By P. M. Kahn

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An upper isfies v0(n) < v0(n) for all n > 0 and 1 - n0( bound on the error in each of the density terms is 2cl, but this bound is very conservative. The first trimming method is undesirable in repeated applications of the algorithm, as needed in computations for the GaltonWatson process, since the computed density corresponding to {n0(n)} plays the role of the mixing density a for the next generation. The number of terms computed varies from one generation COMPUTATIONAL PROBLEMS RL~LATED TO THE GALTON-WATSON PROCESS 25 to the next and the accumulated error grows in a generally unpredictable manner.

Even in examples where k ran up to one hundred, all computed probabilities agreed to at least four decimal places, but on computers with a shorter word length it is probably advisable to perform the latter method in double precision. } is of phase type, except for J the easy computation of the terms of the density. A minor drawback of the Gauss elimination lies in the substantial storage requirements. If we allow values of k up to one hundred, two storage arrays of size 10000 are required, one to store the quantities {r R } and the second one to store the entries of the matrices Kk and Hk.

1(t) is asymptotically normal with mean ai and variance 2laN. 19). 23), but we find, instead lN (aN + c,4'1) = laN + (lc+k) 3N + 0( 1), mN (aN + c3N) = laN + (lx+n) 3N + 0(1). 3), Iglehart's statement of Stone's result is sufficiently general to guarantee the weak convergence of Z1(t) to a diffusion process whenever the initial conditions converge. 22). The drift is upward or downward accordin g as k > v or k< m , and the conditional distribution of Z(t) given Z(0) was derived by Chandrasekhar (1943).

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