K.B. Athreya and P.E. Ney. Branching Processes. Springer, 1972. Home/Search Document Details and Download
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Asymptotic Loss Probability in a Finite Buffer Fluid.. - Jelenkovic, Momcilovic (2002) (Correct)
....of heavy tailed distributions was introduced by Chistyakov [5] De nition A.2 A nonnegative r.v. X is called subexponential, X 2 S, if 1 F (x) 1 F (x) 2; where F denotes the 2 fold convolution of F with itself, i.e. F (x) R 1 0 F (x y)F (dy) It is well known that S L [3]. A survey on subexponential distributions can be found in [13] The class of intermediately regularly varying distributions IR is a subclass of S. De nition A.3 A nonnegative r.v. X is called intermediately regularly varying, X 2 IR, if 1 1 F (x) 1: Regularly varying distributions R ....
K. B. Athreya and P. E. Ney. Branching Processes. Springer-Verlag, 1972.
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....a formalism that originally described a one particle system. One can extract the extinction probability for such a branching and dying process via the following construction. Denote the total mass process 13 of the branching Brownian motion X by N t : X t (R ) It is well known (see e.g. [16]) that fN t : t 0g is a continuous time Galton Watson branching process whose moment generating function is given by (t) Ef N t j N 0 = 1g = k=0 PfN t = k j N 0 = 1g; 15) for 0 (and 0 1) The function solves the initial value problem t (t) W ( ....
....for the process to have died completely at temporal in nity 1: lim 1 1 j1 2 j 2 2 1 for 1 for (130) with almost sure extinction for , as one would intuitively expect. Indeed, 130) is a classical result for branching Brownian motion [16]. As the function D(x; y) only depends on the di erence y x of the space time points x and y, we may write the recurrence relation for D(x; y) in the form D(z) B(z) I Z z wB(w)A(z w )D(z w) 131) Requiring D to be integrable, standard arguments using Banach s xpoint theorem, ....
K. B. Athreya and P. E. Ney (1972), Branching Processes, Springer 52
The Asymptotic Behavior of a - Network Multiplexer With (Correct)
....F (x) 1; cx ;a;1 ;b log x (a 2b log x)# a 0#b 0# and c appropriately chosen. V) Benktander Type II distribution [28] F (x) 1; cax ; 1;b) expf; a=b)x b g# a 0# 0 b 1, and c appropriately chosen. The general relation between S and L is the following. Lemma 3. 3 (Athrey and Ney, [3]) SaeL. The following lemma [8] clearly shows that for long tailed distributions Cram er type conditions are not satisfied. Lemma 3.4 If F 2Lthen (1 ; F (x) e ffx 1as x 1, for all ff 0. An extensive treatment of subexponential distributions (and further references) can be found in Cline ....
K. B. AthreyaandP.E.Ney. Branching Processes. Springer-Verlag, 1972.
Asymptotic Results for Multiplexing Subexponential On-Off.. - Jelenkovic, Lazar (1997) (15 citations) (Correct)
....for her helpful comments. 24 6. Appendix A: Basic Results on Subexponential and Long Tailed Distributions In what follows we will state a few important results from the literature on subexponential distributions. The general relation between S and L is the following. Lemma 3 (Athreya and Ney, [ATN72]) S ae L. Lemma 4 If F 2 L then (1 Gamma F (x) e ffx 1 as x 1, for all ff 0. Note: Lemma 4 clearly shows that for long tailed distributions Cram er type conditions are not satisfied. The proof of the following result can be found in Embrechts, Goldie and Veraverbeke [EGV79] Lemma 5 Let ....
....If G is a probability distribution such that G(x) o( F (x) as x 1, then F G(x) F (x) as x 1. ii) If lim x 1 G(x) F (x) c 2 (0; 1) where G is a distribution function on [0; 1) then G 2 S, and F G(x) F (x) G(x) as x 1. The next result is due to Athreya and Ney (see [ATN72], pp. 147 150) Lemma 6 If F 2 S, then (i) F (x) F (x) n as x 1, for all n 2 N. ii) For each ffl 0 there exists a constant C ffl ( 1) such that F (x) C ffl (1 ffl) F (x) for all x and n. This lemma directly gives the asymptotics of a renewal measure with the following ....
K. B. Athreya and P. E. Ney, Branching Processes. Springer-Verlag, 1972.
Large Deviation Analysis of Subexponential Waiting Times.. - Jelenkovic, Momcilovic (2001) (Correct)
....was introduced by Chistyakov [11] De nition A.2 A nonnegative random variable X (or its d.f. F ) is called subexponential X 2 S (F 2S) if 1 F 1 F (x) 2; where F denotes the 2 fold convolution of F with itself, i.e. F R [0;1) F (x y)F (dy) It is well known that S L [6]. A survey on subexponential distributions can be found in [14] The class of intermediately regularly varying distributions IR is a subclass of S. De nition A.3 A nonnegative random variable X (or its d.f. F ) is called intermediately regularly varying X 2 IR (F 2 IR) if 1 1 F (x) ....
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