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

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3.8 Complements and Details 293

the Cantor distribution µ with support C. Let µ

be the distribution of

X + 1/3

(n = 1, 2,...). It is simple to see that µ

(R\C) = 1 and the

d.f. F

of µ

converges to the d.f. F of µ uniformly, i.e., in the distance

= d

(since, considered as probability measures on R, µ

converges

weakly to µ, and the distribution function F of µ is continuous). On the

other hand µ(S) = 0, so that µ/∈ P(S).

Remark C5.3 The metric d

(see (C5.5), (C5.6)) was introduced in

Bhattacharya and Lee (1988). If (P(S), d

) is complete, then one can

dispense with Lemmas C5.1, C5.2 for the proof of Theorem C5.1, and

) for i.i.d. monotone increasing maps sufﬁces. But the completeness

of (P(S), d

) is in general a delicate matter as shown in Chakraborty and

Rao (1998), where a comprehensive analysis is given for dimensions 1

and 2.

Section 3.6. The transition probability in Example 6.3 does not sat-

isfy the stochastic monotonicity condition assumed in Hopenhayn and

Prescott (1992). To see this consider the increasing function h(x) = x

on [a, b], and note that

Th(x) ≡

h(y)p(x, dy) = ph (F(x)) + (1 − p)h(G(x)) (C6.1)

= pF(x) + (1 − p)G(x) = p(3/4)x

1/2

+ (1 − p)θx(1 − x).

Take p = 1/2. Then, writing the right-hand side of (C6.1) as φ(x), one

has

φ(x) = 3/8x

1/2

+ 1.55(x − x



(x) = 3/16x

−1/2

+ 1.55(1 −2x).

It is simple to check that φ



(r) ≡ φ



(9/16) > 0, and φ



(0.7) < 0. Thus φ

is not monotone. Hence the results of Hopenhayen and Prescott (1992)

do not imply Theorem 5.1 or its special cases considered in Subsection

3.6.3.

Subsection 3.6.4. Theorem 6.2 follows from Kifer (1986, Theorem 1.1).

An analogous result is contained in Stokey and Lucas (1989, p. 384),

but, unlike the latter, we do not require that X

has a unique invariant

probability.

294 Random Dynamical Systems

Section 3.7. Theorem 7.2 is in Silverstrov and Stenﬂo (1998). See also

Stenﬂo (2001) and Elton (1990). For additional results on iterations of

i.i.d. contractions, including measurability questions, rates of conver-

gence to equilibrium, and a number of interesting applications, we re-

fer to Diaconis and Freedman (1999). Random iterates of linear and

distance-diminishing maps have been studied extensively in psychology:

the monograph by Norman (1972) is an introduction to the literature with

a comprehensive list of references. In the context of economics, the liter-

ature on learning has more recently been studied and surveyed by Evans

and Honkapohja (1995, 2001). The “reduced form” model of Honkapo-

hja and Mitra (2003) is a random dynamical system with a state space

S ⊂



. They use Theorem 7.3 to prove the existence and stability of an

invariant distribution.

3.9 Supplementary Exercises

(1) On the state space S ={1, 2} consider the maps γ

,γ

S deﬁned as γ

(1) = γ

(2) = 1, γ

(1) = γ

(2) = 2, γ

(1) = 2, and

(2) = 1. Let {α

: n ≥ 1} be i.i.d. random maps on S such that

P(α

= γ

) = θ

( j = 1, 2, 3), θ

+ θ

= 1.

(a) Write down the transition probabilities of the Markov process {X

n ≥ 0} generated by iterations of α

(n ≥ 1) (for an arbitrary initial state

independent of {α

: n ≥ 1}). [Hint: p

= θ

+ θ

(b) Given arbitrary transition probabilities p

(i, j = 1, 2) satisfying

+ p

≤ 1 (i.e., p

≤ p

), ﬁnd θ

,θ

to construct the corre-

sponding Markov process. [Hint: p

= θ

, p

= θ

be deﬁned by γ

(1) = 1,

(2) = 2. Let {α

: n ≥ 1} be i.i.d. with P(α

= γ

) = θ

( j =

1, 2, 3, 4). Given arbitrary transition probabilities p

(i, j = 1, 2), ﬁnd

( j = 1, 2, 3, 4) to construct the corresponding Markov process (by

iteration of α

(n ≥ 1)). [Hint: p

= θ

+ θ

, p

= θ

+ θ

(2) Consider the model in Example 4.2 on the state space S = [−2, 2]

with {ε

: n ≥ 1} i.i.d. uniform on [−1, 1].

(a) Write down the transition probability density p(x, y), i.e., the prob-

ability density of X

(x) = f (x) +ε

. [Hint: Uniform distribution on

[ f (x) − 1, f (x) + 1].]

(b) Compute the distribution of X

(x) = f (X

(x)) + ε

. [Hint: Sup-

pose x ∈ [−2, 0]. Then X

(x) is uniform on [x, x + 2], and f (X

(x))

3.9 Supplementary Exercises 295

is uniform on [−1, 1]. For f maps [x, 0] onto [x + 1, 1], and it maps

(0, x + 2] onto (−1, x + 1], the maps being both linear and of slope 1.

Finally, f (X

(x)) + ε

is the sum of two independent uniform random

variables on [−1, 1] and, therefore, has the triangular distribution on

[−2, 2] with density (2 −|y|)/4, −2 ≤ y ≤ 2 (by direct integration).]

Random Dynamical Systems: Special Structures

But a Grand Uniﬁed Theory will remain out of the reach of economics, which

will keep appealing to a large collection of individual theories. Each one of them

deals with a certain range of phenomena that it attempts to understand and to

explain.

Gerard Debreu

The general equilibrium models used by macroeconomists are special.

Thomas J. Sargent

4.1 Introduction

In this chapter, we explore in depth the qualitative behavior of several

classes of random dynamical systems that have been of long-standing

interest in many disciplines. Here the special structures are exploited

to derive striking results on the existence and stability of invariant dis-

tributions. A key role is played by the central results on splitting and

contractions, reviewed in Chapter 3. In Sections 4.2 and 4.3 our focus is

on linear time series, especially autoregressive time series models. Next,

we consider in Section 4.4 random iterates of quadratic maps. We move

on to nonlinear autoregressive (NLAR(k)) and nonlinear autoregressive

conditional heteroscedastic (NLARCH(k)) models, which are of consid-

erable signiﬁcance in modern econometric theory (Section 4.5). Section

4.6 is devoted to random dynamical systems on the state space

called random continued fractions, deﬁned by

n+1

= Z

n+1

(n ≥ 0). (1.1)

296

4.2 Iterates of Real-Valued Afﬁne Maps (AR(1) Models) 297

Two short ﬁnal sections concern stability in distribution of processes

under additive shocks that satisfy nonnegativity constraints (Section 4.7):

n+1

= max(X

+ ε

n+1

, 0) (1.2)

and the survival probability of an economic agent in a model with mul-

tiplicative shocks (Section 4.8).

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

We turn to a random dynamical system where S = R and {ε

}

∞

n=1

is a

sequence of i.i.d. random variables (independent of a given initial random

variable X

), and the evolution of the system is described by

n+1

= bX

+ ε

n+1

(n ≥ 0), (2.1)

where b is a constant. This is the basic linear autoregressive model of

order 1, or the AR(1) model extensively used in time series analysis. It

is useful to recall Example 4.2 of Chapter 1, which suggests (2.1) rather

naturally. Also, with b = 1, the main result applies to models of random

walk.

The sequence X

given by (2.1) is a Markov process with initial dis-

tribution given by the distribution of X

and the transition probability

(of going from x to some Borel set C of

R in one step)

p(x, C) = Q(C − bx ),

where Q is the common distribution of ε

. To be sure, in terms of our

standard notation, the i.i.d. random maps α

(n ≥ 1) are given by

(x) = bx + ε

. (2.2)

From (2.1) we get

= b

+ b

n−1

+···+bε

n−1

+ ε

(n ≥ 1). (2.3)

Then, for each n, X

has the same distribution as Y

where

= b

+ ε

+ bε

+···+b

n−1

. (2.4)

Note that the distributions of the two processes {X

: n ≥ 0} and {Y

n ≥ 0} are generally quite different (Exercise 2.1).

Assume that

|b| < 1 (2.5)

298 Random Dynamical Systems: Special Structures

and |ε

|≤η with probability 1 for some constant η. Then it follows from

(2.4) that

→

∞



n=0

n+1

a.s., (2.6)

regardless of X

. Let π be the distribution of the random variable on

the right-hand side of (2.6). Then Y

converges in distribution to π as

n →∞.AsX

= Y

, X

converges in distribution to π. Hence π is the

unique invariant distribution for the Markov process {X

The assumption that the random variable ε

is bounded can be relaxed;

indeed, it sufﬁces to assume that

∞



n=1

P(|ε

| > mδ

) < ∞ for some δ<

|b|

and for some m > 0.

(2.7)

Since the sum on the left equals



∞

n=1

P(|ε

n+1

| > mδ

), one has, by the

(ﬁrst) Borel–Cantelli lemma,

P(|ε

n+1

|≤mδ

for all but ﬁnitely many n) = 1.

This implies that, with probability 1, |b

n+1

|≤m(|b|δ)

for all but

ﬁnitely many n. Since |b|δ<1, the series on the right-hand side of (2.6)

converges and is the limit of Y

Exercise 2.1 Show that, although X

= Y

separately for each n (see

(2.3), (2.4)), the distributions of the processes {X

: n ≥ 0} and {Y

n ≥ 0} are, in general, not the same. [Hint: Let X

≡ 0. Compare the

distributions of (X

, X

) ≡ (ε

, bε

+ ε

) and (Y

, Y

) ≡ (ε

,ε

+ bε

noting that X

− X

≡ (b − 1)ε

+ ε

and Y

− Y

≡ bε

have different

distributions, unless b = 1orε

= 0 a.s. For example, their variances

are different (if ﬁnite).]

For the sake of completeness, we state a more deﬁnitive characteri-

zation of the process (2.1). Recall that x

= max{x, 0}, and we write

log

x = max{log x, 0}≡ (log x)

(x ≥ 0).

Theorem 2.1

(a) Assume that ε

is nondegenerate and

E(log

|ε

|) < ∞. (2.8)

4.2 Iterates of Real-Valued Afﬁne Maps (AR(1) Models) 299

Then the condition

|b| < 1 (2.9)

is necessary and sufﬁcient for the existence of a unique invariant proba-

bility π for {X

}

∞

n=0

. In addition, if (2.8) and (2.9) hold, then X

converges

in distribution to π as n →∞, no matter what X

is.

(b) If b = 0 and

E(log

|ε

|) =∞, (2.10)

then {X

}

∞

n=0

does not converge in distribution and does not have any

invariant probability.

Proof. One may express X

= b

+ b

n−1

+ b

n−2

+···+bε

n−1

+ ε

= b



j=1

n−j

. (2.11)

Now, recalling the notation

= for equality in distribution, note that for

each n ≥ 1,

n−1



j=0

j+1



j=1

n−j

(n ≥ 1). (2.12)

(a) To prove sufﬁciency in part (a), assume (2.8) and (2.9) hold. Then

→ 0asn →∞. Therefore, we need to prove that

n−1



j=0

j+1

converges in distribution as n →∞. (2.13)

Now choose and ﬁx δ such that

1 <δ<

|b|

. (2.14)

Then δ|b| < 1, and writing Z = (log |ε

/ log δ, one has (see

Lemma 2.1 for the last step)

∞



j=0

P(|b

j+1

|≥(δ|b|)

) =

∞



j=0

P(|ε

j+1

|≥δ

)

∞



j=0

P(|ε

|≥δ

) =

∞



j=0

P(log |ε

|≥j log δ)

300 Random Dynamical Systems: Special Structures

∞



j=1

P((log

|ε

|)≥ j log δ)+P(log |ε

|≥0)

∞



j=1

P(|Z|≥j) + P(log |ε

|≥0)

= E[Z ] + P(log |ε

|≥0), (2.15)

where [Z] is the integer part of Z. By (2.8), E[Z ] < ∞. Hence, by the

(ﬁrst) Borel–Cantelli lemma,

P(|b

j+1

| < (δ|b|)

for all but ﬁnitely many j) = 1. (2.16)

Since δ|b| < 1, this implies the existence of a (ﬁnite) random variable X

such that

n−1



j=0

j+1

→ X a.s. as n →∞. (2.17)

Therefore,



n−1

j=0

j+1

converges in distribution to X . Hence, by (2.12)

(and the fact b

→ 0), X

converges in distribution to X . That is, if π

denotes the distribution of the a.s. limit in (2.17), the distribution of X

converges weakly to π, irrespective of X

. Since the transition probability

p(x, dy) of the Markov process (which is the distribution of bx + ε

)is

a weakly continuous function of x (on

R into P(R)), it follows that π

is invariant (see Theorem 11.1, Chapter 2), and it is the unique invariant

probability.

To prove the necessity of (2.9), ﬁrst, let |b| > 1. Choose

θ such that

1 <

θ<|b|. Then

∞



j=0

P(|b

j+1

|≥

) =

∞



j=0





|ε

j+1

|≥



|b|







∞



j=0





|ε

|≥



|b|







=∞, (2.18)

since P(|ε

|≥(

|b|

)

) ↑ P(|ε

| > 0) > 0as j ↑∞, by the assumption of

nondegeneracy of ε

. But (2.18) implies, by the (second) Borel–Cantelli

Lemma, that

P(|b

j+1

|≥

for inﬁnitely many j) = 1, (2.19)

4.2 Iterates of Real-Valued Afﬁne Maps (AR(1) Models) 301

so that



n−1

j=0

j+1

diverges a.s. as n →∞. By a standard result on the

series of independent random variables (see Lo`eve 1963, p. 251), it fol-

lows that



n−1

j=0

j+1

does not converge in distribution. By (2.12), there-

fore,



j=1

n−j

does not converge in distribution. If X

= 0, we see

from (2.11) that X

≡



j=1

n−j

does not converge in distribution.

Now, if X

= 0 then, conditionally given X

, the same argument

applies to the process Z

:= X

− X

, since Z

n+1

= bZ

+ η

n+1

, with

{η

: n ≥ 1} i.i.d., η

:= ε

+ (b − 1)X

. Since Z

= 0, we see that Z

does not converge in distribution (see (2.11)). Therefore, X

≡ Z

+ X

does not converge in distribution.

To complete the proof of necessity in part (a), consider the case |b|=1.

Let c > 0 be such that P(|ε

| > c) > 0. Then

∞



j=0

P(|b

j+1

| > c) =

∞



j=0

P(|ε

j+1

| > c)

∞



j=0

P(|ε

| > c) =∞. (2.20)

By the (second) Borel–Cantelli lemma,



n−1



j=0

j+1

converges a.s.



= 0.

Therefore,



n−1

j=0

j+1

does not converge in distribution (Lo`eve, 1963,

p. 251). Hence,



j=1

n−j

does not converge in distribution. Since

= X

in case b =+1, and b

=±X

in case b =−1, X

≡



n−1

j=1

n−j

does not converge in distribution.

(b) Suppose b = 0 and (2.10) holds. First, consider the case |b| < 1.

Then −log |b| > 0 and, using Lemma 2.1,

∞



j=0

P(|b

j+1

| > 1) =

∞



j=0

P(|ε

j+1

| > |b|

−j

)

∞



j=0

P(|ε

| > |b|

−j

)

∞



j=1

P(log |ε

| > −j log |b|) + P(log |ε

| > 0)

302 Random Dynamical Systems: Special Structures

∞



j=1



log |ε

−log |b|

> j



+ P(log |ε

| > 0)

= E



log

|ε

−log |b|



+ P(log |ε

| > 0)

=∞. (2.21)

For the last step we have used the fact that [x] ≥ x − 1. Hence, by the

second Borel–Cantelli lemma, |b

j+1

| > 1 for inﬁnitely many j outside

a set of 0 probability. This implies that the series on the left-hand side

in (2.12) diverges almost surely. Since b

→ 0asn →∞, it follows

that b



n−1

j=0

j+1

diverges almost surely and, therefore, does

not converge in distribution (Lo`eve 1963, p. 251). But the last random

variable has the same distribution as X

. Thus X

does not converge in

distribution.

Next, consider the case |b| > 1. Write

= b





j=1

−j



. (2.22)

Using the preceding argument,



j=1

−j

diverges almost surely, since

−1

| < 1. Since |b

|→∞as n →∞, it follows that X

diverges almost

surely and, therefore, does not converge in distribution.

Finally, suppose |b|=1. Then

∞



j=0

P(|b

j+1

| > 1) =

∞



j=0

P(|ε

j+1

| > 1)



P(|ε

| > 1) =∞,

since P(|ε

| > 1) > 0 under the hypothesis (2.10). Therefore, X

does

not converge in distribution.

Example 2.1 An example of a distribution (of ε

) satisfying (2.10) is

given by the probability density function

f (x) =



0ifx < 2,

x(log x )

if x ≥ 2,

(2.23)

where c = (log 2)

−1

. The exponent 2 of log x in the denominator may

be replaced by 1 + δ,0<δ≤ 1 (with c = (log 2)

−δ

); but this exponent

cannot be larger than 2.