Bhattacharya R., Majumdar M. Random Dynamical Systems: Theory and Applications

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4.2 Iterates of Real-Valued Afﬁne Maps (AR(1) Models) 303

Remark 2.1 The case b = 1 leads to the general random walk with

arbitrary step size distribution: X

= X

+ ε

+···+ε

(n ≥ 1), which

does not converge in distribution unless ε

= 0 with probability 1.

Remark 2.2 The case b = 0 in (2.1) yields X

= ε

(n ≥ 1), which is

trivially stable in distribution and has as its unique invariant probability

the distribution of ε

Remark 2.3 The case |b| < 1 is generally referred to as the stable case

of the model (2.1). Part (b) of Theorem 2.1 shows, however, that stability

fails no matter how small a nonzero |b| is, if the innovations ε

cross a

threshold of heavy-tailed distributions.

The following lemma has been used in the proof:

Lemma 2.1 Let Z be a nonnegative random variable. Then

∞



j=1

P(Z ≥ j) = E[Z ], (2.24)

where [Z] is the integer part of Z .

Proof.

E[Z ] = 0P(0 ≤ Z < 1) +1P(1 ≤ Z < 2) + 2P(2 ≤ Z < 3)

+3P(3 ≤ Z < 4) +···+ jP( j ≤ Z < j + 1) +···

={P(Z ≥ 1) − P(Z ≥ 2)}+2{P(Z ≥ 2) − P(Z ≥ 3)}

+3{P(Z ≥ 3) − P(Z ≥ 4)}+···+ j {P(Z ≥ j)

−P(Z ≥ j + 1)}+···

= P(Z ≥ 1) + P(Z ≥ 2) + P(Z ≥ 3)

+···+P(Z ≥ j) +···. (2.25)

Remark 2.4 The hypothesis “ε

is nondegenerate” in the theorem is not

quite needed, except for the fact that if P(ε

= c) = 1 and b = 1, then

= b

+ c(b

− 1)/(b − 1), and −c/(b − 1) = z,say,isaﬁxed

point, i.e., δ

is an invariant probability. But even in this last case, if

> 1 then X

does not converge if X

= z.Ifb =+1 and ε

= 0 a.s.,

304 Random Dynamical Systems: Special Structures

then x is a ﬁxed point for every x, so that δ

is an invariant probability

for every x ∈

R.Ifb =−1 and ε

= c a.s., then x =

is a ﬁxed point

and {x, −x + c}is a periodic orbit for every x =

; it follows in this case

that

−x +c

are invariant probabilities for every x ∈ R. These are

the only possible extremal invariant probabilities for the case

≥ 1. In

particular, if (2.8) holds (irrespective of whether ε

is a degenerate or

not), then (2.9) is necessary and sufﬁcient for stability in distribution of

4.3 Linear Autoregressive (LAR(k )) and

Other Linear Time Series Models

Although the results of this section hold with respect to other norms also,

for speciﬁcity, we take the norm of a matrix A to be deﬁned by



= sup

, (3.1)

where

denotes the Euclidean norm of the vector x.

Theorem 3.1 Let S =



, and (A

,ε

)(i ≥ 1) be an i.i.d. sequence with

’

s random (k × k) matrices and ε

’

s random vectors (-dimensional).

If, for some r ≥ 1,

−∞ ≤ E log



···A



< 0, E log

< ∞, (3.2)

and

E log



< ∞, (3.3)

then the Markov process deﬁned recursively by

n+1

= A

n+1

+ ε

n+1

(n ≥ 0) (3.4)

has a unique invariant probability π and it is stable in distribution.

Proof. To apply Theorem 7.2 of Chapter 3 with α

(x) = A

x + ε

, and

Euclidean distance for d, note that

d(α

···α

x,α

···α

y) = d(A

···A

x, A

···A

r−1

···A

(x − y)

≤



···A



x − y

. (3.5)

4.3 Linear Autoregressive (LAR(k)) 305

Thus the ﬁrst condition in (3.2) implies (7.2) of Chapter 3 in this case. To

verify (7.9) of Chapter 3, take x

= 0. If r = 1 then d(α

(0), 0) =

and the second relation in (3.2) coincides with (7.9) of Chapter 3. In this

case (3.3) follows from the ﬁrst relation in (3.2): E log



is ﬁnite

because E log



exists and is negative. For r > 1,

d(α

···α

0, 0) =

···A

+ A

···A

+···

+ A

r−1

+ ε

≤



j=2



···A

·|ε

j−1

log

d(α

···α

(0), 0) ≤



j=2

log(



···A

·|ε

j−1

|+1)

+ log(

+ 1),

log(



···A

·|ε

j−1

|+1) ≤ log(



+ 1) +···+log(A

+1)

+ log(|ε

j−1

|+1). (3.6)

Now E log(



+1) ≤E log



+1, and E log(

+ 1) ≤E(log

) + 1), and both are ﬁnite by (3.2) and (3.3). Hence (3.6) implies

E log

d(α

···α

(0), 0) < ∞, i.e., the hypotheses of Theorem 7.2 of

Chapter 3 are satisﬁed.

Remark 3.1 For the case r = 1, Theorem 3.1 is contained in a result

of Brandt (1986). For r > 1, the above result was obtained by Berger

(1992) using a different method.

As an immediate corollary of Theorem 3.1, we have the following mul-

tidimensional extension of Theorem 2.1. To state it consider the Markov

process deﬁned recursively by

n+1

= AX

+ ε

n+1

(n ≥ 0), (3.7)

where A is a (constant) k × k matrix, ε

(n ≥ 1) an i.i.d. sequence of

k-dimensional random vectors independent of a given k-dimensional

random vector X

Corollary 3.1 Suppose

A

 < 1for some positive integer n

(3.8)

306 Random Dynamical Systems: Special Structures

and

E log

|ε

| < ∞. (3.9)

Then the Markov process (3.7) has a unique invariant probability π and

it is stable in distribution.

For applications, it is useful to know that the assumption (3.8) holds if

the maximum modulus of eigenvalues of A, also known as the spectral

radius r(A) of A, is less than 1. This fact is implied by the following

result from linear algebra, whose proof is given in Complements and

Details.

Lemma 3.1 Let A be an m × m matrix. Then the spectral radius r(A)

satisﬁes

r(A)

 lim

n→∞

A



1/n

. (3.10)

Two well-known time series models will now be treated as special

cases of Corollary 3.1. These are the kt h-order autoregressive (or AR(k))

model and the autoregressive moving-average model (ARMA(p, q)).

Example 3.1 (AR(k) model) Let k > 1 be an integer and β

,β

,...,

k−1

real constants. Given a sequence of i.i.d. real-valued random vari-

ables {η

: n ≥ k}, and k other random variables U

, U

,...,U

k−1

inde-

pendent of {η

}, deﬁned recursively

n+k

k−1



i=0

n+i

+ η

n+k

(n ≥ 0). (3.11)

The sequence {U

}is not, in general, a Markov process, but the sequence

of k-dimensional random vectors

:= (U

, U

n+1

,...,U

n+k−1

)



(n ≥ 0) (3.12)

is Markovian. Here the prime (



) denotes transposition, so X

is to be

regarded as a column vector in matrix operations. To prove the Markov

property, consider the sequence of k-dimensional i.i.d. random vectors

:= (0, 0,...,0,η

n+k−1

)



(n ≥ 1), (3.13)

4.3 Linear Autoregressive (LAR(k)) 307

and note that

n+1

= AX

+ ε

n+1

, (3.14)

where A is the k × k matrix

A: =







0100... 00

0010... 00

........ .

0000... 01

... β

k−2

k−1







. (3.15)

Hence, as in (3.4), {X

} is a Markov process on the state space R

. Write

A − λI =







−λ 100... 00

0 −λ 10... 00

........ .

0000... −λ 1

... β

k−2

k−1

− λ







Expanding det(A − λI ) by its last row, and using the fact that the determi-

nant of a matrix in triangular form (i.e., with all off-diagonal elements

on one side of the diagonal being zero) is the product of its diagonal

elements (see Exercise 3.1), one gets

det(A − λI ) = (−1)

k+1

(β

+ β

λ +···+β

k−1

− λ

). (3.16)

Therefore, the eigenvalues of A are the roots of the equation

+ β

λ +···+β

k−1

− λ

= 0. (3.17)

Finally, in view of (3.10), the following proposition is a consequence of

Theorem 3.1:

Proposition 3.1 Suppose that the roots of the polynomial Equation

(3.17) are all strictly inside the unit circle in the complex plane, and

that the common distribution G of {η

} satisﬁes

E log

|η

| < ∞. (3.18)

Then (i) there exists a unique invariant distribution π for the Markov pro-

cess {X

}deﬁned by (3.12) and (ii) no matter what the initial distribution,

converges in distribution to π.

308 Random Dynamical Systems: Special Structures

It is simple to check that (3.18) holds if G has a ﬁnite absolute moment

of some order r > 0 (Exercise).

An immediate consequence of Proposition 3.1 is that the time series

: n ≥ 0} converges in distribution to a steady state π

given, for all

Borel sets C ⊂

, by

: x

(1)

∈ C}). (3.19)

Here x

(1)

is the ﬁrst coordinate of x = (x

(1)

,...,x

(k)

). To see this, simply

note that U

is the ﬁrst coordinate of X

, so that X

converges to π in

distribution implies U

converges to π

in distribution.

Example 3.2 (ARMA(k, q) model) The autoregressive moving-average

model of order (k, q), in short ARMA(k, q), is deﬁned by

n+k

k−1



i=0

n+i



j=1

n+k−j

+ η

n+k

(n ≥ 0), (3.20)

where k, q are positive integers, β

(0 ≤ i ≤ k − 1) and δ

(1 ≤ j ≤ q)

are real constants, {η

: n ≥ k − q} is an i.i.d. sequence of real-valued

random variables, and U

(0 ≤ i ≤ k − 1) are arbitrary initial random

variables independent of {η

}. Consider the sequences {X

}, {ε

} of

(k + q)-dimensional vectors

= (U

,...,U

n+k−1

,η

n+k−q

,...,η

n+k−1

)



= (0, 0,...,0,η

n+k−1

, 0,...,0,η

n+k−1

)



(n ≥ 0), (3.21)

where η

n+k−1

occurs as the kth and (k + q)th elements of ε

. Then

n+1

= HX

+ ε

n+1

(n ≥ 0), (3.22)

where H is the (k + q) ×(k + q) matrix

H:=











....a

0 .....00

...... . . ......

....a

q−1

....δ

0 .... 00 10... 00

0 .... 00 01... 00

...... . . ......

00... . . 00... 01

00... . . 00... 00











the ﬁrst k rows and k columns of H being the matrix A in (3.15).

4.3 Linear Autoregressive (LAR(k)) 309

Note that U

,...,U

k−1

,η

k−q

,...,η

k−1

determine X

, so that X

independent of η

and, therefore, of ε

. It follows by induction that X

and ε

n+1

are independent. Hence {X

} is a Markov process on the state

space

k+q

In order to apply the Lemma 3.1, expand det(H − λI ) in terms of the

elements of its last row to get (Exercise)

det(H − λI ) = det(A − λI )(−λ)

. (3.23)

Therefore, the eigenvalues of H are q zeros and the roots of (3.17). Thus,

one has the following proposition:

Proposition 3.2 Under the hypothesis of Proposition 3.1, the

ARMA(k, q) process {X

} has a unique invariant distribution π, and

converges in distribution to π, no matter what the initial distribution

is.

As a corollary, the time series {U

} converges in distribution to π

given, for all Borel sets C ⊂ R

, by

(C): = π ({x ∈ R

k+q

: x

(1)

∈ C}), (3.24)

no matter what the distribution of (U

, U

,...,U

k−1

) is, provided the

hypothesis of Proposition 3.2 is satisﬁed.

In the case that ε

is Gaussian, it is simple to check that the random

vector X

in (3.7) is Gaussian, if X

is Gaussian. Therefore, under the hy-

pothesis (3.8), π is Gaussian, so that the stationary vector-valued process

} with initial distribution π is Gaussian (also see Exercise 3.2). In

particular, if η

are Gaussian in Example 3.1, and the roots of the poly-

nomial equation (3.17) lie inside the unit circle in the complex plane,

then the stationary process {U

}, obtained when (U

, U

,...,U

k−1

) has

distribution π in Example 3.1, is Gaussian. A similar assertion holds for

Example 3.2.

Exercise 3.1 Prove that the determinant of an m × m matrix in tri-

angular form equals the product of its diagonal elements. [Hint: Let

, 0, 0,...,0) be the ﬁrst row of triangular m × m matrix. Then its

determinant is a

times the determinant of an (m − 1) ×(m − 1) trian-

gular matrix. Use induction on m.]

Exercise 3.2 Under the hypothesis of Corollary 3.1, (a) mimic the steps

(2.3)–(2.6) to show that the invariant distribution π of X

(n ≥ 0) is given

310 Random Dynamical Systems: Special Structures

by the distribution of Y :=



∞

n=0

n+1

. (b) In the special case ε

(n ≥ 1)

are Gaussian, show that the invariant distribution in (a) is Gaussian, or

normal, and compute its mean and dispersion matrix (assume those of

are µ and



, respectively).

4.4 Iterates of Quadratic Maps

Recall that the family {F

:0≤ θ ≤ 4} of quadratic, or logistic maps, is

deﬁned by

(x) = θ x(1 − x), 0 ≤ x ≤ 1. (4.1)

To study iterations of random (i.i.d.) quadratic maps, we will make use

of Proposition 7.1 of Chapter 1, as well as the following lemma:

Lemma 4.1 Let 1 <µ<λ<4. Let u = min{1 −

, F

(

)}, v =

Then, for every θ ∈ [µ, λ], [u,v] is invariant under F

Proof. We need to prove that (i) max{F

(x): u ≤ x ≤ v}≤v for

θ ∈ [µ, λ] and (ii) min{F

(x): u ≤ x ≤ v}≥u for θ ∈ [µ, λ]. The

ﬁrst of these follows from the relations max{F

(x): u ≤ x ≤ v}≤

max{F

(x): u ≤ x ≤ v}≤

. For (ii) note that by unimodality,

min{F

(x): u ≤ x ≤ v}=min{F

(u), F

(v)}≥min{F

(u), F

(v)}.If

u = 1 −

≤ F

(

), then the last minimum is F

(u) = 1 −

= u.If

u = F

(

) < 1 −

, then min{F

(u), F

(

)}=F

(

) = u, since on

(0, 1 −

), F

(x) > x.

Let C

(n ≥ 1) be a sequence of i.i.d. random variables with values in

[µ, λ], where 1 <µ<λ<4. Then

:= F

, n ≥ 1 (4.2)

is a sequence of random quadratic maps. We consider the Markov process

deﬁned recursively by

, X

n+1

= α

n+1

(n ≥ 0), (4.3)

where X

takes values in (0, 1), and is independent of {C

: n ≥ 1}.For

Theorem 4.1 on the stability in distribution of this Markov process, recall

that if ν is a probability measure on

R with compact support, then there

is a smallest point and a largest point of its support. As always, B(S)

denotes the Borel sigmaﬁeld of a metric space S.

4.4 Iterates of Quadratic Maps 311

Theorem 4.1 Let P(µ

≤ C

≤ λ

) = 1, where 1 <µ

<λ

< 4. As-

sume that the distribution Q of C

has a nonzero absolutely contin-

uous component with density h (with respect to the Lebesgue mea-

sure) which is bounded away from zero on an interval [c, d] with

≤ c < d ≤ 3,d ≤ λ

. Then the Markov process X

(n ≥ 0) deﬁned by

(4.3) on S = (0, 1) has a unique invariant probability π whose support is

contained in [u

] with u

as in Lemma 4.1 for µ = µ

and λ = λ

;

and for every interval [u,v] with 0 < u ≤ u

<v<4, one has

sup

x∈[u,v],B∈B([u,v])

(n)

(x, B) − π(B)|≤(1 −χ )

[n/m]

,n≥ 1, (4.4)

where χ = χ (u,v) is a positive constant and m = m(u,v) is a positive

integer, depending on u,v.

Proof. By Lemma 4.1, if X

∈ S

≡ [u

], X

∈ S

n ≥ 1. Hence

(n ≥ 0) is a Markov process on S

. We will show that the hypothe-

sis of Corollary 5.3 of Chapter 3 is satisﬁed. For this ﬁrst assume, for

the sake of simplicity, that the distribution of C

is absolutely continu-

ous with a density h(θ ) which is continuous on [µ

,λ

] and is positive

on [c, d]. By (4.3), the transition probability density of X

(n ≥ 0) is

given by

p(x, y) =

x(1 − x)



x(1 − x)



, x, y ∈ [u

]. (4.5)

Let p

= 1 − (1/θ) be the ﬁxed point of F

on (0, 1). For θ ∈ [c, d],

p(p

, p

) = ( p

(1 − p

))

−1

h(θ) > 0. In particular, for θ

= (c + d)/2,

writing x

= p

, one has p(x

, x

) > 0. Since (x, y) → p(x, y) is con-

tinuous on S

× S

, there exists an interval [x

, x

], x

< x

such that

:= inf

x,y∈[x

]

p(x, y) > 0. (4.6)

It follows from Proposition 7.4(b) of Chapter 1 that, for each y ∈ [u

there exists an integer n(y) such that

(y) ∈ (x

, x

)(y ∈ [u

]), ∀n ≥ n(y).

The set 0(y):={z ∈ [u

]: F

n(y)

(z) ∈ (x

, x

)} is open in [u

and y ∈ 0(y). By the compactness of [u

], there exists a ﬁnite set

, y

,...,y

} such that ∪

i=1

0(y

) = [u

]. Let N := max{n(y

312 Random Dynamical Systems: Special Structures

1 ≤ i ≤ k}. Then F

(z) ∈ (x

, x

) ∀z ∈ [u

]. Since the function

g(θ

,θ

,...,θ

, z):= F

N −1

···F

(z) is (uniformly) continuous

on [c, d]

× [u

], g(θ

,θ

,...,θ

):= min{F

N −1

···F

(z):

z ∈ [u

]} and g(θ

,θ

,...,θ

):= max{F

N −1

(z): z ∈

]} are continuous on [c, d]

. Also, g(θ

,θ

,...,θ

) =

min{F

(z): z ∈ [u

]} > x

and g(θ

,θ

,...,θ

) = max{F

(z):

z ∈ [u

]} < x

. Therefore there exists ε

> 0 such that

N −1

···F

(z) ∈ (x

, x

) ∀θ

∈ (θ

− ε

,θ

+ ε

)

⊂ [c, d], 1 ≤ i ≤ N , and ∀z ∈ [u

]. (4.7)

Hence

(z) = F

N −1

···F

(z) ∈ (x

, x

if C

∈ (θ

− ε

,θ

+ ε

) ∀i,1≤ i ≤ N . (4.8)

Therefore, ∀x ∈ [u

] and for every Borel set B ⊂ [u

], one has

by (4.8), and using (4.6) in the second inequality below,

(N +1)

(x, B) = P

N +1

∈ B)

≥ P

N +1

∈ B ∩ (x

, x

), {C

∈ (θ

− ε

,θ

+ ε

), ∀i,1 ≤ i ≤ N })

= E

N +1

∈ B ∩ (x

, x

) | X

) · 1

∈(θ

−ε

,θ

+ε

), 1≤i≤N }

]

≥ E

B∩(x

)

dy · 1

∈(θ

−ε

,θ

+ε

), 1≤i≤N }

= δ

m(B ∩ (x

, x

))δ







(θ

−ε

,θ

+ε

)

h(θ) dθ







, (4.9)

where P

, E

denote probability and expectation under X

= x, and

m denotes the Lebesgue measure. Thus Corollary 5.3 of Chapter 3

applies on the state space [u

], with N + 1 in place of m, λ(B) =

m(B ∩ (x

, x

)),χ = χ(u

) = δ

− x

). Hence there ex-

ists a unique invariant probability π on [u

], and (4.4) holds for the

case u = u

, v = v

Now note that the same proof goes through if we replace [u

]by

any larger interval [u

] corresponding to µ = µ

,λ = λ

in Lemma

3.1 with 1 <µ

<µ

<λ

< 4. By letting µ

,λ

be sufﬁciently

close to 1 and 4, respectively, one can choose u

as close to 0, 1,

respectively, as we like, so that (4.4) holds with u = u

, and v = v

and

χ = χ(u

). If one is given arbitrary 0 < u <v<1, one may ﬁnd