Harmonic Moments for Supercritical Multi-type Galton–Watson Processes

Abstract

In this paper, the convergence rates of harmonic moment E[(1 · Z_n)^−r] of a supercritical multi-type Galton–Watson process {Z_n; n ≥ 0} are studied. It is shown that there exists a phase transition in the convergence rate of the harmonic moments, which extends the results from single type cases to multi-type cases. Based on above result, under some moment hypothesis on Z_n, the large deviations of $\boldsymbol{l} \cdot \boldsymbol{Z}_{n+1}\over \boldsymbol{1}\cdot\boldsymbol{Z}_{n}$ are studied for l > 0. In particular, in the case that the offspring distributions are of Pareto-type, an explicit large deviation result is obtained.

Keywords

Supercritical / multi-type / Galton–Watson processes / harmonic moments / large deviations

References

[1] Athreya KB. Large deviation rates for branching processes. I. Single type case. Ann. Appl. Probab., 1994, 4(3): 779-790.

[2] Athreya KB, Ney PE. Branching Processes. Die Grundlehren der Mathematischen Wissenschaften, Band 196, 1972, New York: Springer-Verlag.

[3] Athreya KB, Vidyashankar AN. Large deviation results for branching processes. Stochastic Processes, 1993, New York: Springer-Verlag, 7-12.

[4] Athreya KB, Vidyashankar AN. Large deviation rates for branching processes, II. The multitype case. Ann. Appl. Probab., 1995, 5(2): 566-576.

[5] Barral J. Continuité, moments d’ordres négatifs et analyse multifractale des cascades multiplicative de Mandelbrot, 1997, Orsay: Univ. Paris 11.

[6] Barral J. Moments, continuité, et analyse multifractale des martingales de Mandelbrot. Probab. Theory Related Fields, 1999, 113(4): 535-569.

[7] Brualdi RA, Lewin M. On powers of nonnegative matrices. Linear Algebra Appl., 1982, 43: 87-97.

[8] Cline D.B.H., Hsing T., Large deviation probabilities for sums of random variables with heavy or subexponential tails. 2022, arXiv:2211.16340

[9] Heyde C. On large deviation probabilities in the case of attraction to a non-normal stable law. Sankhya Ser. A, 1968, 30(3): 253-258.

[10] Heyde C, Brown B. An invariance principle and some convergence rate results for branching processes. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 1971, 20: 271-278.

[11] Hoppe FM. Supercritical multitype branching processes. Ann. Probability, 1976, 4(3): 393-401.

[12] Kurtz T, Lyons R, Pemantle R, Peres Y. A conceptual proof of the Kesten–Stigum theorem for multi-type branching processes. Classical and Modern Branching Processes (Minneapolis, MN, 1994), 1997, New York: Springer, 181-185.

[13] Liu QS. Asymptotic properties and absolute continuity of laws stable by random weighted mean. Stochastic Process. Appl., 2001, 95(1): 83-107.

[14] Liu Y, Hu Y. Large deviations for heavy-tailed random sums of independent random variables with dominatedly varying tails. Sci. China Ser. A, 2003, 46(3): 383-395.

[15] Minc H. Nonnegative Matrices, 1988, New York: John Wiley & Sons, Inc..

[16] Ney PE, Vidyashankar AN. Harmonic moments and large deviation rates for supercritical branching processes. Ann. Appl. Probab., 2003, 13(2): 475-489.

[17] Peña VH, Lai TL, Shao QM. Self-normalized Processes—Limit Theory and Statistical Applications, 2009, Berlin: Springer-Verlag.

[18] Petrov VV. Sums of Independent Random Variables, 1975, New York: Springer-Verlag.

[19] Royden HL. Real Analysis, 1988, Third Edition, New York: Macmillan Publishing Company.

[20] Seneta E. Regularly varying functions in the theory of simple branching processes. Adv. in Appl. Probab., 1974, 6: 408-420.

[21] Seneta E. Non-negative Matrices and Markov Chains, 1981, New York: Springer.