The Annals of Mathematical Statistics

Saddle-point Methods for the Multinomial Distribution

I. J. Good

Full-text: Open access


Many problems in the theory of probability and statistics can be solved by evaluating coefficients in generating function, or, for continuous differentiable distributions, by an analogous process with Laplace or Fourier transforms. As pointed out for example by H. E. Daniels [2], these problems can often be solved by asymptotic series derived by the saddle-point method from integrals containing a large parameter. Daniels gave a form of saddle-point theorem that is convenient for applications to probability and statistics. In the present paper we extend the theorem in various directions and give some applications to distributions connected with the multinomial distribution, especially to the distribution of $\chi^2$ and to the distribution of the maximum entry in a multinomial distribution.

Article information

Ann. Math. Statist., Volume 28, Number 4 (1957), 861-881.

First available in Project Euclid: 27 April 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier



Good, I. J. Saddle-point Methods for the Multinomial Distribution. Ann. Math. Statist. 28 (1957), no. 4, 861--881. doi:10.1214/aoms/1177706790.

Export citation


  • See Correction: I. J. Good. Corrections Notes: Corrections to "Saddle-point Methods for the Multinomial Distribution. Ann. Math. Statist., Volume 32, Number 2 (1961), 619--619.