Physica 28D (1987) 143-154 North-Holland, Amsterdam NOISE AND STATISTICAL PERIODICITY Andrzej LASOTA Institute of Mathematics, Silesian University, 40-007 Katowice, Poland Michael C. MACKEY Department of Physiology, McGill University, Montreal, Quebec, Canada H3G 1 Y6 Received 19 April 1986 Revised manuscript received 20 October 1986 In this paper we study the properties of a broad class of discrete time dynamical systems in the presence of added external noise. We prove three theorems showing that added noise will either induce a type of statistical periodicity or statistical stability in the asymptotic behaviour of these systems. This result is based on the fact that, in the presence of noise, the Markov operator describing the evolution of densities has smoothing properties which allow the application of a recently discovered asymptotic decomposition theorem. Using this result it is possible to evaluate the (limiting) period of the sequence of densities. This effect is numerically illustrated by the addition of noise to a discontinuous map studied by Keener. 1. Introduction noise to dynamical systems with highly irregular trajectories (Axiom A systems) will not result in An immense effort has been invested in the an alteration of the statistical behaviour of sys- study of stable, periodic, and chaotic dynamical tems since the invariant measure changes continu- systems over the past 15 years. It is generally ously with the noise level. However, in other cases agreed that all real systems are subjected to the addition of noise results in evident changes in "noise" of one variety or another, and many dis- dynamical behaviour of trajectories and may make crete and continuous time systems with stochastic the dynamics more regular by creating absolutely perturbations have been examined numerically. continuous invariant measures ,1 .2 A general The discrete time systems that have been the most answer to the question of how trajectories behave studied in the presence of noise are the quadratic in the presence of perturbation appears to be quite map ,3 4, 8, 9, 18, 19, 22, the circle map 6, the difficult and involves the use of sophisticated topo- standard mapping 11, 20, and the two-dimen- logical methods in the definition of attracting sets sional Kaplan-Yorke map ,7 10. .112 These studies have clearly highlighted an im- This paper considers the effect of stochastic portant and difficult question: What role does perturbation and, more generally, randomly ap- extraneous noise play in the development of plied stochastic perturbation on discrete time dy- asymptotic behaviour of the system? Or, put dif- namical systems from a statistical point of view. ferently, how is the behaviour of the unperturbed Thus we consider the behaviour of sequences of dynamics altered by noise? densities, or ensembles of trajectories, correspond- The answer to this question appears to depend ing to a given system. We show that for every on the properties of the unperturbed system. The system which concentrates trajectories in a results of Kifer 13, 41 indicate that addition of bounded region of phase space see condition ,1)2( 0167-2789/87/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) 441 A. Lasota and M.C. Mackey / Noise and statistical icidoirep ,O the addition of noise always has the effect of for all n making the sequence of densities asymptotically periodic. prob (~,, ~ B) = fg(x) dx In the next section we specify a class of sto- chastically perturbed systems to be considered. for B c R ,d B a Borel set. Section 3 contains a theorem relating the onset of (iii) The density g has a finite first moment, i.e. asymptotic periodicity with the addition of noise to a deterministic system. Section 4 extends this with a second theorem on the existence of asymp- m = LJxlg(x)dx < oo. (3) totic stability, while section 5 points out three important special cases of the theorems of the Finally, in addition to assumptions (i)-(iii), we preceding two sections. There we also illustrate assume that the initial condition x 0 is indepen- the onset of asymptotic periodicity in a map that dent of the sequence of perturbations (~, .} is normally capable of only very irregular behav- The set of all densities in R d is denoted by D, ior. The results of sections 3 and 4 are generalized and thus in section 6 where we prove a theorem concerning the occurrence of asymptotic periodicity in a broad D= ( fELI(Ra): f>O, IlfllL, = 1}, class of stochastically perturbed discrete time dy- namical systems. The paper concludes with some If we let the density of the distributions of the x~ brief comments in section 7. be denoted by fn, then it is straightforward 17 to show that fn+l(X) ~ fR/n(y)g(X-- S(y)) dy 2. Mathematical preliminaries for n = 0,1 ..... (4) In the next three sections of this paper, we Thus, given an arbitrary initial density f0, the consider a stochastically perturbed d-dimensional evolution of densities by the system (1) is de- discrete dynamical system scribed by the sequence of iterates (P n f0 ,} where x,+l=g(x,)+~, for n = 0,1 ..... (1) Pf(x) = fR/(y)g(x- S(y))dy (s) In eq. (1), S is a transformation that maps R d into itself and the quantities ~o, 1~ .... are inde- is a linear (Markov) operator from L 1 into itself. pendent d-dimensional random vectors. In consid- ering the system (1), we assume the following: (i) The transformation S: R d-~ R d is Borel 3. Asymptotic periodicity measurable and such that Our first step in the study of the sequence Is(x)l~lxl+B forxeR u, (2) { P'f } is to show that the operator P is weakly constrictive. By definition, an operator P is weakly constrictive if there exists a weakly precompact set where a and fl are non-negative constants, tc < 1, ~c L 1 such that and -I is the norm in R .d (ii) The vectors ~o, ~1,-.. are independent and lim p(P"f, ~) = 0 forfc D, (6) all have the same distribution with density g, i.e., A. Lasota and M.C. Mackey / Noise and statistical periodicity 145 where 0 (f, o~) denotes the distance, in L 1 norm, where ( ~0- "(i) ) denotes the inverse permutation between the element f and the set o~'. The impor- of (00"(i)). tance of weak constrictiveness is a consequence of Rewriting the summation portion of (8) in this the following theorem of Komornik 15, also way reveals precisely what is happening with every proved in a more restricted case in ref. 16: successive application of the operator P. As the densities gi(x) all have disjoint supports, each Spectral decomposition theorem. Let P be a weakly successive application of P leads to a new set of constrictive Markov operator. Then there is an scaling coefficients Xo,-.(f ) associated with each integer r, two sequences of non-negative functions density g,(x). Because r is finite, the summation ~g ~ D and ~k ~ L ,~° i = 1,..., r, and an operator portion of (8) is periodic with period less than r !, Q:L 1--*L 1 such that for all f~L 1,Pf may be and since IQ,fl -o 0 as n ~ oo we say that for a written in the form weakly constrictive Markov operator the sequence ( p nf ) is asymptotically periodic. Pf(x) = ~ Xi(f)gi(x ) + Qf(x), (7) Having developed this background, we are ready i=1 to state our first main result: where Theorem 1. If the transformation S: R d ~ R d and X~(f)=fsf(x)k,(x)dx. the density of the distribution of the stochastic perturbation respectively satisfy inequalities (2) and (3), then the Markov operator defined by eq. The functions g, and the operator Q have the (5) is weakly constrictive. following properties: (1) gi(x)gj(x) = 0 for all i ~=j, so that the den- sities gi have disjoint supports. Proof. We define (2) For each integer i there exists a unique integer c0(i) such that Pg, = g~(i). Further, ~o(i) =4 E(f) = fRJxlg(x)dx (9) o:(j) for i ~j and thus the operator P just serves to permute the functions g,. (3) IIP"Q_fl --} 0 as n -~ oo for every f~ L .1 and consider the sequence { E(P"f)) for an f~ D. From eq. (7) it is clear that pnf may be written From eq. (5) and inequalities (2) and (3), it follows immediately that as )/I+°P(E P"f= ~ Xi(f)g~.ti)+ Q.(f), (8) i=1 = f.ff.JxlP"f(y)g(x- S(y))dxdy where Q, = P" ,Q1 and w"(i) = co(w"-l(i)) = ..., and IQ,f ~ 0 as n -o oo. The terms in the sum- mation in eq. (8) are permuted with each applica- = fRafRJz + S(y)iP'f(y)g(z)dzdy tion of P. Since {w"(1) ..... o:"(r)} is just a permutation of (1 ..... r}, there is a unique i corresponding to each w"(i). Hence, the summa- -< fR~fRaZP~f(y)g(z)dzdy tion portion of (8) may be rewritten as + fRafRa(ay+ fl)P~f(y)g(z)dzdy ~,X ,(o(f)gi(x), = m + fl + aE(P~f). i=1 146 A. Lasota and M. .C Mackey / Noise and statistical periodicity As a consequence, (5) we have m + fl + a"E(f). E(P"f)< 1-a fBP~fo(x)dx = {fR "Pu lfo(Y)g(x-S(y))dy}dx Choose an arbitrary M> (m + fl)/(1 - a). If E(f) < oc then, since 0 <_ a < 1, for sufficiently n > no(f), large n, say we have = S(y))dx}P"-lfo(y)dy E( P'f ) < M. (10) }zd)z(g)vLS "yd)y(°fl-n, = fR~{fB For e > 0, denote by 3(e) a positive number such that When B satisfies ~t(B)< 3(e), then the set B- 1g(x)dx<e whenever ~(B) < 3(e), (11) S(y) has the same property and, as a consequence of (11), where # denotes the standard Lebesgue measure on R. Let ~'cD be the set of all densities f fBP~fo(x)dx <efRP" lfo(Y)dy=e. satisfying the following two conditions: n > no(fo ) Thus for every fixed the function f= fx / (x)dx<M I- > _-7- forr>O (12) P"f0 satisfies both (12) and (13), and as a conse- P"fo~ quence for n > n0(fo) + 1 whenever E(fo) E(fo) < oo. Since the set of all f satisfying and < oo is dense in D, this implies (6), and the proof is complete. f/(x)dx <e foreverye>0if#(B)<3(e). Remark. The constrictiveness of integral op- (13) erators defined by a" kernel has been shown previ- ously 15-17, but the operators were not derived From standard and well-known criteria for weak by considering the stochastic perturbation of sys- precompactness 5, it follows from (12) and (13) tems. that the set ~" is weakly precompact. In order to verify (6), and demonstrate that P is As a consequence of Theorem 1 in conjunction weakly constrictive, consider an f0 c D such that with our comments and remarks of the previous E(fo) < oo. From inequality (10) and the section, we know that the addition of any stochas- Chebyshev inequality it follows that tic perturbation with a continuous distribution to R a a deterministic transformation on will make M ). that transformation asymptotically periodic from "fo(x)dx<-- forr>0andn>n0(f0 > r a statistical point of view. The only requirements (14) are that the density of the distribution of the stochastic perturbation must possess a finite first Further, let B c R d and e > 0 be given, and let moment inequality (3), and the transformation S /~(B) < 6(e). Then, from the definition of P in eq. must satisfy the growth condition (2). A. Lasota and M.C. Mackey / Noise and statistical periodicity 147 4. Asymptotic stability Since, by Theorem 1, we know that P is weakly constrictive, we need only to demonstrate that P In eq. (8) of the previous section for (pnf ,} if satisfies the rest of the assumptions of the Lemma. r = 1 so that the summation is reduced to a single Let f ~ D be arbitrary. Since f is integrable there term, then for every f~ D the sequence (Pnf) is a bounded subset B c R d such that converges to the same limit as n ~ oo independent of f. In such situations we say that the operator P ff(x) dx = .½ is asymptotically stable. For applied situations, this is extremely interesting and useful because of the possibility of calculating statistical properties Define fl(X) = 2f(x)lB(x), where B1 denotes the characteristic function of the set B. Clearly, fl ~ D characterizing the dynamics. and E(fa) < oo. Define Using Theorem 1 we may prove the following result concerning the appearance of asymptotic ma + fl 1 m + fl stability of the Markov operator P defined by eq. o=r 0 1-a ' M=z°+ 1-a (5). and r=o+ m+13 1--Or " Theorem 2. Assume that the Borel measurable transformation S: R *--d R d and the density g From inequality (14) it follows that satisfy inequalities (2) and (3). Further assume that there exists a point z o ~ R d and a number "f,(x)dx <-- forn>no(fl ). fxl> r P M r ma + fl ro> --Z----_l A- Thus, f f½_>xd)x(f"P xd)X(lfnP such that <-1'~:I r r_<lxI g(x)>O a.e. for Ix-z01<r o. (15) Then the Markov operator defined by eq. (5) is asymptotically stable. >½1-M/r>0 forn>no(f, ). (17) Proof. To prove this theorem, we employ the fol- Now we may write lowing. P"f(x) = fRP"-y(y)g(x - S(y)) dy Lemma. Let P be a weakly constrictive Markov operator. Assume there is a set A c R d of nonzero measure, /~(A)>0, with the property that for -> fl y"-lf(y)g(x- S(y))dy. (18) vl -< every f~ D there is an integer nl(f) such that Define e=(1-a)o. If Ix-zol<e and lYl<_r, P'f(x) > 0 (16) then for almost all x~A and all n>nx(f). Then (x- S(y)) - oZ < Ix- loZ + IS(y)I { P"f ) is asymptotically stable. < I x- zol + alyl+ fl The proof of this lemma may be found in ref. 17. <e+ar+fl=r .o 148 .4, Lasota and M.C. Mackey / Noise and statistical periodicity Hence, according to inequality (15) we have Case .1 Assume that S maps +dR into itself and that n~ > 0, n = 0,1, 2 .... , with probability g(x-S(y))>O for Ix-z01<eandlyl <r. one. Here, R+ denotes 0,~). In this case, for (19) every initial x 0 > 0, the sequence {x~} is well defined with probability one. Thus we may pro- From (17) and (19) it follows that for every x ceed with all of our arguments and calculations satisfying lx - zol < e the product precisely as in Theorems 1 and 2, noting simply that in the definition of the operator P given by P" af(y)g(x- S(y)) for n > n0(A) + 1, eq. (5) the domain of integration must be altered so that as a function of y, does not vanish in the ball defined by lyl < r. As a consequence, inequality Pf(x): fA, f(y)g(x- S(y))dy, (20) (18) implies (16) with A=(x: Ix-zo<e } and nl(f)=no(fl)+l. where Thus, the proof of the theorem is complete. A(x) = ( y~ :+dR x- S(y)~ +dR }. Therefore, having S: Ra_.~Ra+ which satisfies inequality (2), and the sequence (~, } with ~, _> 0 5. Some special cases all having the same density g with a finite first moment (inequality (3)), Theorem 1 is im- In Theorems 1 and 2 of the two previous sec- mediately applicable. If, in addition, g(x) > 0 for tions, it was assumed that the transformation S is a sufficiently large subset of Ra+, e.g., if defined on the entire space R a, which is quite unrestrictive in that further specifications of the nature of the perturbing vectors ~, are not re- g(x) > 0 for x ~ R~, Ixl ~ 0r quired. However, if the transformation S is de- with r 0 > fd(ma + fl)/(1 - a), fined only on a subset G c R a, then it is quite possible that, for some x,0 ~ G, the point then Theorem 2 holds. There is, however, another way to view this Xno+l = S(Xno) q-~no case. Thus, we could consider that S is indeed defined on the entire space R a (by assuming, for may not belong to G and as a consequence example, that S(x)=0 for x )+dREf and that S(xno+l ) may not be defined. Should this be the fo(X) = 0 for x ~ .+dR Then all of the successive f,, case, the sequence (x n ) cannot be calculated for have the same property, and this situation is merely n>n0+l. a special case of the more general situation with S However, in some special cases that may be defined on R .a important from an applications point of view these Case .2 Now consider the situation where S difficulties may be easily circumvented. We discuss maps an interval 0, a into itself, and S is such three such special situations, which are not ex- that haustive but do serve as good illustrations of how one may proceed under such circumstances. In our supS(x) =b< a. discussion we will always assume that S is Borel measurable, and this assumption will ton be re- Now we may consider the sequence (x n } defined peated. by eq. (1), assuming that x 0 ~ 0, a and 0 < (~ < A. Lasota and M.C. Mackey / Noise and statistical periodicity 149 a - b with probability one. In this case, Theorem 1 holds since inequality (2) is automatically satisfied with a = 0 and fl = b, and inequality (3) holds due to the fact that the space is a finite interval. The operator P is again given by eq. (20) wherein the domain of integration A(x) is given by A(x) = {y ~ 0, a: x > S(y)}. 0 % . For Theorem 2 to hold it is sufficient that g(x) > 0 on a sufficiently large subset of 0, a, e.g., g(x) > 0 a.e. on 0, a - b with b < .a½ Case 3. Consider a transformation S that maps the d-dimensional torus T a into itself. Recall that OH,m,,i T a may be obtained from R d (as a quotient space) if we identify all points x, y ~ R a such that x -y is a sequence of integers. In this case, as in the previous one, inequalities (2) and (3) are trivially satisfied and the arguments in the proof of The- orem 1 proceed exactly as written. Thus, assuming o that we have S: T a~ T a and (~n} independent Fig. 1. Asymptotic periodicity illustrated. Here we show the with values in T a and all having the same density histograms obtained after iterating 10'* initial points uniformly g: T aW R, we obtain Theorem 1. As before, to distributed on 0,1 with a = ~, ,? = 731 , and 8 = s l in eq. (23). (a) n = 10; (b) n = 11; (c) n = 12; and (d) n = 13. The corre- obtain Theorem 2 we need only assume that g(x) spondence of the histograms for n = 10 and n = 13 indicate > 0) on a sufficiently large subset of T ,a e.g., that, with these parameter values, numerically the sequence of g(x) > 0 a.e. on T .a densities has period 3. Theorem 1 implies that, for a very broad class of transformations, the addition of a stochastic perturbation will cause the limiting densities to This transformation is an example of a class of become asymptotically periodic. For some trans- transformations considered by Keener 12. From formations, this would not be at all surprising, e.g. the results for general Keener transformations, for the addition of a small stochastic perturbation to a a ~ (0,1) there exists an uncountable set A such transformation with an exponentially stable peri- that for each X ~ A the rotation number corre- odic orbit gives asymptotic periodicity. However, sponding to the transformation (21) is irrational. the surprising content of Theorem 1 is that even in For each such ~? the sequence (x n } is not periodic a transformation S that has aperiodic limiting and the invariant limiting set behaviour, the addition of noise will result in asymptotic periodicity. 6 )1,O(~S )22( This phenomenon is rather easy to illustrate k=O numerically by considering is a Cantor set. The proof of Keener's general Xn+ 1 = S(Xn) (modl), (21) result offers a constructive technique for numeri- cally determining values of ~? that approximate where S(x)=ax+M O<a<l and 0<~<1. elements of the set A. 150 A. Lasota and M.C. Mackev/ Noise and statistical periodici 'O From our remarks above (see Case 3), the trans- ability (1 - e). In addition, with probability e, the formation (21) clearly satisfies the conditions of value of x.+ 1 is uncertain. If x. =y is given then, Theorem 1 and is, therefore, an ideal candidate to in this case, x. + 1 may be considered as a random illustrate the induction of asymptotic periodicity variable distributed with a density K(x, y) which by noise in a transformation whose limiting be- depends on y. haviour is neither periodic nor asymptotically Our first goal in the description of this process periodic in the absence of noise. is the derivation of an equation for the operator P To be specific, we pick a = 1/2 and use the which gives the prescription for passing from the results of Keener to show that )t = 17/30 is close density f~ of x,, to the density fn+i of x~+ .I We to a value in the set A for which the invariant assume that S maps a Borel measurable set G c limiting set (22) should be a Cantor set. Asymp- R ,a /~(G) > 0, into itself and that S is nonsingu- totic periodicity is illustrated by studying lar. Recall that S: G--* G is nonsingular if S is measurable and such that x,,+~ = (ax. +)t + ~.)(mod 1) (23) /~(A)=0 implies/~(S i(A))=0 where the n~ are random numbers uniformly dis- for all measurable A c G. tributed on 0, 0. Fig. 1 shows the numerically calculated effect of this stochastic perturbation for The requirement that S is nonsingular allows us 0 = 1/15 and these values of a and .X Figs. la to introduce the Frobenius-Perron operator sP through ld, respectively, show the histograms ob- which describes the evolution of densities by the tained after 10, 11, 12, and 13 successive iterations transformation S. The operator sP is given im- of 10000 initial points uniformly distributed on plicitly by 0, 1. The 13th histogram is identical with the =xd)x(fsPAf sf 10th, as is the 14th with the 11th, etc., thus ~(A/(X)dx )52( indicating that numerically the sequence of densi- ties is asymptotically periodic with period 3. for every Borel subset A of G. The Radon- Nikodym theorem guarantees that for every f c LI(G) there exists a unique Psf~ Li(G) such that (25) holds. 6. A generalization Now assume that the density f~ of x~ is given and that a Borel set A c G is given. We would like To this point we have considered a relatively to calculate the probability that x.+ i E A. As we special class of situations in which perturbations outlined the randomly applied perturbation pro- were added to consecutive values of S(x.). This cess, x.+ i may be reached in one of the two ways: could be extended by considering 'multiplicative deterministically with probability (1 - e) and sto- noise' with x.+ 1 = S(x.)~. or, even more gener- chastically with probability .e Thus, in the de- ally, terministic case x. + i = S(x.) and x.+ 1 = S(x., ~.). (24) Probi(x.+lEA)=Probi(S(x.)EA ), (26) where the index I is used to denote the determin- Instead, however, we choose to describe a different istic case. From the definition of the process characterized by randomly applied per- Frobenius-Perron operator, the density of S(x,,) turbations. We assume that, in general, our system is Psf~ and, as a consequence, evolves according to a given transformation S(x.). The qualifying phrase 'in general' means that the transition x. ~ S(x.) occurs with prob- Prob I (S(x.) ~A) = Psf~(x)dx. (27) JA A. Lasota and M. .C Mackey / Noise and statistical periodicity 151 If the stochastic perturbation occurs and if y = x n satisfies then K(x,y)>O fGK(x,y)dx=l. and (x,+l ~Alx,--y)-- fAK(x, y)dx. ProbH These two conditions in conjunction with the re- Since x, is a random variable with density fn, we quirement that K: G X G ~ R as a function of the also have two variables is measurable means that K is a K(x, y) stochastic kernel. We assume that is uni- (x,+ 1 ~A) Prob n formly integrable in x, i.e., for every ~/> 0 there is HbOrPGf a 8 > 0 such that =y)f,(y) dy, -- (Xn+ 1 E A IXn fAK(x, y) dx _< and combining this relation with the previous one we have for every y ~ G and A such that ~t(A)< 8. Fi- . x d } y d ) y ( . f ) Y , x ( K o f { L = ) A ~ 1 + . x ( u b o r P nally, we assume that lcf xlK(x,y)dx<alyl+fl (28) for y c G, (30) From eqs. (26)-(28) we have where a and B are non-negative constants and Prob (x,+ 1 ~A) a < 1. Note that condition (30) is automatically satisfied if G is bounded. However if G is un- (x,+ 1 ~ A) + = (1 - e) Prob I e Probii (Xn+ 1 E A) bounded, for example if G is the entire space R ,d = f (l- )Psfo(x) then condition (30) is quite important since it prevents divergence of trajectories to infinity. fcK(x, + y)L(Y)dydx. Then we have: meroehT 3. If S: G~ G is nonsingular and satisfies inequality (2) and K: G X G ~ R is a Since A is an arbitrary Borel set this relation f.+l uniformly integrable stochastic kernel satisfying implies that the density exists whenever f. (30), then with 0 < e < 1 the operator P given by exists, and is given by (29) is weakly constrictive. e)Psf~(x ) + elK(x, y)f~(y) d y. f~+x(X) = (1 - Proof. sP To simplify our notation we set P0 = and Thus, the expression for the operator P that de- scribes the evolution of densities by our process is Paf(x) = fGK(X, y)f(y)dy. Pf(x) e)Psf(x) + e fcK(x, y)f(y)dy. = (1 - Furthermore, set e o = 1 - e and e 1 = e. Then P = eoP o + elP1, and as a consequence (29) K(x,y) P"f= ~_,ei,eo'"e;Pi, Pi,"'Pi.f, Since with fixed y is a density, it 152 A. Lasota and M.C. Mackey / Noise and statistical periodicitv where the summation is taken over all possible However, if f ~ o~0, then sequences (i 1 ... i,) such that i k = 0 or 1. Next we define the set o~ 0 to be all non-nega- P"f - e;P~f = ff"f c~ tive functions f~ L ~ such that and, as a consequence, Ilfll~l and fa Ixlf(x)dx~-- M forr>O, p(e" _<_ . r J' g0~ where M >/3/(1- a) is a fixed constant. Using so (6) is verified for Y. inequalities (2) and (30) it is easy to verify that To show that Y is weakly precompact is much ~P (.~-0) c ~-0, i = 0, ,1 and as a consequence of the more difficult. However, note that since o~ c ,0~o fact that o~ 0 is convex, P'(o~-0) co~ .0 and owe wo 0 as a consequence, to prove the weak With o~ thus defined, for f~o~ 0 we set precompactness of .,~ it is sufficient to show that all functions f~o~ are uniformly integrable 5. Thus for every ~ > 0 pick n o = n0(7/) such that P~f= ~'eie,2 ... ei°P~P~2 ... P~,f, (31) o)q <_ ,7/2 where the prime (') on the summation indicates and consider the set that the term e~P~f is omitted. With these oper- ators P" defined from (31), we define a sequence It o of subsets in L 1 by "'10 (~il'''~ikri, " Piko'~O)) " a=l o~,, = P"(o~0) , n = 1,2 ..... Note that the primed summation is a summation of sets of functions (not just functions), and that all of the terms in the primed summation contain It is clear that ~,~ c ~0. From the sequence o~ we the operator P, = K. From the uniform continuity define .~- by of K it easily follows that the set o~0 is weakly precompact. Thus given n001 ) we may choose a ^ ^ 8(~) > 0 such that for all fe~,, tl E 1 ff(x) dx _< taking care to note that n -- 0 is not included. We (32) are going to prove that the operator P is weakly constrictive, i.e. that condition (6) is fulfilled and for every a c G satisfying /,(A)< 8(~). Now fix that o ~ is weakly precompact. ~/> 0 and pick an arbitrary f~o ~. Then there To demonstrate that the set o ~ satisfies (6) is exists an integer n > 1 such that f~o~. Consider relatively easy. Since P is a Markov operator, it is two cases. sufficient to demonstrate that (6) holds for a dense Case 1. n<n o . In this case o~co~,, and in- subset D ocD. We take D o to be the set of all equality (32), with f= f, is satisfied. densities with a finite first moment, i.e., Case 2. n > n o . For this case set f= P"g for some g ~ .~-0. Then f = P"g may be written in the ~l form xlf(x) dx < o~. f= El8q "'" E,,,eil ''' ez.g Proceeding as in the proof of Theorem 1, for every f~ D o we have P'f~'o if n is sufficiently large. "'Gg' Therefore, it is sufficient to verify (6) for f ~ o~ .0 + g '''