A central question for an IRFS (Mn)n>o is under which conditions it sta- bilizes, that is, converges to a stationary distribution n. Elton (1990) showed in the more general situation of a stationary sequence ( Fn)n> i t h a t this holds true whenever E l o g+/ ( F i ) and Elog+cJ(Fi(a;o),a;o) are both finite for some (and then all) Xo G X and the Liapunov exponent I* := limn—ooft_1log/(.F„:i) which exists by Kingman's subadditive er- godic theorem, is a.s. negative. His results for i.i.d. F\,F2,--- under the slightly stronger assumptions Elog+Z(Fi) < 0, E l o g+ d(Fi(xo),x0) < oo for some i0e X are restated in Theorem 2.1. The basic idea is to consider the backward iterations M% = Fi:n(x) and to prove their a.s. convergence to a limit M^ which does not depend on x and which has distribution IT.
The obvious inequality
d(M*+ m,M*) < lf[l(Fk)Jd(Fn+lm+m(x),x) a.s., (6) valid for all n, m > 0 and i £ X , forms a key tool in the analysis. Alsmeyer and Fuh (2001) embarked on that same inequality together with the simple observation that
( n \ n
I I W = ^ogl(Fk), n>0, (7)
fc=i / fc=i
is an ordinary zero-delayed random walk and thus perfectly amenable to renewal theoretic (regeneration) arguments. Under the mean contraction assumption Elog+ l(Fi) < 0, it has negative drift whence, for arbitrary 7 6 (0,1), the level log 7 ladder epochs 00(7) > 0,
an(7) := i n f | f e > ( 7 „ - i : £ log^F,) < l o g7| , n > 1, (8) are all a.s. finite and constituting an ordinary discrete renewal process. As a consequence, the subsequence (-^ffn(7))n>o forms again an IRFS of i.i.d.
Lipschitz maps which further is strictly contractive because, by construc- tion,
KFl:*lW) < 7 < 1-
For the associated backward iterations M* ,-, — i?i;<Tn(T)(x), inequality (6) hence takes the very strong form
^ ^ ( 7 ) ^ ( 7 ) ) <ind(F,n+lh):,n+rnh)(x),x) (9) for all n, m > 0 and x £ X and suggests the following procedure to prove
convergence results for (Mn)n>o and its associated sequence of backward iterations:
Step 1. Given a set of conditions, find out what kind of results hold true for the strictly contractive IRFS (M(Tn^)n>o for any 7 e (0,1).
Step 2. Analyze the excursions of (Mn)n>o between two successive ladder epochs 0-4.(7) a n (i o"fc+i(7) a nd adjust the results with respect to (Mn)n>o if necessary.
The stability results in this section are taken from Alsmeyer and Fuh (2001, 2002). They focused on estimates for ^M^,^) under Wx, x£X, and d(M*,M%) for x,y € X. The latter distance may be viewed as the coupling rate of the forward iterations at time n when started at different values x and y. The two sets of conditions we will consider are that, for some p > 0 and some XQ £ X, either
Elogp + 1(l + Li) < o o and Elogp + 1(l + d(F1(x0),x0)) < 00 (10) or
E Lp< o o and Ed(Fl(x0),x0)p < 00 (11) holds. Two major conclusions will concern the distance of Pn(x, •) for
i £ X and IT in the Prokhorov metric associated with d. Following Diaconis and Freedman (1999), the latter is also denoted d and defined, for two probability measures Ai, A2 on X, as the infimum over all 5 > 0 such that
Ai(B) <\2(B5) + 6 and X2(B) < X^B5) + 5
for all B G B(X), where Bs :={x£X: d{x,y) < 6 for some y € B}. They showed that, for all x € X and n > 0,
d(P"(a;,-),7r) < Ax(n + l)~p, (12)
if (10) holds, and
d(Pn{x,-),n) < Axrn (13)
for some r £ (0,1) not depending on x and n, if (11) is true.
Now let 0-1(7) be as defined in (8) for 7 G (0,1), i.e.
0-1(7) := inf{n > 1 : L1:n < 7} = inf j n > 1 : ^ S o g Lf c < l o g7 1(14) Providing E log+ L\ < 0, a condition which will always be in force through- out, 0-1(7) is an a.s. finite first passage time with finite mean n(f). It has also finite variance 0(7)2, say, if Elog(l + L{)2 < 00. Let further
log 7* := inf fe (15)
7e(o,i) /i(7)
If E| log L i | < 00, then it is well known from renewal theory, that
l 0 g 7 < M7) < l 5 ^ T - ( l + o(l)) ( 7 - 0 ) . (16) E l o g i i ~ ^ " - ElogLi
It is now easily checked that in this case
log7* = l i m ^ = E l o g L i . (17) Theorem 2.1. Given an IRFS (M„)„>o of i.i.d. Lipschitz maps, suppose
E l o g+L i < 0 and Elog+d(Fi(x0),x0) < 00 (18) for some XQ € X. Then the following assertions hold:
(a) Mn converges a.s. to a random element M^ with distribution n which does not depend on the initial distribution.
(b) For each 7 e (7*, 1), l i m ^ o o PI(d(M0 0, Mn) > 7") = 0 for all x£X.
(c) Mn converges in distribution to n under every Px,x € X.
(d) n is the unique stationary distribution of (Mn)n>o and (Mn)n>o a sta- tionary sequence under Pn.
(e) (Mn)„>0 is ergodic under F^.
Theorem 2.2. Given the situation of Theorem 2.1 and additionally con- dition (8) for some p > 0, the following assertions hold:
(a) For each 7 6 (7*, 1),
n > l
and
^ np-1Px( d ( M0 0, M „ ) >7" ) < c7( l + l o gp( l + d(x,xo)))
lim npFx(d{M00,Mn)>jn) = 0
for all x e X and some Oy G (0, oo).
(b) For each 7 G (7*, 1),
l i m s u p n ^ f - l o g ^ M o c M n ) - l o g 7 j < 0 Fx-a.s.
n—+00 \ ^ /
for all x G X. /n case 0 < p < 1 i/iis remains true for 7 = 7*.
fcj Ifp=l, then limri^0 07_ nd(M0 0,M„) = 0 Px-a.s. for all x G X and a H7e ( 7 M ) -
fdj d(Pn(x, •), IT) < Ax(n+l)~p for a£Z n > 0, x G X and a positive constant Ax of the form max.{A,2d(x, Xo)}, where A does neither depend on x nor on n.
(e) J0°° logp(l + d{x, x0)) 7r(dx) = /0°° ptP-^ix : log(l + d(x, x0)) >t)dt<
0 0 .
Theorem 2.3. Given the situation of Theorem 2.1 and additionally con- dition (11) for some p > 0, the following assertions hold:
(a) For each 7 G (7*, 1),
lim a rnPIK M0 O )Mn) >7" ) = 0
n—ằoo '
for all x G X and some a7 G (0,1).
(b) There exists rj > 0) such that for each q G (0, rj),
]imsvva-n{l+d{x,xo))-q^xd{M00,Mn)q = 0 for some ctg G (0,1). The same holds true for q — 77 with aq = 1.
(c) d(Pn(x, -),7r) < Axrn for all n > 0, some r G (0,1) and a constant Ax
of the form m&x{A, d(x, XQ)}. The constants r and A do not depend on x nor n.
(d) fxd(x,xo)'nn(dx) = J0°°nt'n''1TT(x : d(x,xo) > t)dt < 00 for some 77 > 0.
Let us mention that the constants Cj, a7, ag,, Ax and r in the previous theorems generally further depend on p > 0 of the supposed respective moment condition.
The assertions of the previous two theorems on d^^jMn) are eas- ily translated into similar results on d(M%,M%) for the forward iterations started at different values x and y. Essentially, this only takes the ob- servation that (M£,M%) and {M*,Mv) are identically distributed for all x, y e X and n > 0 and that
d{MZ,M*) < d{M£,M*) + d(M£,MZ).
We summarize the results in the following two corollaries.
Corollary 2.1. Given the situation of Theorem 2.2, the following asser- tions hold:
(a) For each 7 G (7*, 1),
Y,r?-lnd{M*,M%)>1n) < cr( l + ] Q g * ( ^ s , x0) ) + l o g * ( l + < %l* o ) ) )
n > l
and
lim npP ( d ( M * , M £ ) > 7n) = 0
n—>oo
for all x, y G X and some c^ G (0,00).
(7>j For each 7 G (7*, 1),
l i m s u p n V " I - log d(M*, M£) - log7 ) < 0 a.s.
n—>oo \n J
for all x, y G X. /n case 0 < p < 1 this remains true for 7 = 7*.
fcj 7/p = 1, tfien lim„w o o7~nd(M*,M^) = 0 a.s. /or all x,y G X and aft
76 (7M ) .
C o r o l l a r y 2.2. Given the situation of Theorem 2.3, the following asser- tions hold:
(a) For each 7 G (7*, 1),
limapP(d(M*,M2)>7") = 0
/or a// x, y G X and some a7 G (0,1).
(b) There exists n > 0 suc/i t/iai /or- each q G (0,77),
lim sup a7n(l + d(x,x0)d{y,x0)-qExd(MZ,My)i = 0
n~*cox,y€X
for some aq G (0,1). The same holds true for q — rj with aq — 1.