Functional central limit theorem for ARCH(∞) models with weakly dependent innovations^†

Functional central limit theorem for ARCH(∞) models with weakly dependent innovations ^†

Oesook Lee ¹

1 Department of Statistics, Ewha Womans University

Received 22 January 2021, revised 25 February 2021, accepted 26 February 2021

Abstract

Since the seminal work of Engle (1982) and Bollerslev (1986), many ARCH-type models have been suggested and examined to explain a variety of stylized facts of finan- cial and economic time series. Various popular ARCH-type models can be expressed as ARCH(∞) models. In this paper, we study the stationarity and functional central limit theorem (FCLT) for ARCH(∞) models, because statistical inferences for ARCH(∞) se- quences require the study of asymptotics of various statistics concerned. Most previous results are obtained under independent and identically distributed (i.i.d.) innovation processes. But the i.i.d. assumption on innovations substantially restricts the flexibil- ity of the models. In addition, many authors have shown that the i.i.d. assumption can be weakened to mild conditions. We consider the ARCH(∞) model where the innovation processes are strictly stationary and λ-weakly dependent instead of inde- pendent and identically distributed. We provide sufficient conditions for the existence of a unique stationary and λ-weakly dependent Volterra series type solution to the given ARCH(∞) process. The FCLT for the stationary and λ-weakly dependent solution is also obtained by adding weak dependence coefficients condition on innovations and condition on ARCH(∞) parameters. The FCLT for GARCH(p, q) model with λ-weakly dependent innovations is considered as an example.

Keywords: Functional central limit theorem, λ-weakly dependent process, stationarity.

1. Introduction

After Engle (1982) and Bollerslev (1986), autoregressive conditional heteroscedastic (ARCH) model and its various modified ARCH-type models have been suggested and studied in order to explain a variety of stylized facts of financial and economic time series and related fields (see, for example, Francq and Zako¨ıan, 2010; Jeong and Lee, 2019; Kim and Lee, 2018 etc.).

† This research was supported by Basic Science Research Program through the NRF funded by the Ministry of Education, Science and Technology ( 2018R1D1A1B07044546).

1

Professor, Department of Statistics, Ewha Womans University, Ewhayeodaegil 52, Seoul 03760, Korea.

E-mail: [email protected]

General nonnegative ARCH(∞) sequence is one of those and various popular ARCH-type models can be represented in this framework. The ARCH(∞) process is given by

X t = ρ t ξ t , ρ t = a +

∞

X

j=1

b j X t−j , (1.1)

where a > 0, b j ≥ 0, j = 1, 2, · · · and ξ t is a sequence of nonnegative stationary random variables. Giraitis et al. (2000), Zaffaroni (2004), Douc et al. (2008), Doukhan et al. (2006) and Lee (2018) among many others examined the ARCH(∞) model under independent and identically distributed (i.i.d.) assumptions on ξ t . Strict stationarity, weak stationarity, existence of moments, and asymptotic behaviors of the process including the FCLT are investigated. The FCLT is of great importance in recent research on the process because statistical inferences for ARCH sequences require the study of the asymptotics of various statistics concerned. For example, the FCLT can be used to investigate such as the structural breaks of the data via e.g., CUSUM or MOSUM. The FCLT for ARCH-type processes has been studied by using various mixing properties, association, near epoch dependent (NED) properties, and (θ, η, λ)-weakly dependent properties(see Billingsley, 1968; Davidson, 2002;

Dedecker and Doukhan, 2003; Bardet et al., 2006; Doukhan and Wintenberger, 2007; Berkes et al., 2008; Hwang and Shin, 2016, and references therein).

Nze et al. (2002), Bardet et al. (2006), and Doukhan et al. (2007) showed the (θ, η, λ)-weak properties of infinite ARCH-type model with i.i.d. centered errors. When a given process is λ-weakly dependent, the FCLT for the process can be derived by adopting the following theorem.

Theorem (Doukhan and Wintenberger, 2007) Let (X t , t ∈ Z) be real valued stationary and λ-weakly dependent sequence of random variables satisfying E|X 0 | ^m < ∞ for a real number m > 2. If λ(r) = O(r ^−λ ) (as r → ∞) for λ > 4 + 2/(m − 2), then σ ² < ∞ and

W n (t) = 1

√ n

[nt]

X

i=1

(X i − E(X i )) −→ σW t , t ∈ [0, 1], n ≥ 0

in the Skorohod space D([0, 1]), where W t is the standard Brownian motion.

Most of the aforementioned results about ARCH(∞) processes assumed i.i.d. standardized innovations, but the i.i.d. assumption on innovations restricts substantially the flexibility of ARCH-type models. Lee and Hansen (1994) observed rescaled parameter e _t = u _t /σ _t , where σ _t ² = a + P ∞

j=1 b _j u ² _t−j (a > 0, b _j ≥ 0) is not need to be independent. Probabilistic and statistical properties of ARCH-type models under mild conditions on the innovations have been studied by for example, Drost and Nijman (1993), Berkes et al. (2003), Linton et al.

(2010), Li and Wu (2018), and Lee (2020) etc.

In this paper, we study the λ-weak dependence properties of ARCH(∞) process gener- ated by the equation (1.1) with strictly stationary and λ-weakly dependent innovations {ξ t }.

Conditions for the existence of a strictly and weakly stationary Volterra series type solution

to (1.1) are given. The (functional) central limit theorem for stationary and λ-weakly depen-

dent process X t with proper dependence coefficients is also obtained. The results obtained

help to better understand the behavior of the volatility.

2. Main results

Consider the following ARCH(∞) model given by

X t = ρ t ξ t , (2.1)

ρ _t = a +

∞

X

j=1

b _j X _t−j , t ∈ Z, (2.2)

where {ξ _t , t ∈ Z} is a sequence of nonnegative strictly stationary random variables defined on some probability space (Ω, F , P ) and a > 0, b _j ≥ 0, j = 1, 2, · · · .

Define F t = σ(· · · , ξ t−2 , ξ t−1 , ξ t ) and kXk p = (E|X| ^p ) ^1/p , 1 ≤ p < ∞.

We make the assumption:

(A1) E(ξ _t ^p |F t−1 ) ≤ G p < ∞, p > 0.

We first establish sufficient conditions for the existence of the stationary solution to (2.1) and (2.2) in L _p .

Let p > 0 and we define

ϕ = X

j≥1

b ^p∧1 _j G ^1/(p∨1) _p . (2.3)

Theorem 2.1 (1) If the assumption (A1) and ϕ < 1 hold, then a stationary solution in L ^p to the equation (2.1)-(2.2) is given by

X t = aξ t + a

∞

X

k=1

∞

X

j

1

,··· ,j

k

=1

b j

₁

· · · b j

_k

ξ t ξ t−j

₁

· · · ξ t−j

₁

−···−j

k

. (2.4)

(2) Assume that with p ≥ 1, (A1) and ϕ = P

j≥1 b _j G ^1/p p < 1 hold. If Y _t is a F _t measurable stationary solution to the equation (2.1) and (2.2) with E|Y t | ^p < ∞, then X t = Y t a.s.

Proof: (1) For 0 < p < 1,

E|

∞

X

k=1

∞

X

j

1

,··· ,j

k

=1

b j

₁

· · · b j

_k

ξ t ξ t−j

₁

· · · ξ t−j

₁

−···−j

k

| ^p

≤

∞

X

k=1

∞

X

j

₁

,··· ,j

_k

=1

E|b j

₁

· · · b j

_k

ξ t ξ t−j

₁

· · · ξ t−j

₁

−···−j

_k

| ^p

≤ G p

∞

X

k=1

ϕ ^k < ∞,

where ϕ = P ∞

j=1 b ^p _j G _p .

Now let 1 ≤ p < ∞. From (A1), kξ _t ξ _t−j

₁

· · · ξ _t−j

₁

_{−···−j}

_k

k ^p _p ≤ G ^k+1 _p . Use Minkowski’s inequality to have that

kX t k p ≤ akξ t k p + a

∞

X

k=1

∞

X

j

1

,··· ,j

k

=1

b j

₁

· · · b j

_k

kξ t ξ t−j

₁

· · · ξ t−j

₁

−···−j

k

k p

≤ akξ t k p + a

∞

X

k=1

∞

X

j

1

,··· ,j

k

=1

b j

₁

· · · b j

_k

G ^(k+1)/p _p

≤ aG ^1/p _p (1 +

∞

X

k=1

(

∞

X

j=1

b j G ^1/p _p ) ^k

= aG ^1/p _p (1 +

∞

X

k=1

ϕ ^k ) < ∞,

where ϕ = P ∞

j=1 b j G ^1/p p .

It is obvious that X t in (2.4) is a solution of (2.1) and (2.2) and it is stationary, since ξ t

is stationary.

(2) We write recursively that

Y _t = ξ _t (a + a

m

X

k=1

X

j

₁

,··· ,j

_k

≥1

b _j

₁

ξ _t−j

₁

· · · b _j

_k

ξ _t−j

₁

_{−···−j}

_k

)

+ξ _t ( X

j

₁

,··· ,j

_m+1

≥1

b _j

₁

ξ _t−j

₁

· · · b _j

_m

ξ _t−j

₁

_{−···−j}

_m

b _j

_m+1

Y _t−j

₁

_{−···−j}

_m

).

Then

X t − Y t = ξ t (a X

k>m

X

j

1

,··· ,j

k

≥1

b j

₁

ξ t−j

₁

· · · b j

_k

ξ t−j

₁

−···−j

_k

)

−ξ t ( X

j

1

,··· ,j

m+1

≥1

b j

₁

ξ t−j

₁

· · · b j

_m

ξ t−j

₁

−···−j

_m

b j

_m+1

Y t−j

₁

−···−j

_m

),

and

kX _t − Y _t k _p ≤ ϕ ^m+1

1 − ϕ aG ^1/p _p + kY ₀ k _p ϕ ^m+1 → 0

as m → ∞. Thus X t = Y t a.s. 

Recall that X t in (2.4) can be rewritten as

X _t = aξ _t + a

∞

X

k=1

∞

X

j

_k

<j

_k−1

<···<j

₁

<t

b _t−j

₁

· · · b j

k−1

−j

k

ξ _t ξ _j

₁

· · · ξ j

k

.

Remark 1. (Stationarity for i.i.d. case) Suppose that {ξ t , t ∈ Z} is a sequence of

independent and identically distributed random variables. Then X _t in (2.4) is a solution to

(2.1) and (2.2). (1) If E(ξ ₀ ) < ∞ and E(ξ ₀ ) P b j < 1, then X _t in (2.4) is strictly stationary

with E(X _t ) < ∞. (2) If E(ξ ₀ ² ) < ∞ and kξ ₀ k ₂ P b j < 1, then X _t is also weakly stationary (Giraitis et al., 2000).

We recall the definition of λ-weakly dependent introduced by Doukhan and Louhichi (1999).

For a function h : R ^d → R, we set

Lip(h) = sup _(x

₁

_{,··· ,x}

_d

_)6=(y

₁

_{,··· ,y}

_d

₎ |h(x 1 , · · · , x d ) − h(y 1 , · · · , y d )|

P d

i=1 |x _i − y _i | .

Let L be the set of real valued functions h on R ^u for some positive integer u such that khk ∞ ≤ 1 and Lip(h) < ∞. A process (X t , t ∈ Z) with values in R is (, ψ) weakly dependent process if there exist a sequence ( _r , r ∈ N ) such that  _r → 0 as r → ∞ and a function ψ : N ² × (R ⁺ ) ² → R ⁺ such that

|Cov(f (X _i

₁

, · · · , X _i

_u

), g(X _j

₁

, · · · , X _j

_v

))| ≤ ψ(u, v, Lip(f ), Lip(g)) _r ,

for any r ≥ 0, any (u + v)-tuples such that i 1 ≤ · · · ≤ i u ≤ i u + r ≤ j 1 ≤ · · · ≤ j v , and any functions f, g ∈ L which are defined on R ^u and R ^v respectively. (X t , t ∈ Z) is said to be λ-weakly dependent if ψ(u, v, a, b) = ua + vb + uvab. In this case we write  r = λ r or λ X

_t

(r).

With X t given by (2.4), define

U = (X i

₁

, X i

₂

, · · · , X i

_u

), V = (X j

₁

, X j

₂

, · · · , X j

_v

). (2.5) Choose s ≥ 1 arbitrarily and fix. Define

U ˆ ^(s) = ( ˆ X _i ^(s)

₁

, ˆ X _i ^(s)

₂

, · · · , ˆ X _i ^(s)

_u

), ˆ V ^(s) = ( ˆ X _j ^(s)

₁

, ˆ X _j ^(s)

₂

, · · · , ˆ X _j ^(s)

_v

), (2.6) where ˆ X _t ^(s) is given by

X ˆ _t ^(s) = aξ t + aξ t

∞

X

k=1

X

j

₁

+···+j

_k

<s

b j

₁

· · · b j

_k

ξ t−j

₁

· · · ξ t−j

₁

−···−j

_k

. (2.7)

We set A(x) = P

j≥x b _j and assume that f, g ∈ L.

Lemma 2.1 Suppose that with p > 1 (A1) and ϕ < 1 hold. Then |E(f (U ) − f ( ˆ U ^(s) ))| ≤ Lip(f )uθ s and |E(g(V ) − g( ˆ V ^(s) ))| ≤ Lip(g)vθ s where

θ s = aG ^1/p _p G ^1/p _p

s−1

X

k=1

kϕ ^k−1 A(s/k) + ϕ ^s /(1 − ϕ)

!

. (2.8)

Proof: Note that for any t ∈ Z, E|X t − ˆ X _t ^(s) | = E|aξ t

∞

X

k=1

X

j

₁

+···+j

_k

≥s

b j

₁

· · · b j

_k

ξ t−j

₁

· · · ξ t−j

₁

−···−j

k

|

≤ aG ^1/p _p

s−1

X

k=1

k X

j

₁

≥s/k

b _j

₁

X

j

₂

,···j

_k

≥1

b _j

₂

· · · b j

k

G ^k/p _p

+aG ^1/p _p

∞

X

k=s

X

j

1

,··· ,j

k

≥1

b j

₁

· · · b j

_k

G ^k/p _p

≤ aG ^1/p _p G ^1/p _p

s−1

X

k=1

kϕ ^k−1 A(s/k) + ϕ ^s /(1 − ϕ)

!

= θ s . Thus, |E(f (U ) − f ( ˆ U ^(s) ))| ≤ Lip(f ) P u

j=1 E|X _i

_j

− ˆ X _i ^(s)

j

| ≤ Lip(f )uθ s . Similarly, we have that

|E(g(V ) − g( ˆ V ^(s) ))| ≤ Lip(g)

v

X

i=1

E|X _j

_i

− ˆ X _j ^(s)

i

| ≤ Lip(g)vθ _s .

 Remark 2. (1) Note that θ s → 0 as s → ∞.

(2) It is known that if b, C > 0 and 0 ≤ ρ < 1, then for suitable choice of K, K ⁰ , we have that θ s ≤ K ^{(log s)} _s

b^b∨1

for A(x) ≤ Cx ^−b and θ s ≤ K ⁰ (ρ ∨ ϕ)

√ s for A(x) ≤ Cρ ^x (Doukhan et al., 2006).

We make the additional assumption:

(A2) {ξ t } is λ-weakly dependent with ψ(u, v, a, b) = (ua + vb + uvab) and the weak depen- dence coefficients λ ξ (r).

Lemma 2.2 Consider the process ˆ X _t ^(s) defined by the equation (2.7). Suppose that the assumptions (A1) and (A2) hold and there is (m, m ⁰ ) with E|ξ t | ^m

⁰

< ∞, for m > 2 and m ⁰ ≥ (s + 1)m. Then ˆ X _t ^(s) is λ-weakly dependent with weak dependence coefficients

λ ⁰ _ˆ

X

_t^(s)

(r) = c



 X

j≥[r/3]

ja ^(s) _j + ([2r/3] + 1) ² λ _ξ ([r/3]) ^1/2



 .

Here c is some constant, a ^(s) _j is defined in the proof below, and [x] denotes the largest integer less than or equal to x.

Proof: For x = (x i ≥ 0, i ∈ Z), define

H s (x) = ax 0 + ax 0 s−1

X

k=1

X

j

₁

+···+j

_k

<s

b j

₁

· · · b j

_k

x j

₁

x j

₁

+j

₂

· · · , x j

₁

+···+j

_k

.

Then ˆ X _t ^(s) = H s (ξ t−j , j ∈ Z). Let

B _0,k ^(s) = X

j

₁

+···+j

_k

<s

b _j

₁

· · · b _j

_k

,

B _l,k ^(s) =

k

X

u=1

X

j1+···+ju=l

j

u+1

+···+j

k

<s−l

b _j

₁

· · · b _j

_u

b _j

_u+1

· · · b _j

_k

, 1 ≤ l ≤ s − 1,

and

a ^(s) ₀ = a + a

s−1

X

k=1

B _0,k ^(s) , a ^(s) _l = a

s−1

X

k=1

B ^(s) _l,k , 1 ≤ l ≤ s − 1, a ^(s) _l = 0, l ≥ s.

Then for x, y ∈ (R ⁺ ) ^Z with x _l 6= y l , x _i = y _i (i 6= l), z _l = 0, z _i = x _i (i 6= l) and kzk = sup _{0≤i≤s−1} |z _i |,

|H s (x) − H s (y)| ≤ a ^(s) _l (kzk ^s ∨ 1)(x l − y l ).

Since a ^(s) _l = 0, l ≥ s, P

l≥1 la ^(s) _l < ∞. Apply Lemma 3.2 in Doukhan and Wintenberger (2007) to obtain that ˆ X _t ^(s) is strictly stationary with E| ˆ X _t ^(s) | ^m < ∞, ˆ X _t ^(s) is λ-weakly dependent, and there exists a constant c > 0 such that

λ _{X ˆ}

(s) t

(r) ≤ c inf

q≤[r/2] ( X

j≥q

ja ^(s) _j + (2q + 1) ² λ ξ (r − 2q)

^{m0 −1−sm0 −1}

). (2.9)

Take q = [r/3] and use (m ⁰ − 1 − s)/(m ⁰ − 1) ≥ 1/2 in (2.9) to obtain that

λ _{X ˆ}

(s) t

(r) ≤ λ ⁰ _ˆ

X

^(s)_t

(r) = c



 X

j≥[r/3]

ja ^(s) _j + ([2r/3] + 1) ² λ _ξ ([r/3]) ^1/2



 , (2.10)

and

Cov(f ( ˆ U ^(s) ), g( ˆ V ^(s) )) ≤ ψ(u, v, Lip(f ), Lip(g))λ ⁰ _ˆ

X

_t^(s)

(r).



Theorem 2.2 Suppose that with p > 2, the assumptions (A1), (A2) and ϕ < 1 hold. If (1) E|ξ t | ^m

⁰

< ∞ for all m ⁰ > 0,

(2) (2r/3 + 1) ² λ ξ (r/3) ^1/2 → 0 as r → ∞,

Then X t in (2.4) is stationary, λ-weakly dependent and E|X t | ^m < ∞, m > 2. This means that for any f, g ∈ L and U, V given in (2.5),

|Cov(f (U ), g(V ))| ≤ ψ(u, v, Lip(f ), Lip(g))λ ^∗ _r , with λ ^∗ _r = 2θ _[r/4] + λ ⁰ _ˆ

X

_t^[r/4]

(r), where θ _[r/4] and λ ⁰ _ˆ

X

_t^[r/4]

(r) are given in (2.8) and (2.10) with

s = [r/4] respectively.

Proof: Let U, V and ˆ U ^(s) , ˆ V ^(s) be given in (2.5) and (2.6). Then

|Cov(f (U ), g(V ))| ≤ |Cov(f (U ) − f ( ˆ U ^(s) ), g(V ))|

+|Cov(f ( ˆ U ^(s) ), g(V ) − g( ˆ V ^(s) ))| + |Cov(f ( ˆ U ^(s) ), g( ˆ V ^(s) ))|. (2.11) Let s = [r/4]. Use Lemma 1 to obtain that

|Cov(f (U ) − f ( ˆ U ^[r/4] ), g(V ))| ≤ 2kgk _∞ Lip(f )uθ _[r/4] (2.12) and

|Cov(f ( ˆ U ^[r/4] ), g(V ) − g( ˆ V ^[r/4] ))| ≤ 2kf k ∞ Lip(g)vθ [r/4] . (2.13) By Lemma 2,

|Cov(f ( ˆ U ^[r/4] ), g( ˆ V ^[r/4] ))| ≤ (uLip(f ) + vLip(g) + uvLip(f )Lip(g))λ ⁰ _ˆ

X

_t^[r/4]

(r). (2.14) Combine (2.8), (2.10)-(2.14) to obtain that

|Cov(f (U ), g(V ))| ≤ (uLip(f ) + vLip(g) + uvLip(f )Lip(g))λ ^∗ (r), where λ ^∗ (r) = 2θ _[r/4] + λ ⁰ _ˆ

X

_t^[r/4]

(r). Here θ _[r/4] → 0 as r → ∞, λ ⁰ _ˆ

X

_t^[r/4]

(r) ≤ c(2r/3 + 1) ² λ _ξ (r/3) ^1/2 → 0 as r → ∞ by assumption (2) above, and a ^(s) _l = 0 if l ≥ s.  Corollary 2.1 Suppose that the assumptions in Theorem 2 hold. Fix λ > 4+2/(m−2), m >

2. If one of the following (1)-(4) holds, then λ ^∗ (r) = O(r ^−λ ) as r → ∞ where λ ^∗ (r) is given in Theorem 2.

(1) A(x) ≤ cρ ^x and λ _ξ (r) = O(e ^−ra ), a > 0.

(2) A(x) ≤ cρ ^x and λ _ξ (r) = O(r ^−a ), a > λ.

(3) A(x) = cx ^−b , b > λ, and λ _ξ (r) = O(e ^−ra ), a > 0.

(4) A(x) = cx ^−b , b > λ, and λ _ξ (r) = O(r ^−a ), a > λ.

Proof: (1) For fixed m > 2, let λ > 4 + 2/(m − 2). Then for sufficiently large r, (ϕ ∨ ρ) ^{√ r/2} ≤ r ^−λ and (2r/3 + 1) ² (e ^−(r/3)a ) ^1/2 ≤ r ^−λ . Therefore λ ^∗ (r) = O(r ^−λ ) for λ > 4 + 2/(m − 2).

(2) If a > λ, then λ ^∗ (r) = O(r ^−λ ).

(3)and (4) If A(x) ≤ cx ^−b , b > λ, then θ _[r/4] = c (log([r/4])/[r/4]) ^b ≤ cr ^−λ . Hence λ ^∗ (r) =

O(r ^−λ ). 

Theorem 2.3 Consider the ARCH(∞) model defined by the equation (2.4) with λ-weakly dependent innovation {ξ t }. Suppose that with p > 2, the assumptions (A1), (A2), ϕ < 1, and E|ξ 1 | ^m

⁰

< ∞ for all m ⁰ > 0. Then the FCLT holds for X t under the condition that one of the (1)-(4) in above Corollary 1 is satisfied.

Proof: Combine the Theorem in Section 1, Theorem 2, and Corollary 1 to get the

results. 

Remark 3.( FCLT for i.i.d. case) Suppose that {ξ t , t ∈ Z} is a sequence of independent

and identically distributed random variables. If kξ ₀ k 2 P b j < 1, then X _t and ρ _t in (2.1) and

(2.2) obey the FCLT (Lee, 2018).

One of the best-known model in economics and financial time series field is the GARCH model. As an example, consider GARCH(p, q) process with λ-weakly dependent errors:

u t = σ t e t , σ _t ² = ω +

p

X

i=1

α i u ² _t−i +

q

X

j=1

β j σ _t−j ² ,

where ω > 0, α _i ≥ 0, β j ≥ 0(i = 1, 2, · · · , p, j = 1, 2, · · · , q, p ≥ 0, q ≥ 0) are model parameters. It is known that under proper constraints on the polynomials α(x) = 1 − P p

i=1 α _i x ⁱ , β(x) = 1 − P q

j=1 β _j x ^j , the conditional variance σ _t ² can be written as σ ² _t = a + P ∞

j=1 b j u ² _t−j , where b j decays geometrically ( i.e., b j = O(ρ ^j ), 0 < ρ < 1) (see Nelson and Cao, 1992). Let λ > 4 + 2/(m − 2), m > 2 and ξ t = e ² _t , E(ξ _t ^m

⁰

) < ∞ for all m ⁰ ≥ 0.

If λ ξ (r) = O(e ^−ra ), a > 0 or λ ξ (r) = O(r ^−a ), a > λ, then λ ^∗ (r) = O(r ^−λ ) and hence FCLT holds for u ² _t . For GARCH(1,1), b j = α 1 (β 1 ) ^j−1 . For GARCH(2,2), b j is a solution to the difference equation b ₀ = α ₁ , b ₁ = β ₁ b ₀ + α ₂ and for j ≥ 2, b _j = β ₁ b _j−1 + β ₂ b _j−2 . If P α i + P β j < 1, then b _j ≤ ρ ^j , 0 ≤ ρ = (β ₁ + β ₂ ) ^1/2 < 1.

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