2024/06/19 by Goodman, Jesse
#41A60 (Primary) 60J80 (Secondary) #60E10 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2406.13182
This paper presents an identity between the multivariate and univariate saddlepoint approximations applied to sample path probabilities for a certain class of stochastic processes. This class, which we term the recursively compounded processes, includes branching processes and other models featuring sums of a random number of i.i.d. terms; and compound Poisson processes and other Lévy processes in which the additive parameter is itself chosen randomly. For such processes, fX1,\dotsc,XN | X0=x0(x1,…,xN) = ∏n=1N f_Xn | X0=x0,…,Xn-1=xn-1(xn), where the left-hand side is a multivariate saddlepoint approximation applied to the random vector (X1,…,XN) and the right-hand side is a product of univariate saddlepoint approximations applied to the conditional one-step distributions given the past. Two proofs are given. The first proof is analytic, based on a change-of-variables identity linking the functions that arise in the respective saddlepoint approximations. The second proof is probabilistic, based on a representation of the saddlepoint approximation in terms of tilted distributions, changes of measure, and relative entropies.