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References
Courant, R., andD. Hilbert: Methods of mathematical physics, Vol. II. New York: Interscience 1962.
Fukushima, M. andM. Hitsuda: On a class of Markov process taking values on lines and the central limit theorem. Nagoya math. J.30, 47–56 (1967).
Goldstein, S.: On diffusion by discontinuous movements and on the telegraph equation. Quart J. Mech. appl. Math.IV, 129–156 (1950).
Hille, E.: Analytic function theory, Vol. II. London: Ginn 1959.
Ito, K., andH. P. McKean, Jr.: Diffusion processes and their sample paths, Berlin-Heidelberg-New York: Springer 1965.
Kac, M.: Some stochastic problems in physics and mathematics. Magnolia Petrolum Co., Lectures in Pure and Applied Science, No. 2 (1956).
Kaplan, S.: Differential equations in which the Poisson process plays a role. Bull. Amer. math. Soc. 70, 264–268 (1964).
Kielson, J., andD. Wishart: A central limit theorem for processes defined on a finite Markov chain, Proc. Cambridge philos. Soc.,60, 547–567 (1964).
Khas'minskii, R., andA. Il'in: On equations of Brownian motion. Theor. Probab. Appl.,9, 421–438 (1964).
Vishik, M. I., andL. A. Lusternik: Regular degeneration and boundary layer for differential equations with a small parameter. Amer. math. Soc. Translat., II. Ser.,20, 173–239 (1962).
Volkov, I.: On the distribution of sums of random variables defined on a homogeneous Markov chain with a finite number of states. Theor. Probab. Appl.,31, 413–429 (1958).
Zlamal, M.: Sur l'équation des télégraphistes avec un petit paramètre, Atti Accad. naz. Lincei Rend. Cl. Sci., fis. mat. natur. VIII. Ser.27, 324–332 (1959).
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Research supported in part under contract N0014-67-A-0112-0015 at Stanford University, Stanford, California.
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Pinsky, M. Differential equations with a small parameter and the central limit theorem for functions defined on a finite Markov chain. Z. Wahrscheinlichkeitstheorie verw Gebiete 9, 101–111 (1968). https://doi.org/10.1007/BF01851001
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DOI: https://doi.org/10.1007/BF01851001