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- ItemLarge deviations for sums defined on a Galton-Watson processes(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Fleischmann, Klaus; Wachtel, VitaliIn this paper we study the large deviation behavior of sums of i.i.d. random variables Xi defined on a supercritical Galton-Watson process Z. We assume the finiteness of the moments EX2 1 and EZ1 log Z1 . The underlying interplay of the partial sums of the Xi and the lower deviation probabilities of Z is clarified. Here we heavily use lower deviation probability results on Z we recently published in [FW06].
- ItemCritical Galton-Watson processes: The maximum of total progenies within a large window(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Fleischmann, Klaus; Valutin, Vladimir A.; Wachtel, VitaliConsider a critical Galton-Watson process Z=Z_n: n=0,1,... of index 1+alpha, alpha in (0,1]. Let S_k(j) denote the sum of the Z_n with n in the window [k,...,k+j), and M_m(j) the maximum of the S_k with k moving in [0,m-j]. We describe the asymptotic behavior of the expectation EM_m(j) if the window width j=j_m satisfies that j/m converges in [0,1] as m tends to infinity. This will be achieved via establishing the asymptotic behavior of the tail probabilities of M_infinity(j).