Lindley  studies the topic in our title. By using Fisher's conditional-Poisson approach to the multinomial and the logarithmic transformation of gamma variables to normality, he showed that linear contrasts in the logarithms of the cell probabilities $\theta_i$ are asymptotically jointly normal and suggested that the approximation can be improved by applying a "correction" to the sample. By studying the asymptotic series for the joint distribution in Section 2 an improved correction procedure is found below. A more detailed expansion is given in Section 3 for the distribution of a single contrast in the $\log \theta_i$. In many problems a linear function of the $\theta_i$ is of interest. The exact distribution is obtained and is of a form familiar in the theory of serial correlation coefficients. A beta approximation is given. For three cells, a numerical example is given to show the merit of this approximation. A genetic linkage example is considered which requires the joint distribution of two linear functions of the $\theta_i$. The exact joint distribution is found but is too involved for practical use. A normal approximation leads to Lindley's results .
"A Bayesian Study of the Multinomial Distribution." Ann. Math. Statist. 38 (5) 1423 - 1435, October, 1967. https://doi.org/10.1214/aoms/1177698697