## Journal of Applied Probability

### On degenerate sums of m-dependent variables

Svante Janson

#### Abstract

It is well known that the central limit theorem holds for partial sums of a stationary sequence (Xi) of m-dependent random variables with finite variance; however, the limit may be degenerate with variance 0 even if var(Xi) ≠ 0. We show that this happens only in the case when Xi - EXi = Yi - Yi-1 for an (m - 1)-dependent stationary sequence (Yi) with finite variance (a result implicit in earlier results), and give a version for block factors. This yields a simple criterion that is a sufficient condition for the limit not to be degenerate. Two applications to subtree counts in random trees are given.

#### Article information

Source
J. Appl. Probab., Volume 52, Number 4 (2015), 1146-1155.

Dates
First available in Project Euclid: 22 December 2015

https://projecteuclid.org/euclid.jap/1450802758

Digital Object Identifier
doi:10.1239/jap/1450802758

Mathematical Reviews number (MathSciNet)
MR3439177

Zentralblatt MATH identifier
1334.60021

Subjects
Primary: 60G10: Stationary processes
Secondary: 60F05: Central limit and other weak theorems 60C05: Combinatorial probability

#### Citation

Janson, Svante. On degenerate sums of m -dependent variables. J. Appl. Probab. 52 (2015), no. 4, 1146--1155. doi:10.1239/jap/1450802758. https://projecteuclid.org/euclid.jap/1450802758

#### References

• Aaronson, J., Gilat, D., Keane, M. and de Valk, V. (1989). An algebraic construction of a class of one-dependent processes. Ann. Prob. 17, 128–143.
• Bradley, R. C. (1980). A remark on the central limit question for dependent random variables. J. Appl. Prob. 17, 94–101.
• Bradley, R. C. (2007a). Introduction to Strong Mixing Conditions, Vol. 1. Kendrick Press, Heber City, UT.
• Bradley, R. C. (2007b). Introduction to Strong Mixing Conditions, Vol. 2. Kendrick Press, Heber City, UT.
• Bradley, R. C. (2007c). Introduction to Strong Mixing Conditions, Vol. 3. Kendrick Press, Heber City, UT.
• Burton, R. M., Goulet, M. and Meester, R. (1993). On 1-dependent processes and $k$-block factors. Ann. Prob. 21, 2157–2168.
• Devroye, L. (1991). Limit laws for local counters in random binary search trees. Random Structures Algorithms 2, 303–315.
• Devroye, L. (2002). Limit laws for sums of functions of subtrees of random binary search trees. SIAM J. Comput. 32, 152–171.
• Diananda, P. H. (1955). The central limit theorem for $m$-dependent variables. Proc. Camb. Phil. Soc. 51, 92–95.
• Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edn. John Wiley, New York.
• Hoeffding, W. and Robbins, H. (1948). The central limit theorem for dependent random variables. Duke Math. J. 15, 773–780.
• Holmgren, C. and Janson, S. (2015). Limit laws for functions of fringe trees for binary search trees and random recursive trees. Electron. J. Prob. 20, 51 pp.
• Holst, L. (1981). Some conditional limit theorems in exponential families. Ann. Prob. 9, 818–830.
• Ibragimov, I. A. (1975). A note on the central limit theorems for dependent random variables. Theory Prob. Appl. 20, 135–141.
• Ibragimov, I. A. and Linnik, Yu. V. (1971). Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff, Groningen.
• Janson, S. (2012). Simply generated trees, conditioned Galton–Watson trees, random allocations and condensation. Prob. Surveys 9, 103–252.
• Janson, S. (2014). Asymptotic normality of fringe subtrees and additive functionals in conditioned Galton–Watson trees. Random Structures Algorithms 10.1002/rsa.20568.
• Le Cam, L. (1958). Un théorème sur la division d'un intervalle par des points pris au hasard. Publ. Inst. Statist. Univ. Paris 7, 7–16.
• Leonov, V. P. (1961). On the dispersion of time-dependent means of a stationary stochastic process. Theory Prob. Appl. 6, 87–93.
• Robinson, E. A. (1960). Sums of stationary random variables. Proc. Amer. Math. Soc. 11, 77–79.
• Schmidt, K. (1977). Cocycles on Ergodic Transformation Groups (Macmillan Lectures Math. 1). Macmillan Company of India, Delhi. \endharvreferences