Advances in Applied Probability

Marcinkiewicz law of large numbers for outer products of heavy-tailed, long-range dependent data

Hui He

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Given a supercritical Galton‒Watson process {Zn} and a positive sequence {εn}, we study the limiting behaviors of ℙ(SZn/Zn≥εn) with sums Sn of independent and identically distributed random variables Xi and m=𝔼[Z1]. We assume that we are in the Schröder case with 𝔼Z1 log Z1<∞ and X1 is in the domain of attraction of an α-stable law with 0<α<2. As a by-product, when Z1 is subexponentially distributed, we further obtain the convergence rate of Zn+1/Zn to m as n→∞.

Article information

Source
Adv. in Appl. Probab., Volume 48, Number 3 (2016), 672-690.

Dates
First available in Project Euclid: 19 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.aap/1474296309

Mathematical Reviews number (MathSciNet)
MR3511765

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60F10: Large deviations

Keywords
Galton‒Watson process domain of attraction stable distribution slowly varying function large deviation Lotka‒Nagaev estimator Schröder constant

Citation

He, Hui. Marcinkiewicz law of large numbers for outer products of heavy-tailed, long-range dependent data. Adv. in Appl. Probab. 48 (2016), no. 3, 672--690. https://projecteuclid.org/euclid.aap/1474296309


Export citation

References

  • Athreya, K. B. (1994). Large deviation rates for branching processes. I. Single type case. Ann. Appl. Prob. 4, 779–790.
  • Athreya, K. B. and Vidyashankar, A. N. (1997). Large deviation rates for supercritical and critical branching processes. In Classical and Modern Branching Processes (IMA Vol. Math. Appl. 84), Springer, New York, pp. 1–18.
  • Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1989). Regular Variation. Cambridge University Press.
  • Chu, W., Li, W. V. and Ren, Y.-X. (2014). Small value probabilities for supercritical branching processes with immigration. Bernoulli 20, 377–393.
  • Cline, D. B. H. and Hsing, T. (1998). Large deviation probabilities for sums of random variables with heavy or subexponential tails. Tech. Rep., Texas A&M University.
  • Denisov, D., Dieker, A. B. and Shneer, V. (2008). Large deviations for random walks under subexponentiality: the big-jump domain. Ann. Prob. 36, 1946–1991.
  • Dubuc, S. (1971). Problèmes relatifs à l'itération de fonctions suggérés par les processus en cascade. Ann. Inst. Fourier (Grenoble) 21, 171–251.
  • Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edn. John Wiley, New York.
  • Fleischmann, K. and Wachtel, V. (2007). Lower deviation probabilities for superciritcal Galton–Watson processes. Ann. Inst. H. Poincaré Prob. Statist. 43, 233–255.
  • Fleischmann, K. and Wachtel, V. (2008). Large deviations for sums indexed by the generations of a Galton–Watson process. Prob. Theory Relat. Fields 141, 445–470.
  • Jacob, C. and Peccoud, J. (1996). Inference on the initial size of a supercritical branching processes from migrating binomial observations. C. R. Acad. Sci. Paris I 322, 875–880.
  • Jacob, C. and Peccoud, J. (1998). Estimation of the parameters of a branching process from migrating binomial observations. Adv. Appl. Prob. 30, 948–967.
  • Nagaev, A. V. (1967). On estimating the expected number of direct descendants of a particle in a branching process. Theory Prob. Appl. 12, 314–320.
  • Nagaev, S. V. (1979). Large deviations of sums of independent random variables. Ann. Prob. 7, 745–789.
  • Ney, P. E. and Vidyashankar, A. N. (2003). Harmonic moments and large deviation rates for supercritical branching processes. Ann. Appl. Prob. 13, 475–489.
  • Ney, P. E. and Vidyashankar, A. N. (2004). Local limit theory and large deviations for superciritcal branching processes Ann. Appl. Prob. 14, 1135–1166.
  • Piau, D. (2004). Immortal branching Markov processes: averaging properties and PCR applications. Ann. Prob. 32, 337–364.
  • Pruitt, W. E. (1981). The growth of random walks and Lévy processes. Ann. Prob. 9, 948–956.
  • Rozovskiĭ, L. V. (1998). Probabilities of large deviations of sums of independent random variables with a common distribution function from the domain of attraction of an asymmetric stable law. Theory Prob. Appl. 42, 454–482.
  • Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman & Hall, New York.