## Local limit theory and large deviations for supercritical branching processes

Venue: | Ann. Appl. Probab |

Citations: | 7 - 0 self |

### BibTeX

@ARTICLE{Ney_locallimit,

author = {E. Ney and N. Vidyashankar},

title = {Local limit theory and large deviations for supercritical branching processes},

journal = {Ann. Appl. Probab},

year = {},

pages = {1135--1166}

}

### OpenURL

### Abstract

In this paper we study several aspects of the growth of a supercritical Galton–Watson process {Zn:n ≥ 1}, and bring out some criticality phenomena determined by the Schröder constant. We develop the local limit theory of Zn, that is, the behavior of P(Zn = vn) as vn ր ∞, and use this to study conditional large deviations of {YZn:n ≥ 1}, where Yn satisfies an LDP, particularly of {Z −1 n Zn+1:n ≥ 1} conditioned on Zn ≥ vn. 1. Introduction. In

### Citations

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Large Deviations Techniques and Applications
- Dembo, Zeitouni
- 1998
(Show Context)
Citation Context ...variables and Nn (ր ∞ in probability) is a sequence of random variables. When one normalizes the sequence {S(Nn):n ≥ 1} by n then the large deviation behavior of n −1 S(Nn) is easily established [see =-=[11]-=-, Exercise 2.3.19(b)]. In fact, the conclusion is that {n −1 S(Nn):n ≥ 1} satisfies an LDP with the rate function being the Legendre–Fenchel transform of Λ(ΛX(θ)), where ΛX(θ) is the logarithmic momen... |

453 | Branching processes
- Athreya, Ney
- 1972
(Show Context)
Citation Context ...negative random variable W; under the further assumption that E(Z1 logZ1) < ∞, the limit random variable is nontrivial with an absolutely continuous density w(·), except for a possible atom at 0 (see =-=[3]-=- for details). The behavior of the density near 0 has been investigated by Dubuc [13] who showed that for 0 < x < 1 there exists universal constants 0 < C1 < C2 < ∞ such that (4) C1x α−1 ≤ w(x) ≤ C2x ... |

54 |
Assessing the accuracy of the maximum likelihood estimator: Observed verses expected Fisher information., Biometrica 65(3
- Efron, Hinkley
- 1978
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Citation Context ...TIONAL LDP FOR BP 3 Conditioning as a technique to improve the precision of confidence intervals has long been used in the statistical literature. It has been argued by Efron, Hinkley and others (see =-=[16, 26, 27]-=-) that when one seeks to construct confidence intervals for an unknown parameter it should be conditioned on the observed Fisher information. Motivated by this, Sweeting [28] considered the problem of... |

48 |
Likelihood methods in statistics
- Severini
- 2000
(Show Context)
Citation Context ...TIONAL LDP FOR BP 3 Conditioning as a technique to improve the precision of confidence intervals has long been used in the statistical literature. It has been argued by Efron, Hinkley and others (see =-=[16, 26, 27]-=-) that when one seeks to construct confidence intervals for an unknown parameter it should be conditioned on the observed Fisher information. Motivated by this, Sweeting [28] considered the problem of... |

35 |
Mitochondrial DNA sequences in primates: tempo and mode of evolution
- Brown, Prager, et al.
- 1982
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Citation Context ... It was established in those papers that, under an exponential moment hypothesis, if a > m and γ > 0, then 1 0 < lim n→∞ n logP(Rn (1) > a|Zn > 0) = −logγ. Thus the sequence Rn satisfies the LDP (see =-=[10]-=- for definition) with the constant rate function I(x) = −logγ. The limit is “degenerate” in the sense that the rate function is independent of a. It is also puzzling, at first sight, that though Rn = ... |

19 |
On deviations of the sample
- Bahadur, Rao
- 1960
(Show Context)
Citation Context ... generating function of the random variable Z1 and B is as in the theorem. If n −1 vn−k → ∞, then Ĩ1(x) = B. Remark 6. Under an exponential moment condition on Z1, a sharper result along the lines of =-=[5]-=- can be obtained. Theorem 3 also sheds light on the large deviations of W −1 n W. Namely, we have the following corollary.10 P. E. NEY AND A. N. VIDYASHANKAR Corollary 2. Assume E(e θZ1 ) < ∞, limn→∞... |

17 | Large deviations in the supercritical branching process
- Biggins, Bingham
- 1993
(Show Context)
Citation Context ...tional iterations literature, offspring distributions with this property are said to belong to the Schröder case. A further refinement of the above estimate has been considered by Biggins and Bingham =-=[7]-=-. The process with α = ∞ grows exponentially fast at all times and as such has a different probabilistic structure and is frequently referred to as the Böttcher case. The quantity α shows up in severa... |

16 |
On the limit of a supercritical branching process
- Bingham
- 1988
(Show Context)
Citation Context ...whenever vn < jn 0 . Thus the local limit theorem yields nontrivial results when jn 0 < vn and vn = O(mn ). Dubuc [14] has given a detailed analysis of the moment generating function of W and Bingham =-=[8]-=-, using a Tauberian argument, has elucidated the behavior of P(W ≤ x) as x ↓ 0. A crucial role is played by the parameter β[≡ (logm) −1 logj0] that relates the minimum family size and the mean of the ... |

14 |
La densite de la loi limite d’un processus en casacade expansif
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- 1971
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Citation Context ...imit random variable is nontrivial with an absolutely continuous density w(·), except for a possible atom at 0 (see [3] for details). The behavior of the density near 0 has been investigated by Dubuc =-=[13]-=- who showed that for 0 < x < 1 there exists universal constants 0 < C1 < C2 < ∞ such that (4) C1x α−1 ≤ w(x) ≤ C2x α−1 , where α is a solution to γ = m −α and is assumed to be finite. The finiteness o... |

12 |
Large deviation rates for branching processes. I Single type
- Athreya
- 1994
(Show Context)
Citation Context ....i.d. random variables, it is natural to enquire whether it has large deviation properties along the lines of Cramér’s theorem. Large deviations of {Rn :n ≥ 1} were previously investigated by Athreya =-=[1]-=- and Athreya and Vidyashankar [4]. It was established in those papers that, under an exponential moment hypothesis, if a > m and γ > 0, then 1 0 < lim n→∞ n logP(Rn (1) > a|Zn > 0) = −logγ. Thus the s... |

12 |
Trigonometric Series. 2nd ed., Cambridge Univ
- Zygmund
- 1959
(Show Context)
Citation Context ...Qr(ψs(uη(r,s)))du C ∣ ≤ |t| . (83) r,s≥1 π □LOCAL LIMITS AND CONDITIONAL LDP FOR BP 19 Proof. The proof uses the following estimate whose proof can be constructed using methods similar to the one in =-=[31]-=-, page 45, (4.2). Dubuc ([13], page 481) and Dubuc and Senata ([15], page 494) have used this estimate extensively in their work. For any h ∈ L∞[a,b], (84) ∫ b ∣ e a iut h(u)du ∣ ≤ ‖h‖∞π |t| + (b − a)... |

11 |
Harmonic moments and large deviation rates for supercritical branching processes
- Ney, Vidyashankar
(Show Context)
Citation Context ... has the power series representation Q(s) = ∑ qjs j (6) for 0 ≤ s < 1. j≥0 In the work on large deviations of branching processes (and in other contexts as well; see [19, 20]) it has been established =-=[1, 24]-=- that the integrability of Q(·) near 1, that is, finiteness of I ≡ ∫ 1 0 Q(s)ds, plays an important role. In the embeddable situation 0 < α < 1 and the above-mentioned result of Dubuc readily yields t... |

9 |
Exact distributions of kin numbers in a Galton-Watson process
- JOFFE, WAUGH
- 1982
(Show Context)
Citation Context ...nterior of the unit disc and hence has the power series representation Q(s) = ∑ qjs j (6) for 0 ≤ s < 1. j≥0 In the work on large deviations of branching processes (and in other contexts as well; see =-=[19, 20]-=-) it has been established [1, 24] that the integrability of Q(·) near 1, that is, finiteness of I ≡ ∫ 1 0 Q(s)ds, plays an important role. In the embeddable situation 0 < α < 1 and the above-mentioned... |

6 | Problemes relatifs a l’iteration des fonctions suggeres par les processus en cascade - Dubuc - 1971 |

6 |
Branching Processes and Neutral Evolution
- Taïb
- 1992
(Show Context)
Citation Context ...st structure of the branching population, based on the information about its present (namely Zn and Zn+1). This formulation has practical significance, for example, in the area of molecular evolution =-=[9, 10, 23, 30]-=-. Now the analysis of (3) requires a careful study of the large deviations of {Rn :n ≥ 1} which is carried out in the present paper. The behavior of (3) itself requires additional technical tools whic... |

3 | The local limit theorem and some related aspects of supercritical branching processes - Athreya, Ney - 1970 |

3 |
Non-parametric estimation in the Galton-Watson processes
- Pakes
- 1975
(Show Context)
Citation Context ...on 0 < α < 1 and the above-mentioned result of Dubuc readily yields that I < ∞. Of course, one can establish the finiteness of I, under the assumption α < 1, without invoking any embeddability issues =-=[1, 24, 25]-=-. In this paper we make a detailed study of P(Zn = vn) for various ranges of values of vn. This problem was studied in the 1970s for the extreme cases when vn ∼ mn for a complicated restriction on the... |

2 |
A method of estimating from two-aligned present-day DNA sequences and their ancestral composition and subsequent rates of substitution, possibly different in the two lineages, corrected for multiple and parallel substitutions at the same site
- Blaisdell
- 1985
(Show Context)
Citation Context ...st structure of the branching population, based on the information about its present (namely Zn and Zn+1). This formulation has practical significance, for example, in the area of molecular evolution =-=[9, 10, 23, 30]-=-. Now the analysis of (3) requires a careful study of the large deviations of {Rn :n ≥ 1} which is carried out in the present paper. The behavior of (3) itself requires additional technical tools whic... |

2 |
The local limit theorem for the Galton-Watson process
- Dubuc, Seneta
- 1976
(Show Context)
Citation Context ...hreya and Ney [2] (see also [13]) have established the rate of convergence to 0 of the difference |mnP(Zn = j) − w(jm−n )|, when j(n) ∼ mn . Assuming only the finiteness of the mean, Dubuc and Seneta =-=[15]-=- proved a weaker form of the above result. It is unclear from these works if α plays any role in the behavior of P(Zn = vn). For the early history of the local limit theorem, see the paper of Athreya ... |

2 |
Embedding iterates of analytic functions with two fixed points into continuous groups
- Karlin, McGregor
- 1968
(Show Context)
Citation Context ...red to as the Böttcher case. The quantity α shows up in several deep results in the theory of supercritical branching processes and will play a critical role in our study as well. Karlin and McGregor =-=[21, 22]-=- studied the problem of embeddability of discrete-time branching processes into continuous-time branching processes. The Karlin–McGregor function K(s) ≡ s α Q(φ(s)) where φ(s) ≡ E(e −sW ) and its cons... |

2 |
Information and conditional inference
- Severini
- 1995
(Show Context)
Citation Context ...TIONAL LDP FOR BP 3 Conditioning as a technique to improve the precision of confidence intervals has long been used in the statistical literature. It has been argued by Efron, Hinkley and others (see =-=[16, 26, 27]-=-) that when one seeks to construct confidence intervals for an unknown parameter it should be conditioned on the observed Fisher information. Motivated by this, Sweeting [28] considered the problem of... |

1 |
Large deviation results for branching processes
- Athreya, Vidyashankar
- 1993
(Show Context)
Citation Context ...ural to enquire whether it has large deviation properties along the lines of Cramér’s theorem. Large deviations of {Rn :n ≥ 1} were previously investigated by Athreya [1] and Athreya and Vidyashankar =-=[4]-=-. It was established in those papers that, under an exponential moment hypothesis, if a > m and γ > 0, then 1 0 < lim n→∞ n logP(Rn (1) > a|Zn > 0) = −logγ. Thus the sequence Rn satisfies the LDP (see... |

1 |
Efficient conditional tests for mixture experiments with applications to the birth and branching processes
- Basawa
- 1981
(Show Context)
Citation Context ...hermore, Sweeting [29] demonstrates the improved performance of these conditional confidence intervals through simulations. For further work on conditioning in the context of branching processes, see =-=[6, 18, 17]-=-. We of course focus on the statistic Rn. The observed conditional Fisher information in this case is σ −2 Zn. A further motivation for the present study comes from a class of conditioned limit laws s... |

1 | La fonction de Green d’un processus de Galton–Watson - Dubuc - 1970 |

1 |
Remarks on efficiency in estimation for branching processes
- Heyde
- 1975
(Show Context)
Citation Context ...hermore, Sweeting [29] demonstrates the improved performance of these conditional confidence intervals through simulations. For further work on conditioning in the context of branching processes, see =-=[6, 18, 17]-=-. We of course focus on the statistic Rn. The observed conditional Fisher information in this case is σ −2 Zn. A further motivation for the present study comes from a class of conditioned limit laws s... |

1 |
An optimal property of maximum likelihood with application to branching process estimation
- Heyde
- 1977
(Show Context)
Citation Context ...hermore, Sweeting [29] demonstrates the improved performance of these conditional confidence intervals through simulations. For further work on conditioning in the context of branching processes, see =-=[6, 18, 17]-=-. We of course focus on the statistic Rn. The observed conditional Fisher information in this case is σ −2 Zn. A further motivation for the present study comes from a class of conditioned limit laws s... |

1 |
Exact distributions of kin numbers in multitype Galton–Watson population
- Joffe, Waugh
- 1985
(Show Context)
Citation Context ...nterior of the unit disc and hence has the power series representation Q(s) = ∑ qjs j (6) for 0 ≤ s < 1. j≥0 In the work on large deviations of branching processes (and in other contexts as well; see =-=[19, 20]-=-) it has been established [1, 24] that the integrability of Q(·) near 1, that is, finiteness of I ≡ ∫ 1 0 Q(s)ds, plays an important role. In the embeddable situation 0 < α < 1 and the above-mentioned... |

1 |
Embeddability of discrete-time simple branching processes into continuous-time branching processes
- Karlin, McGregor
- 1968
(Show Context)
Citation Context ...red to as the Böttcher case. The quantity α shows up in several deep results in the theory of supercritical branching processes and will play a critical role in our study as well. Karlin and McGregor =-=[21, 22]-=- studied the problem of embeddability of discrete-time branching processes into continuous-time branching processes. The Karlin–McGregor function K(s) ≡ s α Q(φ(s)) where φ(s) ≡ E(e −sW ) and its cons... |

1 |
A test for DNA evolutionary models
- Kelly
- 1994
(Show Context)
Citation Context ...st structure of the branching population, based on the information about its present (namely Zn and Zn+1). This formulation has practical significance, for example, in the area of molecular evolution =-=[9, 10, 23, 30]-=-. Now the analysis of (3) requires a careful study of the large deviations of {Rn :n ≥ 1} which is carried out in the present paper. The behavior of (3) itself requires additional technical tools whic... |

1 |
On efficient tests for branching processes
- Sweeting
- 1978
(Show Context)
Citation Context ...y and others (see [16, 26, 27]) that when one seeks to construct confidence intervals for an unknown parameter it should be conditioned on the observed Fisher information. Motivated by this, Sweeting =-=[28]-=- considered the problem of maximum likelihood estimation of the offspring mean when the underlying offspring distribution is geometric. Indeed, Sweeting establishes that the statistic Tn ≡ ( ∑ n j=0 Z... |

1 |
Asymptotic conditional inference for the offspring mean of a supercritical Galton–Watson process
- Sweeting
- 1986
(Show Context)
Citation Context ...ed on ∑ n j=0 Zj−1 = Vn, where Vn ∼ m n , appropriately centered and normalized converges to a Gaussian random variable with mean 0 and variance equal to the Fisher information. Furthermore, Sweeting =-=[29]-=- demonstrates the improved performance of these conditional confidence intervals through simulations. For further work on conditioning in the context of branching processes, see [6, 18, 17]. We of cou... |