Source: Ann. Appl. Probab.
Volume 8, Number 4
Large deviation principles are established for some random
combinatorial structures including the Ewens sampling formula and the Pitman
sampling formula. A path-level large deviation principle is established for the
former on the cadlag space D$(o, 1], R)$ equipped with the uniform convergence
topology, and the rate function is the same as for a Poisson process justifying
the Poisson process approximation for the Ewens sampling formula at the large
deviation level. A large deviation principle for the total number of parts in a
partition is obtained for the Pitman formula; here the rate function depends
only on one of the two parameters which display the different roles of the two
parameters at different scales. In addition to these large deviation results,
we also provide an embedding scheme which gives the Pitman sampling formula. A
product of this embedding is an intuitive alternate proof of a result of Pitman
on the limiting total number of parts.
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