K.B. Athreya and A.N. Vidyashankar. Large deviation rates for branching processes II: The multitype case. Ann. Appl. Prob., 5:566--576, 1995. Home/Search Document Not in Database Summary Related Articles Check |
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Large Deviations For Supercritical Multi-Type Branching Processes - Jones (2001) (Correct)
....of Biggins Bingham, but applicable to a more general class of tree based processes. See in particular [Liu96, Liu99] Recently Hambly Jones [HJ01] have extended the main results of [BB93] to a class of general Crump ModeJagers branching processes. Athreya [Ath94] and Athreya Vidyashankar [AV95, AV97] have considered related large deviation questions for ratios of the form x Z n 1 [i, x Z n [i, looking at both single and multi type cases. The multi type case is also considered in Hambly Jones [HJ] in the more general context of varying environments. In most of these papers, the ....
....example f[1] x, y) x 2 y 2 and f[2] x, y) x y . 3. Schroder case Throughout this section, in addition to our underlying assumptions on Z, we will assume that A1 A0 holds, # (0, 1) and # n A converges to a matrix# which is non zero and finite. Theorem 4 (Athreya Vidyashankar [AV95] Theorem 1) Under Assumption A1, for all q F (x) lim # n (f n (x) q # (k 1) a(f k (x) exists and solves the multivariate Schroder equation F (f(x) #F (x) Here a is defined by f(x) q A(x q) a(x q) The convergence of the sum is uniform, whence F is ....
K.B. Athreya and A.N. Vidyashankar. Large deviation rates for branching processes II: The multitype case. Ann. Appl. Prob., 5:566--576, 1995.
Branching Processes - Athreya, Vidyashankar (1999) (113 citations) Self-citation (Athreya Vidyashankar) (Correct)
.... [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 Vidyashankar (see [25] and [26] This leads to interesting questions about the rates of decay of iterates of multidimensional generating functions given in (2.6) This is given in the next Theorem. We first define C p j [0; 1] Theta : Theta [0; 1] the p dimensional unit cube and R p to be the p dimensional ....
....(j l Delta Z n 1 Delta Z n Gamma l:v 1 Delta v j ffl) 3.13) exists and is positive and finite. 2 As in the single type case further refinements and other large deviation results concerning multitype branching processes have been developed and can be found in Athreya and Vidyashankar ([25] and [26] and Vidyashankar( 60] We now describe the critical case. This result was obtained by Joffe and Spitizer (see [60] Theorem 13. Let ae = 1 and u and v be as in (3.3) Let oe 2 ij j E(Z 2 1j jZ 0 = i) Gamma m 2 ij 1 for all i; j: Then for any initial Z 0 6= 0, and any w such ....
Athreya,K.B. and Vidyashankar,A.N.(1995): Large deviation rates for branching processes II-the multitype case, Annals of Applied Probability, 5, No. 2, 566-576.
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