > 0 and sufficiently large x,*
*P{Y > x/z}*

*P{|Y | > x}* *< δz*^{α}

*and hence I*1 *converges to 0 as n goes to infinity.* ¤

*Observe that if |i − j| > 2 then S**i**and S**j* *are independent. For |i − j| ≤ 2 we have the following*
bound:

Lemma 3.12. *The following relation holds for some constant c > 0:*

sup

*i≥1,|i−j|≤2*

*P{|S**i**| > x, |S**j**| > x} ≤ c n*^{0.5}_{1} *x*^{−α}*,* *x > 1 .*

*Proof.* *Assume without loss of generality that i = 1 and j = 2, 3. Then we have*

*|S*1*| ≤* ¡

Π*1,n*1*+ · · · + Π**n*1*,n*1

¢

*×*¡

*|B**n*1+1*| + Π**n*1*+1,n*1+1*|B**n*1+2*| + · · · + Π**n*1*+1,n*1*+n*2*−1**|B**n*2*+n*1*|*¢

*=: U*1*T*_{1}^{0}*,*

*|S*2*| ≤* ¡

Π*n*1*+1,2n*1*+ · · · + Π**2n*1*,2n*1

¢

*×*¡

*|B**2n*1+1*| + Π**2n*1*+1,2n*1+1*|B**2n*1+2*| + · · · + Π**2n*1*+1,2n*1*+n*2*−1**|B**2n*1*+n*2*|*¢

*=: U*2*T*_{2}^{0}*,*

*|S*3*| ≤* ¡

Π*2n*1*+1,3n*1*+ · · · + Π**3n*1*,3n*1

¢

*×*¡

*|B**3n*1+1*| + Π**3n*1*+1,3n*1+1*|B**3n*1+2*| + · · · + Π**3n*1*+1,3n*1*+n*2*−1**|B**3n*1*+n*2*|*¢

*=: U*3*T*_{3}^{0}*.*
*We observe that U*1 *d*

*= η**n*1*, U**i**, i = 1, 2, 3, are independent, U**i* *is independent of T*_{i}^{0}*for each i, and*
*the T*_{i}^{0}*’s have power law tails with index α > 0. We conclude from (3.10) that*

*P{|S*1*| > x, |S*2*| > x} ≤ P{T*_{1}^{0}*> x n*^{−1/(2α)}_{1} *} + P{T*_{1}^{0}*≤ x n*^{−1/(2α)}_{1} *, U*1*T*_{1}^{0}*> x , U*2*T*_{2}^{0}*> x}*

*≤ c n*^{0.5}_{1} *x*^{−α}*+ P{n*^{−1/(2α)}_{1} *U*1*> 1, U*2*T*_{2}^{0}*> x}*

*≤ c n*^{0.5}_{1} *x*^{−α}*+ P{U*1*> n*^{1/(2α)}_{1} *} P{U*2*T*_{2}^{0}*> x}*

*≤ c n*^{0.5}_{1} *x*^{−α}*.*

*In the same way we can bound P{|S*1*| > t, |S*3*| > t}. We omit details.* ¤
Remark 3.13. *In what follows, we will often use bounds for the moments of X*1*, S*1*, Z*1. Elementary
computations show that

*EX*^{α}_{1} *≤ n*^{2 max(α,1)}_{1} *E|B|*^{α}*,*

*E|S*1*|*^{α}*≤ n*^{max(α,1)}_{1} *(2m + 1)*^{max(α,1)}*E|B|*^{α}*,*
(3.18)

*EZ*^{α}_{1} *≤ n*^{max(α,1)}_{1} *(n*3*− n*2)^{max(α,1)}*E|B|*^{α}*,*
*and therefore for any P ∈ {X*_{1}*, S*1*, Z*_{1}*}, P*^{0}*= P 1**{|P |≤y}* *and 1 < α ≤ 2,*

*var(P*^{0}*) ≤ cy*^{2−α}*E|P |*^{α}*.*
(3.19)

*For α > 2, var(P*^{0}*) ≤ cn*^{2}_{1}.

3.3. Counterexamples in the case lim sup_{n,x→∞}*log x/n > 0. The objective of this section is*
*to show that, if n, x → ∞ and log x/n does not tend to 0, one cannot expect that sup** _{n}*in (2.1) and
(2.3) can be replaced by lim

*n→∞*. This fact follows from the next result.

Proposition 3.14. *Assume the conditions of Theorem 2.1. If n = n*1*(x) then*

(3.20) lim

*x→∞*

*P{ eS**n**− d**n**> x}*

*n P{|Y | > x}* *= 0.*

*If n = n*1*(x) + n*2*(x) then*

(3.21) lim

*x→∞*

*P{ eS**n**− d**n**> x}*

*n P{|Y | > x}* = *c**∞**c*^{+}_{∞}*2(c*^{+}*∞**+ c*^{−}*∞*)*.*

*If c*^{+}_{∞}*> 0, x**l**→ ∞ as l → ∞, n**l**≥ n*2*(x**l**) and γ = lim**l→∞**log x**l**/n**l* *exists then*

(3.22) lim

*l→∞*

*P{|Y*0*|*P_{n}_{l}

*i=1*Π*i**> x**l**}*
*n**l**P{Y > x**l**}* = *c*

*c*^{+}*∞*

*γ ,*
*where c is the constant in the tail bound (3.2).*