Survival probabilities of autoregressive processes

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DOI:10.1051/ps/2013031 www.esaim-ps.org

SURVIVAL PROBABILITIES OF AUTOREGRESSIVE PROCESSES

Christoph Baumgarten

¹

Abstract. Given an autoregressive process X of order p (i.e. Xn =a1Xn−1+· · ·+apXn−p+Yn

where the random variablesY1, Y2, . . . are i.i.d.), we study the asymptotic behaviour of the probability that the process does not exceed a constant barrier up to timeN (survival or persistence probability).

Depending on the coeﬃcientsa1, . . . , apand the distribution ofY1, we state conditions under which the survival probability decays polynomially, faster than polynomially or converges to a positive constant.

Special emphasis is put on AR(2) processes.

Mathematics Subject Classification. 60G15, 60G50.

Received May 31, 2012. Revised October 8, 2012.

1. Introduction

For ﬁxed p ≥ 1, deﬁne X_n = _p

k=1a_kX_n−k +Y_n, n ≥ 0 with the convention that X_n = 0 for n ≤ 0.

Throughout the paper, we assume that (Y_n)_n≥1is a sequence of i.i.d. (nondegenerate) random variables. (X_n)_n≥1

is called an autoregressive process of order p (AR(p)–process). We sometimes refer to the random variables (Y_n)_n≥1 as innovations. Denote byp_N(x) the probability that the processX stays belowxuntil timeN,i.e.

p_N(x) :=P

sup

n=1,...,NX_n≤x

, N≥1, x≥0.

We refer top_N as the survival probability up to time N, and we write p_N instead ofp_N(0) in the sequel.

The aim of this paper is to study the asymptotic behaviour ofp_N(x) as N → ∞. Sometimes, the problem of determining the asymptotic behaviour ofp_N(x) is referred to as one-sided exit or one-sided barrier problem sincep_N(x) =P(τ_x> N) whereτ_x:= inf{n≥0 :X_n > x}. Such asymptotic results are known in a number of special cases such as random walks, integrated random walks, fractional Brownian motion and AR(1)–processes.

The study of survival probabilities is motivated by several applications such as the inviscid Burgers equation [16]

or zeros of random polynomials [5]. We refer to the recent survey of Aurzada and Simon [2] and [12] for further information, applications and references. For instance, if X is a random walk (p = 1, a1 = 1), it holds that p_N(x)∼c(x)N⁻¹^/²ifE[Y1] = 0 andE

Y₁²

= 1 (seee.g.Thm. XII.7.1a in [9]). In [14], the authors study AR(1)–

processes with a1∈(0,1) and show thatp_N(x) decays at least exponentially for a large class of distributions.

Bounds on the exponential rate of decay for AR(1)–processes with Gaussian innovations can be found in [1].

Keywords and phrases. Autoregressive process, autoregressive moving average, boundary crossing probability, one-sided exit problem, persistence probablity, survival probability.

1 Technische Universit¨at Braunschweig, Institut f¨ur Mathematische Stochastik, Pockelsstrasse 14, 38106 Braunschweig, Germany.[email protected]

Article published by EDP Sciences c EDP Sciences, SMAI 2013

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Besides, the decay of the survival probability is known for integrated random walks (p= 2, a1= 2, a2=−1): if E[Y1] = 0,E

Y₁²

∈(0,∞), it holds thatp_N(x)N⁻¹^/⁴(see [4] and the references therein).

Taken as a whole, very little is known about the decay of p_N for AR processes except in the few cases mentioned above. As noted in Demboet al.[4], this would be of much interest in view of the frequent appearance of AR-processes and survival probabilities in physical and ecomomic models. Here we investigate the behaviour of the survival probability for such processes under various conditions on the distribution of the innovations.

Since an AR(p)–processX can be written asX_n =_n

k=1c_n−kY_k where the (c_n) solve the diﬀerence equation c_n =a1c_n−1+· · ·+a_pc_n−p with suitable inital conditions, we search criteria for the sequence (c_n) that allow us to characterize the survival probability. Speciﬁcally, we are interested in the following question for AR(p)–

processes: when is p_N of polynomial order, when does p_N converge to a positive limit and when is the decay faster than any polynomial? This classiﬁcation seems natural if one recalls the results for AR(1)–processes X_n =ρX_n−1+Y_n where c_n =ρⁿ for all n. In this case, the behaviour of the survival probability ranges from exponential decay forρ <1, polynomial decay ifρ= 1 andE[Y1] = 0 to convergence to a positive constant if ρ >1.

As we will see, the sequence (c_n) often has a much more complex form ifp≥2 so that results for AR(1)–

processes generally cannot be extended directly to higher order processes. We will derive criteria that allow for the classiﬁcation of the asymptotic behaviour of thep_N as above. Particular emphasis is put on AR(2)–processes.

Let us introduce some notation and conventions: Iff, g:R→Rare two functions, we writef g(x→ ∞) if lim sup_x→∞f(x)/g(x)<∞ and f g iff g andg f. Moreover,f ∼g (x→ ∞) if f(x)/g(x)→1 as x→ ∞. If (Xt)_t≥0 is a stochastic process, it will often be convenient to writeX(t) instead of X_t. If X andY are random variables, we writeX =^d Y to denote equality in distribution.

The remainder of this article is organized as follows. After presenting the main results for AR(2) processes below, we start with some preliminaries on autoregressive processes in Section2. In Section3, we state general conditions ensuring that p_N decays exponentially or at least faster than any polynomial. Special emphasis is put on the case that (c_n)_n≥0 is absolutely summable and AR(2)–processes. We also prove exponential lower bounds for certain classes of AR-processes. We then determine the region where the survival probability decays polynomially for AR(2)–processes in Section4, before brieﬂy treating the case thatp_N converges to a positive constant in Section5.

1.1. Main results for AR(2) processes

Let us illustrate our main result when X is AR(2),i.e. X_n =a1X_n−1+a2X_n−2+Y_n with (Y_n)_n≥1 i.i.d.

Recall thatX_n =_n

k=1c_n−kY_k forn≥1. We decomposeR²into three disjoint regionsC, E andP (see Fig.1) deﬁned as follows:

C:=

(a1, a2) :a1≥2, a²₁+ 4a2>0

∪ {(a1, a2) :a1∈(0,2), a1+a2>1}

∪

(a1, a2) :a²₁+ 4a2= 0, a1>2

∪ {(a1, a2) :a1= 0, a2>1}, P :={(a1, a2) :a1+a2= 1, a2∈[−1,1]},

E:=R²\(C∪P).

Depending on the membership of (a1, a2) to one of these sets, we can characterize the behaviour of the survival probability under certain conditions on the law ofY1.

If (a1, a2)∈P, the survival probability decays polynomially ifE[Y1] = 0 under suitable moment conditions.

The choice a1 = 2, a2 =−1 corresponds to an integrated random walk wherep_N N⁻¹^/⁴ if E[Y1] = 0 and E

Y₁²

∈(0,∞), see [4]. Ifa1+a2= 1 with|a2|<1, we will see thatX can be seen as a perturbed random walk sincec_n=c+Cⁿ where||<1. Moreover,X can also be written as an integrated AR(1)–process. The process corresponding to a1= 0, a2 = 1 describes two independent random walks such that its survival probability is the square of that of a random walk.

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C

E

3 2 1 1 2 3 4 a₁

4 3 2 1 1 2 3a₂

Figure 1. The regions C and E. P corresponds to the dotted line. The dashed line is the boundary ofC whereasEis open.

Theorem 1.1. Let (a1, a2)∈P \ {(2,−1)}. Assume that E[Y1] = 0 and that E e^|Y¹^|^α

<∞for some α >0.

Then

p_N =N⁻¹^/²⁺^o⁽¹⁾ (|a2|<1), p_N N⁻¹ (a2= 1).

Next, we also prove that the survival probability decays faster than any polynomial if (a1, a2) ∈ E under certain conditions on the law ofY1.

Theorem 1.2. Let (a1, a2) ∈ E. Assume that P(Y1>0) ∈ (0,1), E e^|Y¹^|^α

< ∞ for some α > 0 and that the characteristic function ϕ of Y1 satisﬁes ϕ(t) → 0 as |t| → ∞. Then p_N exp(−λN/logN) for some λ=λ(a1, a2)>0.

Actually, we can show that p_N decays at least exponentially on most parts of E under various conditions onY1. The reason for the rapid decay of the survival probability onEcan be explained as follows: eitherc_n→0 exponentially fast or (c_n) oscillates and diverges to±∞.

If (a1, a2)∈C, we will see thatc_n= exp(λn(1 +o(1)) for someλ >0. One therefore expects that the process stays below a constant barrier at all times with positive probability. This is conﬁrmed by the following theorem:

Theorem 1.3. Let (a1, a2)∈C. Assume thatP(Y1<0)>0 andP(Y1≥x)(logx)^−α as x→ ∞for some α >1. Then it holds that

P

supn≥1X_n≤x

= lim

N→∞p_N(x)>0, x≥0.

Note that the assumption E[Y1] = 0 is essential for the polynomial behaviour ofp_N if (a1, a2)∈P. For instance, if (S_n)_n≥1 is a random walk, it is known that the survival probability can decay polynomially or exponentially ifE[S1]>0 [6] whereas it converges to a positive constant ifE[S1]<0. In contrast, if (a1, a2)∈E∪C, the behaviour ofp_N is more stable in the sense that Theorems1.3and1.2do not rely on the conditionE[Y1] = 0.

The best results can be obtained if the innovations are Gaussian. In that case, one can actually prove that p_N admits an exponential upper bound for all (a1, a2)∈E. Summing up, this leads to the following theorem:

Theorem 1.4. If Y1 is Gaussian with zero mean, the following statements hold:

1. lim_N_→∞p_N =p_∞>0 if and only if(a1, a2)∈C,

2. p_N ∼cN⁻¹ iff(a1, a2) = (0,1), andp_N N⁻^1/4 iff (a1, a2) = (2,−1), 3. p_N =N⁻¹^/²⁺ô⁽¹⁾ if and only if(a1, a2)∈P and|a2|<1, and

4. p_N e^−λN for someλ >0 if and only if(a1, a2)∈E.

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The theorems above are mostly corollaries to more general theorems that are also applicable to AR(p)–

processes if p ≥ 3 (see e.g. Thms. 3.2 and 3.10 and Props. 3.17 and 5.1 below). We will indicate possible extensions troughout the article. The main advantage of focussing on AR(2)–processes consists of the fact that we have an explicit solution of the diﬀerence equation for the sequence (c_n)_n≥0. For instance, this allows us to explicitly describe the parameters (a1, a2) such thatc_n→0.

Even for AR(2)–processes, one is forced to distinguish a variety of cases that require diﬀerent treatment. It is clear that this becomes much more complicated for processes of higher order. Finally, let us mention that the class of AR(p)–processes containsp-times integrated centered random walksS⁽^p⁾as a special case (i.e.S⁽¹⁾ is a centered random walk, andS_n^(p)=_n

k=1S_k^(p−¹⁾). Here, the behaviour of the survival probability is not known forp≥3.

2. Autoregressive processes

We begin by recalling a few facts about AR(p)–processes. For ﬁxed p≥1, deﬁneX_n=_p

k=1a_kX_n−k+Y_n, n≥0 with the convention thatX_n = 0 forn≤0 where (Y_n)_n≥1 is a sequence of i.i.d. random variables. One veriﬁes thatX_n=_n

k=1c_n−kY_k where

c_n= 0, n <0, c0= 1, c_n=

p

k=1

a_kc_n−k, n≥1.

In other words, (c_n)_n≥0 solves the linear diﬀerence equation

c_n =a1c_n−1+. . . a_pc_n−p, n≥p, with initial conditions

c0= 1, c1=a1c0, c2=a1c1+a2c0, . . . , c_p−1=a1c_p−2+· · ·+a_p−1c0.

Solving this equation amounts to ﬁnding the roots s1, . . . , s_p ∈C of the characteristic polynomialf_p(·), given byf_p(x) :=x^p−_p

k=1a_kx^p−k, x∈R. Ifp= 2, the rootss1, s2of f2(λ) =λ²−a1λ−a2 are given by s1:= (a1+h)/2, s2:= (a1−h)/2, h:=

a²₁+ 4a2∈C. (1)

Taking into account the inital conditionsc0= 1, c1=a1, one can show that c_n =

h⁻¹

sⁿ⁺¹₁ −sⁿ⁺¹₂

, n≥0, a²₁+ 4a2= 0,

(a1/2)ⁿ(n+ 1), n≥0, a²₁+ 4a2= 0. (2) Ifa²₁+ 4a2<0, writings1=re^iϕ ands2=s1=re^−iϕ in polar form, elementary manipulations show that the solution is given by

c_n= 2|a2|^(n+1)/2sin((n+ 1)ϕ)/˜h, (3) where

h˜=

−(a²1+ 4a2)>0, ϕ=

⎧⎪

⎨

⎪⎩

arctan(˜h/a1)∈(0, π/2), a1>0,

π/2, a1= 0,

π+ arctan(˜h/a1)∈(π/2, π), a1<0.

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Remark 2.1. It holds thatc_n→0 if and only if max{|s1|,|s2|}<1 which is equivalent to the conditions a1+a2<1, a2<1 +a1, a2>−1,

see Theorem 2.37 of [7].

(-2,-1) (2,-1)

(0,1)

Figure 2. The region of parameters (a1, a2) wherec_n→0.

Remark 2.2. Note that the convention thatX_n= 0 forn <0 is not standard to deﬁne autoregressive processes.

It is often customary to deﬁne AR(p)–processes as follows, seee.g.Chapter 3 in [3]: If (Y_n)_n∈Zis a sequence of i.i.d. random variables,X = (X_n)_n∈Z is AR(p) if

X_n=a1X_n−1+· · ·+a_pX_n−p+Y_n, n∈Z.

Moreover, X is called causal if there exists a deterministic sequence (c_n)_n≥0 with

|cn| < ∞ such that X_n =_∞

k=0c_nY_n−k.

By Theorem 3.1.1 of [3],X is causal if and only if the polynomialp(z) = 1−a1z− · · · −a_pz^p has no zeros in {z∈C:|z| ≤1}. In that case, the coeﬃcientsc_n are determined by the following relation_∞

k=0c_kz^k = 1/p(z) for|z| ≤1. Equating the coeﬃcients ofz^k, one easily veriﬁes (or see Sect. 3.3 in [3]) that the sequence (c_n)_n≥0

satisﬁes the same recursion equation with the same initial conditions as above. Hence, ifX is a causal AR(p)–

process, we can decompose it forn≥1 in the following way:

X_n =

n−1 k=0

c_kY_n−k+

∞ k=n

c_kY_n−k =

n k=1

c_n−kY_k+

∞ k=0

c_n+kY_−k =X_n⁽¹⁾+X_n⁽²⁾.

Note that X⁽¹⁾ and X⁽²⁾ are independent and thatX⁽¹⁾ is an AR(p)–process in the sense of this article. The termX⁽²⁾can be seen as a small perturbation since EX_n⁽²⁾

≤E[|Y1|]_∞

k=nc_k→0.

Moreover, the fact thatc_n→0 allows us to apply Theorem3.5below if X AR(p) in the sense of Brockwell and Davis. By using the alternative deﬁnition above, we can also deﬁne autoregressive processes when c_n does not go to zero and we get a much larger class of processes including, for example, random walks.

We will use diﬀerent methods to prove certain statements about the survival probability depending on the parameters (a1, a2). To this end, set

E1:={(a1, a2) :a1<0, a2>0, a2>1 +a1}, E2:= (−∞,0]², E3:=

(a1, a2) :a1>0, a²₁+ 4a2<0 .

Figure3 will be helpful to visualize the regions that will be considered separately below.

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C E₁

E₂ E₃

3 2 1 1 2 3 4 a₁

4 3 2 1 1 2 3a₂

Figure 3.The regionsE1, E2, E3andC.

Let us also comment brieﬂy on the dependence of the survival probability on the barrierxfor AR(p)–processes.

In principle, the behaviour of the survival probability can vary signiﬁcantly for diﬀerent barriers. An extreme example is an AR(1)–process Z_n = ρZ_n−1+Y_n where ρ∈ (0,1) with P(Y1= 1) = P(Y1=−1) = 1/2. It is known thatp_N exp(−λN) for some λ >0 (see Thm. 3.1below), whereas p_N(x) = 1 for all x≥1/(1−ρ) since|Xn|=ⁿ_k=1ρ^n−kY_k≤_∞

k=0ρ^k = 1/(1−ρ).

On the other hand, if c_n ≥ δ > 0 for all n ≥ 0 and P(Y1≤ −) > 0 for some > 0, one can show that p_N(x)p_N asN → ∞for allx≥0. Indeed, note that ifY1≤ −, it follows thatX_n=c_n−1Y1+_n

k=2c_n−kY_k≤

−δ+_n

k=2c_n−kY_k, so that p_N =P

sup

n=1,...,NX_n≤0

≥P(Y1≤ −)P

sup

n=2,...,N n

k=2

c_n−kY_k≤δ

≥P(Y1≤ −)p_N(δ).

Iteration shows thatp_N ≥P(Y1≤ −)^Lp_N(Lδ) forL= 1, . . . , N. Hence, if x≥0, takeLwithLδ≥xto get thatP(Y1≤ −)^Lp_N(x)≤p_N ≤p_N(x) for allN large enough.

3. Exponential bounds

3.1. Exponential upper bounds

Let us begin with a trivial observation: Ifa1≤0, . . . , a_p≤0, we have that P

sup

n=1,...,N

X_n≤0

≤P(Y1≤0)^N,

sinceX1 ≤0, . . . , X_n ≤0 implies that Y_k ≤ −a1X_k−1− · · · −a_pX_k−p ≤0 for allk= 1, . . . , n. If p= 2, this shows thatp_N decays exponentially onE2, see Figure3.

As we will see in the sequel, exponential decay of p_N occurs for two diﬀernt reasons: ﬁrst, if c_n → 0 and second, if (c_n)_n≥0 oscillates and diverges exponentially fast.

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Let us ﬁrst consider the case that c_n goes to zero. Recall that for AR(1)–processes (Z_n)_n≥1 with Z_n = ρZ_n−1+Y_n forρ∈(0,1),c_n=ρⁿ→0 andp_N decays exponentially under mild assumptions on the distribution ofY1by Theorem 1 of [14]:

Theorem 3.1. Let 0 < ρ < 1, x ≥ 0 and assume that E (Y₁⁻)^δ

< ∞ for some δ ∈ (0,1) and P(Y1> x(1−ρ))>0. Then E[exp(ατ_x)]<∞for someα >0.

We now state a similar weaker result that provides a simple criterion for AR(p)–processes to ensure that p_N decays faster to zero than any polynomial.

Theorem 3.2. Let (c_k)_k≥0 denote a sequence with c0 = 1 and A := _∞

k=0|ck| < ∞ such that _∞

k=q|c_k| ≤ Ce^−λq for every q ≥ 1 where C, λ > 0 are constants. Assume that there is γ > 0 with min{P(Y1<−γ),P(Y1> γ)} > 0 and that E

|Y1|^δ

< ∞ for some δ > 0. Let X_n = _n

k=1c_n−kY_k. Then for x∈[0, γA), there isc(x)>0 such that

p_N(x)exp

−c(x)√ N

, N → ∞.

Moreover, if E[exp(|Y1|^α)]<∞for someα >0 andx∈[0, γA), there is c(x)>0 such that p_N(x)exp (−c(x)N/logN), N → ∞.

Proof. Forq≥1, deﬁneZ_q,n=_n

k=n−qc_n−kY_kforn≥q+1. Note thatZ_q,nis measurable w.r.t.σ(Y_n−q, . . . , Y_n) which implies that (Z_q,n(q+1))_n≥1 deﬁnes a sequence of i.i.d. random variables withZ_q,q+1 =X_q+1. We will show thatZ_q,nis a good approximation ofX_n ifqis large. We then obtain an estimate onp_N(x) by computing the survival probability of the independent random variables (Z_q,(q+1)n+1)_n≥1.

First, observe that P

sup

n=q+2,...,N

|X_n−Z_q,n|> u

≤ ^N

n=q+2

P

n−q−1 k=1

c_n−kY_k > u

=

N n=q+2

P

⎛

⎝

n−1 k=q+1

c_kY_k > u

⎞

⎠=:h_N(u). (4)

In the ﬁrst equality, we have used that theY_k are i.i.d., and therefore exchangeable.

Hence,

P

n=1,...,Nsup X_n≤x

≤P

n=q+2,...,Nsup Z_q,n≤x+

+h_N()

≤P

sup

n=1,...,(N−1)/(q+1)Z_q,n(q+1)+1≤x+

+h_N()

=P(Zq,q+2≤x+)^(N⁻^1)/(q+1)+h_N(), (5) where we have used the fact that (Z_q,n(q+1)+1)_n≥1 is an i.i.d. sequence. Since the (Y_n) are i.i.d. (and therefore exchangeable), we get fory∈Rthat

P(Z_q,q+2≤y) =P _q

k=0

c_kY_k+1≤y

→P _∞

k=0

c_kY_k+1≤y

, q→ ∞, (6)

since the series_∞

k=0c_kY_k+1 =:Z converges a.s. by Kolmogorov’s Three Series Theorem. Next,P(Z ≤y)<1 for every 0≤y < γAby Theorem 3.7.5 of [13]. Then for 0≤y < γA, by (6), there isρ=ρ(y)<1 such that P(Zq,q+2≤y)≤ρfor allq suﬃciently large.

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Note that

sup

n=q+1,...,N

n k=q+1

c_kY_k

≤ sup

l=q+1,...,N

|Yk| ^∞

k=q+1

|ck| ≤Ce^−λq sup

l=q+1,...,N

|Yk|,

so we see that

h_N(u)≤ ^N

n=q+1

P

sup

k=q+1,...,N

|Yk|>e^λqu/C

≤N²P

|Y1|>e^λqu/C

. (7)

Let q = q_N := β√

N −1, β > 0. If u > 0 is such that x+u < γA, we deduce from (5) and (7) (using Chebychev’s inequality) that

p_N(x)≤ρ

√N/β+N²E

|Y1|^δ

(u/C)^−δe^−δλβ

√N.

By choosingβ suﬃciently large, the theorem follows under the assumption|Y1|^δ is integrable.

IfE[exp(|Y1|^α)]<∞, the estimate onh_N becomes h_N(u)≤N²P

|Y1|>e^λqu/C

≤N²exp

−e^αλq(u/C)^α

E[exp(|Y1|^α)].

In particular, with q = q_N = κlogN, if κ is large enough, this implies together with (5) that, for some c(x)>0,

p_N(x)N²e^−N²+ρ^N/⁽^q^N⁺¹⁾exp(−c(x)N/logN), N → ∞.

The proof of Theorem 3.2reveals that fast decay of p_N can be explained intuitively as follows: if we write X_n =_n−q−1

k=1 c_n−kY_k+_n

k=n−qc_n−kY_k, the ﬁrst summand is typically small ifqis large andc_n→0. Hence, P

n=1,...,Nsup X_n ≤)

≈P

⎛

⎝ sup

n=q+1,...,N n

k=n−q

c_n−kY_k≤0

⎞

⎠≈P _q+1

k=1

c_n−kY_k ≤0 N/q

.

Remark 3.3. If (c_k)_k≥0 denote a sequence with c0 = 1 and _∞

k=0|ck| < ∞ and |Y1| ≤ M a.s. for some M <∞, one can prove in an analogous way that evenp_N exp(−cN) for somec >0 sinceh_N(u) in the proof of Theorem3.2vanishes forqlarge enough.

Remark 3.4. As it was already remarked in [14], if (c_k)_k≥0 denotes a sequence of positive numbers, one has that

X_n =

n

k=1

c_n−kY_k≥

n

k=1

c_n−kY_k1_{Y_k_≤M_}=

n

k=1

c_n−kY˜_k =: ˜X_n, such that P(X_n ≤x,∀n≤N) ≤P

X˜_n ≤x,∀n≤N

. Hence, if the c_n are positive, one can assume without loss of generality that the innovations are bounded from above in order to establish an upper bound on the survival probability. Hence, the moment conditions of Theorem3.2only apply toY₁⁻ in that case.

For AR(2)–processes, Theorem3.2is applicable ifa1+a2<1,a2< a1+ 1 anda2>−1,cf.Remark2.1and Figure 2. Moreover, the preceding theorem can be generalized easily to cover more general processes (such as autoregressive moving average models ARMA(p,q) and moving average processes of inﬁnte order MA(∞), see Sect. 3 in [3]):

Theorem 3.5. Let (c_k)_k∈Z denote a sequence with c0 = 1, A := _∞

k=−∞|ck| < ∞ and

|k|≥q|ck| ≤ Ce^−λq for all q ≥ 1 and some λ > 0. Let (Y_k)_k∈Z be a sequence of i.i.d. random variables such that

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min{P(Y1> γ),P(Y1<−γ)} > 0 for some γ > 0 and E

|Y1|^δ

< ∞ for some δ > 0. Let X_n :=

_∞

k=−∞c_n−kY_k for n∈Z. If x∈[0, γA), it holds for somec(x)>0 that P

|n|≤Nsup X_n≤x

exp(−c(x)√

N), N → ∞.

Moreover, if E[exp(|Y1|^α)]<∞for someα >0 andx∈[0, γA), there is c(x)>0 such that P

sup

|n|≤NX_n≤x

exp(−c(x)N/logN), N→ ∞.

Proof. X_n is well deﬁned for everyn∈Zby Kolmogorov’s Three Series Theorem. The proof is then very similar to that of Theorem3.2. We deﬁneZ_q,n:=_n+q

k=n−qc_n−kY_k. Note that (Z_q,n(2q+1))_n∈Zare i.i.d. random variables withZq,0=_q

k=−qc_kY_k. The remainder of the proof is along the same lines of the proof of Theorem3.2.

In certain special cases, we can improve Theorem3.2. Namely, if (c_n) is a sequence of positive numbers and c_n=ρⁿ(1 +o(1)) whereρ∈(0,1), it follows from Theorem3.1that p_N goes to zero exponentially fast under mild assumptions onY1:

Proposition 3.6. Let(c_n)_n≥0be a sequence such thatαCρⁿ ≤c_n≤Cρⁿfor alln≥0whereρ∈(0,1),0< α <

1,C >0. Assume thatE (Y₁⁻)^δ

<∞for some δ∈(0,1). Letx≥0 be such that P(Y1≥x(1−ρ)/(αC))>0, andX_n:=_n

k=1c_n−kY_k. Then there is someλ=λ(x)>0 such thatp_N(x)exp(−λN).

Proof. Deﬁne the i.i.d. random variables ˜Y_k:=Y_k1_{Y_k_<0}+αY_k1_{Y_k_>0},k≥1. Sincec_k ≥0 for allk, we obtain that

X_n=

n k=1

c_n−kY_k≥ ⁿ

k=1

Cρ^n−kY_k1_{Y_k_<0}+

n k=1

αCρ^n−kY_k1_{Y_k_>0}=C

n k=1

ρ^n−kY˜_k=:CZ_n,

whereZ_n:=ρZ_n−1+ ˜Y_n. In particular, we conclude that P

sup

n=1,...,N

X_n≤x

≤P

sup

n=1,...,N

Z_n≤x/C

.

Now P

Y˜1> x(1−ρ)/C

= P(Y1≥x(1−ρ)/(αC)) > 0 by the choice of x. Hence, the result follows from

Theorem 1 of [14] (Thm.3.1 above).

The preceding proposition yields the following corollary for AR(2)–processes:

Corollary 3.7. Leta1∈(0,2), a2<0witha1+a2<1anda²₁+ 4a2>0. Assume thatE (Y₁⁻)^δ

<∞for some δ∈(0,1)andP(Y1≥y)>0for everyy. For everyx≥0, there isλ=λ(x)>0such thatp_N(x)exp(−λN).

Proof. It is not hard to check that 0< s2< s1 <1. Hence,c_n =sⁿ₁(s1−s2(s2/s1)ⁿ)/handh⁻¹(s1−s2)sⁿ₁ ≤ c_n≤h⁻¹sⁿ₁⁺¹ for alln. The result follows from Proposition3.6.

If |Y1| ≤ M a.s., the preceding corollary is not applicable. However, we already know that p_N e^−cN for somec >0 in that case, see Remark3.3.

Let us now establish exponential upper bounds forp_N for certain distributions if the sequence (c_n) oscillates and diverges exponentially. The proof relies on the following proposition.

(10)

Proposition 3.8. Let ρ∈(−1,1) (ρ= 0) and set Z:=_∞

n=1ρⁿY_n. Moreover, suppose thatE[|Y1|^a]<∞for some a >0. Let ϕ denote the characteristic function ofY1 and assume that there are Δ∈(0,|ρ|) and t0 >0 such that |ϕ(t)| ≤Δ forall|t| ≥t0. It follows that P(|Z| ≤) as↓0.

Proof. Z is well–deﬁned and its characterisitic function ˜ϕis given by ˜ϕ(t) =_∞

n=1ϕ(ρⁿt), seee.g.Section 3.7 of [13]. Let us show that ˜ϕis absolutely integrable. If this holds, by Theorem 3.2.2 of [13],Z admits a continuous densityggiven by

g(x) := 1 2π

_∞

−∞e^−ixtϕ(t) dt,˜ x∈R.

In particular,g is bounded implying thatP(|Z| ≤)≤C for any≥0.

To prove the integrability of ˜ϕ, letΔandt0 be as in the statement of the proposition and note that

|ϕ(t)|˜ = ∞ n=1

|ϕ(ρⁿt)| ≤Δ^N^(t),

whereN(t) = #{n≥1 :|ρⁿt| ≥t0}. One veriﬁes thatN(t) =(log(t)−log(t0))/log(1/|ρ|)so that

|ϕ(t)| ≤˜ exp

logΔ

log|t| −log(t0) log(1/|ρ|) −1

=C|t|^−α,

where C depends on t0, ρ and Δ only andα:= log(1/Δ)/log(1/|ρ|)>1. This shows that|ϕ(t)˜ | is integrable

overR.

Remark 3.9. Recall that lim_|t|→∞E e^itX

= 0 if X has an absolutely continuous distribution, see e.g.

Section 2.2 in [13]. However, if the distribution of X is purely discrete, lim sup_|t|→∞E

e^itX = 1, and in general, it is a very challenging problem to ﬁnd conditions such that the random series_∞

k=1ρⁿY_n has a den- sity. This question has attracted a lot of attention for so-called inﬁnite Bernoulli convolutions. We refer to [15]

for a survey.

We can now prove the following theorem.

Theorem 3.10. Let X_n := _n

k=1c_n−kY_k where c_n = dρⁿ +β_nrⁿ where d = 0, ρ < −1 and |ρ| > |r| and

|β_n|e^−λn →0 as n→ ∞ for everyλ > 0. Assume E[|Y1|^a]<∞ for some a >0. Moreover, suppose that the characteristic function ϕof Y1 satisﬁes the inequality |ϕ(t)| ≤Δ <1/|ρ|for all|t| large enough. Then there is a constant C >0 such that for every x≥0, it holds that

lim inf

N→∞ −N⁻¹logP

n=1,...,Nsup X_n≤x

≥C.

If E[exp(|Y1|^α)]<∞for someα >0, then C≥

log|ρ/r|, |r|>1, log|ρ|, else.

Proof. Assume w.l.o.g. that d = 1 (write X_n = _n

k=1(c_n−k/d)(dY_k)). Let ˆβ_n := sup{|β0|, . . . ,|βn|} and E_N :={|Y1| ≤f_N, . . . ,|Y_N| ≤f_N}where 1≤f_N → ∞is to be speciﬁed later. OnE_N, it holds forn= 1, . . . , N that

X_n =

n

k=1

c_n−kY_k=

n

k=1

ρ^n−kY_k+

n

k=1

β_n−kr^n−kY_k

≥

n k=1

ρ^n−kY_k−βˆ_nf_N

n k=1

|r|^n−k≥

n k=1

ρ^n−kY_k−βˆ_Nf_N

N k=0

|r|^k.

(11)

Case 1.Consider ﬁrst the case thatβ_n= 0 for somen. LetR_N :=_N

k=0|r|^k. Then p_N(x)≤P(E_N^c ) +P

sup

n=1,...,N n

k=1

ρ^n−kY_k ≤x+ ˆβ_Nf_NR_N, E_N

. (8)

Note thatZ_n :=_n

k=1ρ^n−kY_kis an AR(1)–process satisfyingZ_n=ρZ_n−1+Y_n. Let us begin with the following useful observation: ifZ_N−1≤zandZ_N ≤zfor some largez >0, we have with high probability that|Z_N₋1| ≤z.

This will allow us to reduce the estimation ofp_N(x) to controlling P(|ZN| ≤z_N) wherez_N → ∞as N → ∞.

To be precise, note that

{ZN−1≤z, Z_N ≤z} ⊆ {|ZN−1| ≤z} ∪ {ZN−1<−z, ZN ≤z}

⊆ {|ZN−1| ≤z} ∪ {YN ≤(1− |ρ|)z}. (9) For the last inclusion, we have used that the event{ZN−1<−z, ZN ≤z}implies thatz≥Z_N =ρZ_N₋1+Y_N ≥

−ρz+Y_N. Hence, combining this with (8), we obtain that p_N(x)≤P(E_N^c ) +P

Z_N₋1≤x+ ˆβ_Nf_NR_N, Z_N ≤x+ ˆβ_Nf_NR_N

≤P(E_N^c ) +P

|ZN−1| ≤x+ ˆβ_Nf_NR_N

+P

Y_N ≤(1− |ρ|)(x+ ˆβ_Nf_NR_N)

. (10)

It remains to estimate the three probabilities above. Clearly, P(E_N^c) =P

_N

n=1

{|YN|> f_N}

≤NP(|Y1|> f_N).

Next, since |ρ| >1 and ˆβ_N ≥β >0 for some β >0 and for all N ≥N0 large enough and R_N ≥1, it follows that

P

Y_N ≤(1− |ρ|)(x+ ˆβ_Nf_NR_N)

≤P(|Y1| ≥(|ρ| −1)βf_N), N ≥N0.

For largeN, using the last two inequalities in (10), we arrive at p_N(x)≤(N+ 1)P(|Y1| ≥C1f_N) +P

|ZN−1| ≤2 ˆβ_Nf_NR_N

, (11)

whereC1:= min{1,(|ρ| −1)β}. Set ˜Z_n:=ρ⁻ⁿZ_n=_n

k=1ρ^−kY_k. Then P

|Z_N₋1| ≤2 ˆβ_Nf_NR_N

=PZ˜_N−1≤2|ρ|⁻⁽^N⁻¹⁾βˆ_Nf_NR_N

.

Note that ˜Z_n converges a.s. to a random variable ˜Z_∞by Kolmogorov’s Three Series Theorem. Moreover, foru, v >0,

PZ˜_∞≤u+v

≥PZ˜_∞−Z˜_N≤u+v−Z˜_N,Z˜_N≤u

≥PZ˜_∞−Z˜_N≤v,Z˜_N≤u

=PZ˜_∞−Z˜_N≤v

PZ˜_N≤u

.

The last equality follows from the independence of increments of ˜Z. Hence, PZ˜_N≤u

≤ PZ˜_∞≤u+v 1−PZ˜_∞−Z˜_N> v

, u, v >0, N≥1.