Hi.

I'm not sure I understood your question, but you could look at one bin/category at the time, and merge all others into a second bin.  Then the binomial distribution could give you an upper bound?

Cheers,

Diego

I'm not sure I understand your question either, but on another tack, if apprpriate, you could apply a conjugate prior Bayesian analysis, yielding a Dirichlet posterior distribution for your unknown parameters, and thus relatively straighforward access to probabilities associated with any function of those parameters. This assumes that your N is a fixed parameter. Changing gears again, Johnson, Kotz, and Balakrishan's Discrete Multivariate Distributions is getting dated, but is comprehensive in content and historical literature references.

From your profile, you would have the posterior distribution stuff. I'll try two more suggestions. If your problem is general, Hoeffding's inequality has been used to obtain pretty tight bounds on sums of delimited quantities. And the links from there abound. If you are sort ordering your category probabilities, then you might look at Marshall, Olkin, and Arnold's Inequalities: Theory of Majorization and Its Applications (3rd Ed.).  Majorization is a poweful idea and the book is remarkable. 

Diego, Mike, thank you for your suggestions !

The closest answer i have found so far is the following paper by B. Levin, Annals of Statistics, 1981

http://projecteuclid.org/download/pdf_1/euclid.aos/1176345593

(formula (3) sums up the "obvious" bounds i was referring to in my question)

but i shall definitely have a look at Marshall, Olkin and Arnold book

.

Sharing link:

https://www.google.es/url?sa=t&source=web&rct=j&url=http://www.stat.cmu.edu/~cshalizi/uADA/12/lectures/ch20.pdf&ved=2ahUKEwja8vD-t9PZAhXCPxQKHQbSDx0QFjAFegQIAhAB&usg=AOvVaw0MOkJyv8Dsn8b4M-VrM1yr

Regards!