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

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A5. Convergence 433

A more general version is now stated. For a nice proof, see Etemadi

(1983), also given in Bhattacharya and Waymire (1990, pp. 647–649).

Theorem A.13 Let {X

} be a sequence of independent, identically dis-

tributed real-valued random variables. Let S



i=1

,n = 1, 2,....

Then the sequence {n

−1

} converges as to a ﬁnite limit c if and only if

E|X

| < ∞ and in this case E X

= c.

Corollary A.2 If {X

} is i.i.d. and E X

=∞, then n

−1

→∞a.s.

Proof. Since EX

= EX

− EX

−

, EX

−

< ∞. Hence n

−1



−

→

a.s. EX

−

. Note that n

−1



= n

−1



−

+ n

−1



. We need

to show n

−1



→∞a.s. Let Y

n,M

= min{X

, M}. Then EY

n,M

≤

M < ∞. Therefore, n

−1



i,M

a.s.

−→

n,M

.Nown

−1



≥

−1



i,M

∀n. Therefore, lim inf

−1



≥ EY

1,M

a.s., ∀M. But

1,M

↑ EX

=∞.

Theorem A.14 Let X

be a sequence of i.i.d. random variables with

mean 0 and variance σ

> 0. Then S

√

⇒ N (0,σ

Proof. See Proposition 4.1, Chapter 5 for a more general result.

Bibliography

Akerlof, G. and Main, B. (1981): “Pitfalls in Markov Modelling of Labor Market

Stocks and Flows,” Journal of Human Resources, 16, 141–151.

Allen, R.G.D. (1956): Mathematical Economics, St. Martin’s Press, New York.

An, H.Z. and Huang, F.C. (1996): “The Geometrical Ergodicity of Non-Linear

Autoregressive Models,” Statistica Sinica, 8, 943–956.

Anderson, T.W. (1971): The Statistical Analysis of Time Series, Wiley, New

York.

Arrow, K.J. (1958): “Toward a Theory of Price Adjustment,” in The Allocation

of Economic Resources, (M. Abramovitz, ed.), Stanford University Press,

Stanford, CA, pp. 41–51.

Arrow, K.J. (1974): Essays in the Theory of Risk-Bearing, North-Holland,

Amsterdam.

Arrow, K.J., Block, H.D., and Hurwicz, L. (1959): “On the Stability of Compet-

itive Equilibrium II,” Econometrica, 27, 82–109.

Arrow, K.J. and Hurwicz, L. (1958): “On the Stability of Competitive Equilib-

rium I,” Econometrica, 26, 522–552.

Arrow, K.J. and Kurz, M. (1970): Public Investment: The Rate of Return and

Optimal Fiscal Policy, Johns Hopkins Press, Baltimore, MD.

Athreya, K.B. (2003): “Iterations of IID Random Maps on R

,” in Probabil-

ity, Statistics and their Applications: Papers in Honor of Rabi Bhattacharya

(K. Athreya, M. Majumdar, M. Puri, and E. Waymire, eds.), Institute of

Mathematical Statistics, Beachwood, OH, pp. 1–14.

Athreya, K.B. (2004): “Stationary Measures for some Markov Chain Models in

Ecology and Economics,” Economic Theory, 23, 107–122.

Athreya, K.B. and Dai, J.J. (2000): “Random Logistic Maps – I,” Journal of

Theoretical Probability, 13, 595–608.

Athreya, K.B. and Dai, J.J. (2002): “On the Non-Uniqueness of the Invariant

Probability for i.i.d. Random Logistic Maps,” Annals of Probability, 30, 437–

442.

Athreya, K.B. and Majumdar, M. (2003): “Estimating the Stationary Distribution

of a Markov Chain,” Economic Theory, 27, 729–742.

435

436 Bibliography

Athreya, K.B. and Ney, P. (1978): “A New Approach to the Limit Theory

of Recurrent Markov Chains,” Transactions of the American Mathematical

Society, 245, 493–501.

Azariadis, C. (1993): Intertemporal Macroeconomics, Blackwell, Cambridge,

MA.

Baccelli, F. and Bremaud, P. (1994): Elements of Queuing Theory: Palm-

Martingale Calculus and Stochastic Recurrences, Springer Verlag, Berlin.

Bala, V. and Majumdar, M. (1992): “Chaotic Tatonnement,” Economic Theory,

2, 437–446.

Bandopadhyay, B. (1999): Aparajito: The Unvanquished, Harper Collins, New

Delhi (original publication in Bengal in 1932; English translation by Gopa

Majumdar).

Barnsley, M. (1993): Fractals Everywhere, 2nd edition, Academic Press, New

York.

Barnsley, M. and Elton, J. (1988): “A New Class of Markov Processes for Image

Encoding,” Advances in Applied Probability, 20, 14–32.

Baumol, W.J. (1970): Economic Dynamics, The MacMillan Company, New York.

Baumol, W.J. and Benhabib, J. (1989): “Chaos: Signiﬁcance, Mechanism and

Economic Applications,” Journal of Economic Perspectives, 3, 77–105.

Becker, R. and Foias, C. (1994): “The Local Bifurcation of Ramsey Equilibrium,”

Economic Theory, 4, 719–744 (reprinted in Majumdar, M., Mitra, T., and

Nishimura, K. (2000), Chapter 5).

Becker, R.A. (1985): “Comparative Dynamics in Aggregate Models of Optimal

Capital Accumulation,” Quarterly Journal of Economics, 100, 1235–1256.

Benhabib, J. (ed.) (1992): Cycles and Chaos in Economic Equilibrium, Princeton

University Press, Princeton, NJ.

Benhabib, J. and Day, R. (1981): “Rational Choice and Erratic Behavior,” Review

of Economic Studies, 48, 459–472.

Benhabib, J. and Day, R. (1982): “A Characterization of Erratic Dynamics in

the Overlapping Generations Model,” Journal of Economic Dynamics and

Control, 4, 37–55.

Benhabib, J. and Nishimura, K. (1979): “The Hopf Bifurcation and the Existence

and Stability of Closed Orbits in Multisector Models of Optimal Economic

Growth,” Journal of Economic Theory, 21, 421–444.

Benhabib, J. and Nishimura, K. (1985): “Competitive Equilibrium Cycles,”

Journal of Economic Theory, 35, 284–306.

Berge, C. (1963): Topological Spaces, Oliver and Boyd, Edinburgh, UK.

Berger, M. (1992): “Random Afﬁne Iterated Function Systems: Mixing and

Encoding,” in Diffusion Processes and Related Problems in Analysis, Vol. II.

(Progress in Probability, Vol. 27) Birkhauser, Boston, MA, pp. 315–346.

Berry, D.A. and Friested, B. (1985): Bandit Problems: Sequential Allocation of

Experiments, Chapman and Hall, London.

Bertsekas, D.P. (1995a): Dynamic Programming and Optimal Control, Vol. I,

Athena Scientiﬁc, Belmont, MA.

Bibliography 437

Bertsekas, D.P. (1995b): Dynamic Programming and Optimal Control, Vol. II,

Athena Scientiﬁc, Belmont, MA.

Bhaduri, A. and Harris, D.J. (1987): “The Complex Dynamics of the Simple

Ricardian Model,” Quarterly Journal of Economics, 102, 893–902.

Bhattacharya, R.N. (1978): “Criteria for Recurrence and Existence of Invariant

Measures for Multidimensional Diffusions,” Annals of Probability, 6, 541–

553.

Bhattacharya, R.N. (1982): “On the Functional Central Limit Theorem and the

Law of Iterated Logarithm for Markov Processes,” Zeitschrift fur Wahrschein-

lichkeitstheorie und Verwandte Gebiete, 60, 185–201.

Bhattacharya, R.N. and Goswami, A. (1998): “A Class of Random Continued

Fractions with Singular Equilibria,” in Proceedings of the 3rd Calcutta Trien-

nial Symposium in Probability and Statistics, Oxford University Press, Oxford,

pp. 75–85.

Bhattacharya, R.N. and Lee, O. (1988): “Asymptotics of a Class of Markov

Processes Which Are Not in General Irreducible,” Annals of Probability, 16,

1333–1347 (correction: Bhattacharya, R.N. and Lee, O. (1997), Annals of

Probability, 25, 1541–1543).

Bhattacharya, R.N. and Lee, C. (1995): “On Geometric Ergodicity of Non-Linear

Autoregressive Models,” Statistics and Probability Letters, 22, 311–315

(correction: Bhattacharya, R.N. and Lee, C. (1999), Statistics and Probability

Letters, 41, 439–440).

Bhattacharya, R. and Majumdar, M. (1980): “On Global Stability of Some

Stochastic Processes: A Synthesis,” in Quantitative Economics and Devel-

opment (L. Klein, M. Nerlove, and S.C. Tsiang, eds.), Academic Press, New

York, pp. 19–44.

Bhattacharya, R. and Majumdar, M. (1989a): “Controlled Semi-Markov Pro-

cesses: The Discounted Case,” Journal of Statistical Planning and Inference,

21, 365–381.

Bhattacharya, R. and Majumdar, M. (1989b): “Controlled Semi-Markov Pro-

cesses: The Average Reward Criterion,” Journal of Statistical Planning and

Inference, 22, 223–242.

Bhattacharya, R. and Majumdar, M. (1999a): “On a Theorem of Dubins and

Freedman,” Journal of Theoretical Probability, 12, 1067–1087.

Bhattacharya, R.N. and Majumdar, M. (1999b): “Convergence to Equilibrium

of Random Dynamical Systems Generated by i.i.d. Monotone Maps with

Applications to Economics,” Asymptotics, Nonparametrics and Time Series

(S. Ghosh, ed.), Marcel Dekker, New York, pp. 713–742.

Bhattacharya, R.N. and Majumdar M. (2001): “On a Class of Stable Random

Dynamical Systems: Theory and Applications,” Journal of Economic Theory,

96, 208–239.

Bhattacharya, R.N. and Majumdar M. (2002): “Stability in Distribution of Ran-

domly Perturbed Quadratic Maps as Markov Processes,” Center for Analytical

Economics Working Paper 02–03, Cornell University, Ithaca, NY.

438 Bibliography

Bhattacharya, R.N. and Majumdar M. (2004): “Stability in Distribution of Ran-

domly Perturbed Quadratic Maps as Markov Processes,” Annals of Applied

Probability, 14, 1802–1809.

Bhattacharya, R.N. and Ranga Rao, R. (1976): Normal Approximation and

Asymptotic Expansion, Wiley, New York.

Bhattacharya, R.N. and Rao, B.V. (1993): “Random Iterations of Two Quadratic

Maps,” in Stochastic Processes: A Festschrift in Honor of Gopinath Kallianpur

(S. Cambanis, J.K. Ghosh, R.L. Karandikar, and P.K. Sen, eds.), Springer-

Verlag, New York, pp. 13–21.

Bhattacharya, R.N. and Waymire, E.C. (1990): Stochastic Processes with Appli-

cations, Wiley, New York.

Bhattacharya, R.N. and Waymire, E.C. (2002): “An Approach to the Existence

of Unique Invariant Probabilities for Markov Processes. Limit Theorems,” in

Probability and Statistics-I (I. Berkes, E. Csaki, and M. Csorgo, eds.), Janos

Bolyai Soc, Budapest, pp. 181–200.

Bhattacharya, R.N. and Waymire, E.C. (2006a): A Basic Course in Probability

Theory, Springer-Verlag, Berlin.

Bhattacharya, R.N. and Waymire, E.C. (2006b): Theory and Applications

of Probability and Stochastic Processes, Graduate texts in Mathematics,

Springer-Verlag, Berlin.

Billingsley, P. (1961): “The Lindberg–Levy Theorem for Martingales,” Pro-

ceedings of the American Mathematical Society, 12, 788–792.

Billingsley, P. (1965): Ergodic Theory and Information, Wiley, Huntington, NY.

Billingsley, P. (1968): Convergence in Probability Measures, Wiley, New York

(2nd edition, 1999).

Billingsley, P. (1986): Probability and Measure, 2nd edition. Wiley, New York.

Blackwell, D. (1965): “Discounted Dynamic Programming,” Annals of Mathe-

matical Statistics, 36, 226–235.

Blackwell, D. (1967): “Positive Dynamic Programming,” in Proceedings of the

5th Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1

(L. Le Cam and J. Neyman, eds.), University of California Press, Berkeley,

CA, pp. 415–418.

Blumenthal, R.M. and Corson, H. (1972): “On Continuous Collection of

Measures,” in Proceedings of the 6th Berkeley Symposium of Mathemati-

cal Statistics and Probability, Vol. 5, University of California Press, Berkeley,

CA, pp. 33–40.

Boldrin, M. and Montrucchio, L. (1986): “On the Indeterminacy of Capital

Accumulation Paths,” Journal of Economic Theory, 40, 26–36.

Box, G.E.P. and Jenkins, G.M. (1976): Time Series Analysis Forecasting and

Control, revised edition, Holden-Day, San Francisco, CA.

Brandt, A. (1986): “The Stochastic Equation Y

n+1

= A

n+B

+ B

With Sta-

tionary Coefﬁcients,” Advances in Applied Probability, 18, 211–220.

Brauer, F. and Castillo-Chavez, C. (2001): Mathematical Models in Population

Biology and Epidemiology, Springer-Verlag, New York.

Bibliography 439

Brillinger, D.R. (1981): Time Series Data Analysis and Theory, expanded

edition, Holden-Day, San Francisco, CA.

Brock, W. (1971): “Sensitivity of Optimal Growth Paths With Respect to a

Change in Target Stock,” Zeitschrift fur Nationalokonomie Supplement, 1,

73–89.

Brock, W. and Gale, D. (1969): “Optimal Growth Under Factor Augmentating

Progress,” Journal of Economic Theory, 1, 229–243.

Brock, W.A. and Mirman, L.J. (1972): “Optimal Economic Growth and Uncer-

tainty: The Discounted Case,” Journal of Economic Theory, 4, 479–513.

Brock, W.A. and Mirman, L.J. (1973): “Optimal Economic Growth and Un-

certainty: The No Discounting Case,” International Economic Review, 4,

479–513.

Brockwell, P.J. and Davis, R.A. (1991): Time Series Theory and Methods, 2nd

edition, Springer-Verlag, New York.

Brown, B.M. (1977): “Martingale Central Limit Theorems,” Annals of Mathe-

matical Statistics, 42, 59–66.

Carlsson, N. (2002): “A Contractivity Condition for Iterated Function Systems,”

Journal of Theoretical Probability, 15, 613–630.

Cass, D. and McKenzie, L. (eds.) (1974): Selected Readings on Macroeconomics

and Capital Theory from Econometrica, MIT Press, Cambridge, MA.

Cass, D. and Stiglitz, J.E. (1972): “Risk Aversion and Wealth Effects on Portfolios

With Many Assets,” Review of Economic Studies, 39, 331–354.

Chakravarty, S. (1959): Logic of Investment Planning, North-Holland,

Amsterdam.

Chakraborty, S. and Rao, B.V. (1998): “Completeness of Bhattacharya Metric

on the Space of Probabilities,” Statistics and Probability Letters, 36, 321–

326.

Chassaing, P., Letac, G., and Mora, M. (1984): “Brocot Sequences and Random

Walks in SL(2, R),” in Probability Measures on Groups VII, Lecture Notes in

Mathematics, Springer, Berlin, pp. 36–48.

Chesson, P.L. (1982): “The Stabilizing Effect of a Random Environment,”

Journal of Mathematical Biology, 15, 1–36.

Chipman, J.S. (1973): “The Ordering of Portfolios in Terms of Mean and

Variance,” Review of Economic Studies, 40, 167–190.

Chung, K.L. (1967): Markov Chains with Stationary Transition Probabilities,

2nd edition, Springer, New York.

Clark, C.W. (1971): “Economically Optimal Policies for the Utilization of

Biologically Renewable Resources,” Mathematical Biosciences, 17, 245–

268.

Clark, C.W. (1976): Mathematical Bioeconomics, Wiley, New York.

Collet, P. and Eckmann, J.P. (1980):

Iterated Random Maps on the Interval as

Dynamical Systems, Birkhauser, Boston, MA.

Dai, J.J. (2000): “A Result Regarding Convergence of Random Logistic Maps,”

Statistics and Probability Letters, 47, 11–14.

440 Bibliography

Dana, R. and Montrucchio, L. (1987): “Dynamic Complexity in Duopoly

Games,” in Nonlinear Economic Dynamics (J.M. Grandmont, ed.), Academic

Press, Boston, MA, pp. 40–56.

Danthene, J. and Donaldson, J.B. (1981a): “Stochastic Properties of Fast vs.

Slow Growing Economies,” Econometrica, 49, 1007–1033.

Danthene, J. and Donaldson, J.B. (1981b): “Certainty Planning in an Uncertain

World,” Review of Economic Studies, 48, 507–510.

Dasgupta, P.S. (1982): The Control of Resources, Blackwell, Oxford, UK.

Dasgupta, P.S. and Heal, G. (1979): Economic Theory and Exhaustible

Resources, Cambridge University Press, Cambridge, UK.

Day, R.H. (1982): “Irregular Growth Cycles,” American Economic Review, 78,

406–414.

Day, R.H. (1994): Complex Economic Dynamics, MIT Press, Cambridge, MA.

Day, R.H. and Pianigiani, G. (1991): “Statistical Dynamics and Economics,”

Journal of Economic Behavior and Organization, 16, 37–83.

Debreu, G. (1952): “A Social Equilibrium Existence Theorem,”Proceedings of

the National Academy of Sciences of the USA, 38, 886–893.

Debreu, G. (1959): Theory of Value, Wiley, New York.

Debreu, G. and Herstein, I.N. (1953): “Nonnegative Square Matrices,” Econo-

metrica, 21, 597–607.

Dechert, W.D. and Nishimura, K. (1983): “A Complete Characterization of

Optimal Growth Patterns in an Aggregated Model with a Non-Concave Pro-

duction Function,” Journal of Economic Theory, 31, 332–354.

Demetrius, L. (1971): “Multiplicative Processes,” Mathematical Biosciences,

12, 261–272.

Deneckere, R. and Pelikan, S. (1986): “Competitive Chaos,” Journal of Eco-

nomic Theory, 40, 13–25.

Devaney, R.L. (1986): An Introduction to Chaotic Dynamical Systems,

Benjamin-Cummins, Menlo Park, CA (2nd edition, 1989, Addison-Wesley,

New York).

Diaconis, P. (1988): Group Representations in Probability and Statistics, Insti-

tute of Mathematical Statistics, Hayward, CA.

Diaconis, P. and Freedman, D. (1999): “Iterated Random Functions,” SIAM

Review, 41, 45–79.

Diaconis, P. and Shashahani, M. (1986): “Product of Random Matrices

and Computer Image Generation,” Contemporary Mathematics, 50, 173–

182.

Diaconis, P. and Strook, D. (1991): “Geometrical Bounds for Eigenvalues of

Markov Chains,” Annals of Applied Probability, 1, 36–61.

Diamond, P.A. (1965): “National Debt in a Neoclassical Growth Model,”

American Economic Review, 55

, 1126–1150.

Dieudonne, J. (1960): Foundations of Modern Analysis, Academic Press,

New York.

Diks, C. (1999): Non-Linear Time Series Analysis, World Scientiﬁc, Singapore.

Bibliography 441

Doeblin, W. (1937): “Sur les proprietes asymptotique du movement regis par

certain types de chaines simples,” Bull. Math. Soc. Rom. Sci., 39, no. 1,

57–115; no. 2, 3–61.

Doob, J.L. (1953): Stochastic Processes, Wiley, New York.

Dubins, L.E. and Freedman, D.A. (1966): “Invariant Probabilities for Certain

Markov Processes,” Annals of Mathematical Statistics , 37, 837–838.

Dudley, R.M. (1989): Real Analysis and Probability, Wadsworth and Brooks-

Cole, CA.

Dugundji, J. and Granas, A. (1982): Fixed Point Theory, Vol. 1, Polish Scientiﬁc

Publishers, Warszawa.

Durrett, R. (1996): Probability Theory and Examples , 2nd edition. Wadsworth,

Brooks and Cole, Paciﬁc Grove, CA.

Dutta, P. (1987): “Capital Deepening and Impatience Equivalence in Stochastic

Aggregative Growth Models,” Journal of Economic Dynamics and Control,

18, 1069–1092.

Dutta, P. (1991): “What Do Discounted Optima Converge To? A Theory of Dis-

count Rate Asymptotics in Economic Models,” Journal of Economic Theory,

55, 64–94.

Dutta, P. (1999): Strategies and Games: Theory and Practice, MIT Press, Cam-

bridge, MA.

Dutta, P., Majumdar, M., and Sundaram, R.K. (1994): “Parametric Continuity

in Dynamic Programming Problem,” Journal of Economic Dynamics and

Control, 19, 1069–1092.

Dutta, P. and Sundaram, R. (1993): “How Different Can Strategic Models Be?”

Journal of Economic Theory, 60, 42–61.

Dynkin, E.B. (1961): Theory of Markov Processes, Prentice Hall, Englewood

Cliffs, NJ.

Easley, D. and Kiefer, N.M. (1988): “Controlling a Stochastic Process With

Unknown Parameters,” Econometrica, 56, 1045–1064.

Easley, D. and Kiefer, N.M. (1989): “Optimal Learning with Endogenous Data,”

International Economic Review, 30, 963–978.

Elaydi, S. (2000): An Introduction to Difference Equations, 3rd edition, Springer,

New York.

Ellner, S. (1984): “Asymptotic Behavior of Some Stochastic Difference Equation

Population Models,” Journal of Mathematical Biology, 19, 169–200.

Elton, J.H. (1990): “A Multiplicative Ergodic Theorem for Lipschitz Maps,”

Stochastic Processes and Applications, 34, 39–47.

Etemadi, N. (1983): “On the Laws of Large Numbers for Nonnegative Random

Variables,” Journal of Multivariate Analysis, 13, 187–193.

Evans, G.W. and Honkapohja, S. (1995): “Local Convergence of Recursive

Learning to Steady States and Cycles in Stochastic Nonlinear Models,”

Econometrica, 63, 195–206.

Evans, G.W. and Honkapohja, S. (2001): Learning and Expectations in

Macroeconomics, Princeton University Press, Princeton, NJ.

442 Bibliography

Feller, W. (1968): An Introduction to Probability Theory and Its Applications,

Vol. 1, 3rd edition, Wiley, New York.

Fill, J.A. (1991): “Eigenvalue Bounds on Convergence to Stationarity for Non-

reversible Markov Chains, With an Application to the Exclusion Processes,”

Annals of Applied Probability, 1, 62–87.

Foley, D. and Hellwig, M. (1975): “Asset Management With Trading Uncer-

tainty,” Review of Economic Studies, 42, 327–246.

Frankel J., Kreiss, J.-P., and Mammen, E. (2002): “Bootstrap of Kernel Smooth-

ing in Non-Linear Time Series,” Bernoulli, 8, 1–37.

Frisch, R. (1933): “Propagation Problems and Impulse Problems in Dynamic

Economics,” in Economic Essays in Honor of Gustav Cassel, Allen and

Unwin, London, pp. 171–205 (reprinted in Readings in Business Cycles (R.A.

Gordon and L. Klein, eds., Chapter 9), Richard D. Irwin, Homewood, IL, pp.

155–185).

Frisch, R. and Holme, H. (1935): “The Characteristic Solutions of Mixed Dif-

ference and Differential Equations,” Econometrica, 3, 225–239.

Furukawa, N. (1972): “Markovian Decision Processes With Compact Action

Spaces,” Annals of Mathematical Statistics, 43, 1612–1622.

Futia, C.A. (1982): “Invariant Distributions and Limiting Behavior of Markovian

Economic Models,” Econometrica, 50, 377–408.

Gale, D. (1960): The Theory of Linear Economic Models, McGraw-Hill, New

York.

Gale, D. (1967): “On the Optimal Development in a Multi-Sector Economy,”

Review of Economic Studies, 34, 1–18.

Gale, D. (1973): “Pure Exchange Equilibrium of Dynamic Economic Models,”

Journal of Economic Theory, 6, 12–36.

Galor, O. and Ryder, H.E. (1989): “Existence, Uniqueness and Stability of Equi-

librium in an Overlapping Generations Model With Productive Capital,” Jour-

nal of Economic Theory, 49, 360–375.

Gantmacher, F.R. (1960): The Theory of Matrices, Vols. I and II, Chelsea, New

York.

Gittins, J.C. (1989): Multi-Armed Bandit Allocation Indices, Wiley, New York.

Goldberg, S. (1958): Introduction to Difference Equations, Wiley, New York.

Goodwin, R. (1990): Chaotic Economic Dynamics, Clarendon Press, Oxford,

UK.

Gordin, M.I. and Lifsic, B.A. (1978): “The Central Limit Theorem for Stationary

Markov Processes,” Doklady Akademii Nauk SSR , 19, 392–393.

Gordon, R.A. and Klein, L. (eds.) (1965): Readings in Business Cycles, Richard

D. Irwin, Homewood, IL.

Goswami, A. (2004): “Random Continued Fractions: A Markov Chain Ap-

proach,” Economic Theory, 23, 85–106.

Grandmont, J.M. (1985): “On Endogenous Competitive Business Cycles,”

Econometrica,

53, 995–1045.