K.B. Athreya, Limit theorems for multitype continuous time Markov branching processes. II. The case of an arbitrary linear functional. Z. Wahrsch. Verw. Gebiete 13 (1969), 204--214.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....consequences are # e (2.10) and more generally, for 0 #,k P # e (2.11) and, using (A4) t (2.12) where, as sometimes later, C denotes unspecified constants that may depend on the data q, a i , # i , X 0 . As is well known since decades [33] 5] [6], the asymptotic behavior depends on whether there is any eigenvalue beside # 1 with a real part # 1 2. We thus define # I : # : Re# # 1 2 , # II : # : Re# = # 1 2 , # III : # : Re# # 1 2 ; hence # is the disjoint union # I III . We further define P I : P # , the ....

....a and (3.17) EV (x)V (y) # = mx# I (y x) m 1 A # 6. Relations to previously known results Large parts of Theorem 3.1 are known since Athreya s thesis, at least in the irreducible case: the a.s. convergence is in [4] and [9, Theorem V.7.2] and the limits in Corollary 3. 8 are proved in [5] [6], see also [9, V.8] Our results give more explicit formulas for the asymptotic variances and covariances, and the extension to stochastic processes. The independence between the limit processes in (i) and (ii) seems new too. Also Theorems 3.16 3.19 for urn models are basically due to Athreya ....

K.B. Athreya, Limit theorems for multitype continuous time Markov branching processes. II. The case of an arbitrary linear functional. Z. Wahrsch. Verw. Gebiete 13 (1969), 204--214.

....consequences are # e (2.10) and more generally, for 0 #,k P # e (2.11) and, using (A4) t (2.12) where, as sometimes later, C denotes unspecified constants that may depend on the data q, a i , # i , X 0 . As is well known since decades [33] [5], 6] the asymptotic behavior depends on whether there is any eigenvalue beside # 1 with a real part # 1 2. We thus define # I : # : Re# # 1 2 , # II : # : Re# = # 1 2 , # III : # : Re# # 1 2 ; hence # is the disjoint union # I III . We further define P I : P # , ....

....m 1 a and (3.17) EV (x)V (y) # = mx# I (y x) m 1 A # 6. Relations to previously known results Large parts of Theorem 3.1 are known since Athreya s thesis, at least in the irreducible case: the a.s. convergence is in [4] and [9, Theorem V.7.2] and the limits in Corollary 3. 8 are proved in [5], 6] see also [9, V.8] Our results give more explicit formulas for the asymptotic variances and covariances, and the extension to stochastic processes. The independence between the limit processes in (i) and (ii) seems new too. Also Theorems 3.16 3.19 for urn models are basically due to ....

K.B. Athreya, Limit theorems for multitype continuous time Markov branching processes. I. The case of an eigenvector linear functional. Z. Wahrsch. Verw. Gebiete 12 (1969), 320--332.

....d Gamma N(0; oe 2 ) for some 0 oe 2 1 independent of Z 0 . iii) jj 2 = ae ) for any Z 0 6= 0; l DeltaZ n p u DeltaZ n (ln u DeltaZ n ) d Gamma N(0; oe 2 ) for some 0 oe 2 1 independent of Z 0 . 2 There are extensions of this result to arbitrary vectors l (see Athreya [10 ] and Athreya and Ney [23] Further limit results for functionals of branching process have been established in Asmussesn (see [1] 2] and [3] Asmussen and Keiding [7] and Asmussen and Kurtz [8] The large deviation results for multitype branching processes have been considered by Athreya and ....

Athreya,K.B.(1969): Limit theorems for multitype continuous time Markov branching processes II-the case of an arbitrary linear functional, Z. Wahrsc .Verw. Gebiete, 13, 204-214.

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