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Legendre polynomials and trinomial random walks

Published online by Cambridge University Press:  24 October 2008

I. J. Good
I. J. Good
Affiliation:
25 Scott House Princess Elizabeth Way Cheltenham
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Extract

The main purpose of this note is to show some connexions between Legendre polynomials and random walks. More specifically we shall be concerned with what may be called ‘trinomial’ random walks i.e. walks on the points 0, ± 1, ± 2, …, such that the probabilities of ‘currently’ taking steps of – 1, 0, + 1 are p–1, p0, p1.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 54 , Issue 1 , January 1958 , pp. 39 - 42
Copyright
Copyright © Cambridge Philosophical Society 1958

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References

REFERENCES

(1) Bateman Project Staff. Higher transcendental functions, III (New York, 1955).Google Scholar
(2) Chung, K. L. and Fuchs, W. H. J.On the distribution of sums of random variables. Mem. Amer. Math. Soc. no. 6 (1951), 2–12.Google Scholar
(3) Copson, E. T.The asymptotic expansion of afunction defined by a definite, integral or contour integral (Admiralty Computing Service, London, 1946).Google Scholar
(4) Courant, R. and Hilbert, D.Methoden der Mathematischen Physik, I (Berlin, 1931; English translation, New York, 1953).CrossRefGoogle Scholar
(5) Daniels, H. E.Saddlepoint approximations in statistics. Ann. Math. Statist. 25 (1954), 631–50.CrossRefGoogle Scholar
(6) Domb, C.On multiple returns in the random-walk problem. Proc. Camb. Phil. Soc. 50 (1954), 586–91.CrossRefGoogle Scholar
(7) Feller, W.An introduction to probability theory and ita opplications, I (New York, 1950).Google Scholar
(8) Feller, W.Fluctuation theory of recurrent events. Trans. Amer. Math. Soc. 67 (1949), 98–119.CrossRefGoogle Scholar
(9) Foster, F. G. and Good, I. J.On a generalization of Pólya's random-walk theorem. Quart. J. Math. (2), 4 (1953), 120–6.CrossRefGoogle Scholar
(10) Good, I. J. Saddle-point methods for the multinomial distribution. Annoia Math. Statist. (in the Press).Google Scholar
(11) Harris, T. E.First passage and recurrence distributions. Trans. Amer. Math. Soc. 73 (1952), 471–83.CrossRefGoogle Scholar
(12) Hobson, E. W.Spherical and ettipsoidal harmonics (Cambridge, 1931).Google Scholar
(13) Jeffreys, H. and Jeffreys, B. S.Methods of mathematical physics (Cambridge, 1946).Google Scholar
(14) Kac, M.Random walk and the theory of Brownian motion. Amer. Math. Mon. 54 (1947), 369–91.CrossRefGoogle Scholar
(15) Watson, G. N.Asymptotic expansions of hypergeometric functions. Trans. Camb. Phil. Soc. 22 (1918), 277–308.Google Scholar