K. B. Athreya. Large deviation rates for branching processes - I, single type case. Annals of Applied Probability, 4(3):779--790, 1994. Home/Search Document Not in Database Summary Related Articles Check |
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Single-Source Shortest-Paths on Arbitrary Directed Graphs in.. - Meyer (2001) (13 citations) (Correct)
....only duplicate entries into C i . The additional paths can therefore be discarded. All remaining events are independent. We have E[Yu ] degree(u) 2 i 1 2 and by the elementary theory of branching processes, E[Z l ] max degree(G i ) 2 i 1 ) l 2 l . The tail bounds in [2] additionally show that Z l = O( max deg. G i ) 2 i 1 ) l log n) O(2 l log n) whp. Furthermore, P j l Z j = O(2 l log n) whp. By Lemma 7 we have l l = O( log n log logn ) whp. Hence, jfu : u; w; v] i 2 C ; 0 i dlog 2 2negj = O dlog 2 2ne 2 O( log n ....
K. B. Athreya. Large deviation rates for branching processes - I, single type case. Annals of Applied Probability, 4(3):779-790, 1994.
Branching Processes - Athreya, Vidyashankar (1999) (113 citations) Self-citation (Athreya) (Correct)
....d N(0; oe 2 ) where N(0; oe 2 ) is a normal random variable with mean 0 and variance oe 2 . 2 A law of iterated logarithm associated with the above convergence has been established by Heyde [41] and the large deviation results are contained in Athreya and Vidyashankar [24] and Athreya [14]. Theorem 4 Assume p 0 = 0, p 1 6= 0, and E(Z 2r ffi 1 ) 1 for some r 1 and ffi 0 and m r p 1 1. Then lim n 1 1 p n 1 P (j Z n 1 Z n Gamma mj ffl) C(ffl) j X k1 P (jX k Gamma mj ffl)q k where X k = 1 k P k j=1 X j where X j s are i.i.d. as Z 1 and q k = lim n 1 P ....
....X j s are i.i.d. as Z 1 and q k = lim n 1 P (Zn=k) p n 1 . Furthermore, the limit C(ffl) is a finite positive constant. 2 A number of related large deviation results concerning the rates of convergence of the martingale W n to W and other refinements of Theorem 4 can be found in Athreya [14]. We now move on to describe the critical branching processes. The first result in this direction describes the behavior of the process conditioned on non extinction. Theorem 5. Let m = 1 and oe 2 = P 1 1 j 2 p j Gamma 1 be finite. For any initial Z 0 6= 0, and 0 x 1 lim n 1 P ( ....
Athreya,K.B.(1994): Large deviation rates for branching processes-1, the single type case, Annals of Applied Probability, 4, N0. 3, 779-790.
Directed Single-Source Shortest-Paths in Linear Average-Case Time - Meyer (2001) (Correct)
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K. B. Athreya. Large deviation rates for branching processes - I, single type case. Annals of Applied Probability, 4(3):779--790, 1994.
Eternal branching Markov processes: averaging properties and PCR.. - Piau (Correct)
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Krishna B. Athreya. Large deviation rates for branching processes. I. Single type case. Ann. Appl. Probab., 4(3):779-790, 1994.
On the Harmonic Means of Branching Processes - Piau (Correct)
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Krishna B. Athreya. Large deviation rates for branching processes. I. Single type case. Ann. Appl. Probab., 4(3):779-790, 1994.
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