Uniform saddlepoint approximations

The validity of the saddlepoint expansion evaluated at the point y is considered in the limit y tending to .. This is done for the expansions of the density and of the tail probability of the mean https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20180209072205981-0599:S0001867800018176:S0001867800018176_inline1.gif?pub-status=live of n i.i.d. random variables and also for the expansion of the tail probability of a compound Poisson sum https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20180209072205981-0599:S0001867800018176:S0001867800018176_inline2.gif?pub-status=live , where N is a Poisson random variable. We consider both general conditions that ensure the validity of the expansions and study the four classes of densities for X1 introduced in Daniels (1954).