Central limit theorem for sampled sums of dependent random variables
ESAIM: Probability and Statistics, Tome 14 (2010), pp. 299-314

We prove a central limit theorem for linear triangular arrays under weak dependence conditions. Our result is then applied to dependent random variables sampled by a ℤ-valued transient random walk. This extends the results obtained by [N. Guillotin-Plantard and D. Schneider, Stoch. Dynamics 3 (2003) 477-497]. An application to parametric estimation by random sampling is also provided.

DOI : 10.1051/ps:2008030
Classification : Primary 60F05, 60G50, 62D05, Secondary 37C30, 37E05
Keywords: random walks, weak dependence, central limit theorem, dynamical systems, random sampling, parametric estimation 
@article{PS_2010__14__299_0,
     author = {Guillotin-Plantard, Nadine and Prieur, Cl\'ementine},
     title = {Central limit theorem for sampled sums of dependent random variables},
     journal = {ESAIM: Probability and Statistics},
     pages = {299--314},
     year = {2010},
     publisher = {EDP Sciences},
     volume = {14},
     doi = {10.1051/ps:2008030},
     mrnumber = {2779486},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/ps:2008030/}
}
TY  - JOUR
AU  - Guillotin-Plantard, Nadine
AU  - Prieur, Clémentine
TI  - Central limit theorem for sampled sums of dependent random variables
JO  - ESAIM: Probability and Statistics
PY  - 2010
SP  - 299
EP  - 314
VL  - 14
PB  - EDP Sciences
UR  - https://www.numdam.org/articles/10.1051/ps:2008030/
DO  - 10.1051/ps:2008030
LA  - en
ID  - PS_2010__14__299_0
ER  - 
%0 Journal Article
%A Guillotin-Plantard, Nadine
%A Prieur, Clémentine
%T Central limit theorem for sampled sums of dependent random variables
%J ESAIM: Probability and Statistics
%D 2010
%P 299-314
%V 14
%I EDP Sciences
%U https://www.numdam.org/articles/10.1051/ps:2008030/
%R 10.1051/ps:2008030
%G en
%F PS_2010__14__299_0
Guillotin-Plantard, Nadine; Prieur, Clémentine. Central limit theorem for sampled sums of dependent random variables. ESAIM: Probability and Statistics, Tome 14 (2010), pp. 299-314. doi: 10.1051/ps:2008030

[1] A.D. Barbour, R.M. Gerrard and G. Reinert, Iterates of expanding maps. Probab. Theory Relat. Fields 116 (2000) 151-180. | Zbl

[2] J.M. Bardet, P. Doukhan, G. Lang and N. Ragache, Dependent Linderberg central limit theorem and some applications. ESAIM: PS 12 (2008) 154-172. | Zbl

[3] H.C.P. Berbee, Random walks with stationary increments and renewal theory. Math. Centre Tracts 112, Amsterdam (1979). | Zbl

[4] P. Collet, S. Martinez and B. Schmitt, Exponential inequalities for dynamical measures of expanding maps of the interval. Probab. Theory. Relat. Fields 123 (2002) 301-322. | Zbl

[5] C. Coulon-Prieur and P. Doukhan, A triangular CLT for weakly dependent sequences. Statist. Probab. Lett. 47 (2000) 61-68. | Zbl

[6] D. Dacunha-Castelle and M. Duflo, Problèmes à temps mobile. Deuxième édition. Masson (1993).

[7] J. Dedecker and C. Prieur, New dependence coefficients, Examples and applications to statistics. Probab. Theory Relat. Fields 132 (2005) 203-236. | Zbl

[8] J. Dedecker, P. Doukhan, G. Lang, J.R. Leon, S. Louhichi and C. Prieur, Weak dependence: With Examples and Applications. Lect. Notes in Stat. 190. Springer, XIV (2007). | Zbl

[9] C. Deniau, G. Oppenheim and M.C. Viano, Estimation de paramètre par échantillonnage aléatoire. (French. English summary) [Random sampling and parametric estimation]. C. R. Acad. Sci. Paris Sér. I Math. 306 (1988) 565-568. | Zbl

[10] P. Doukhan and S. Louhichi, A new weak dependence condition and applications to moment inequalities. Stochastic Process. Appl. 84 (1999) 313-342. | Zbl

[11] N. Guillotin-Plantard and D. Schneider, Limit theorems for sampled dynamical systems. Stoch. Dynamics 3 (2003) 477-497. | Zbl

[12] F. Hofbauer and G. Keller, Ergodic properties of invariant measures for piecewise monotonic transformations. Math. Z. 180 (1982) 119-140. | Zbl

[13] I.A. Ibragimov, Some limit theorems for stationary processes. Theory Probab. Appl. 7 (1962) 349-382. | Zbl

[14] J.F.C. Kingman, The ergodic theory of subadditive stochastic processes. J. R. Statist. Soc. B 30 (1968) 499-510. | Zbl

[15] M. Lacey, On weak convergence in dynamical systems to self-similar processes with spectral representation. Trans. Amer. Math. Soc. 328 (1991) 767-778. | Zbl

[16] M. Lacey, K. Petersen, D. Rudolph and M. Wierdl, Random ergodic theorems with universally representative sequences. Ann. Inst. H. Poincaré Probab. Statist. 30 (1994) 353-395. | Zbl | Numdam

[17] A. Lasota and J.A. Yorke, On the existence of invariant measures for piecewise monotonic transformations. Trans. Amer. Math. Soc. 186 (1974) 481-488. | Zbl

[18] F. Merlevède and M. Peligrad, On the coupling of dependent random variables and applications. Empirical process techniques for dependent data. Birkhäuser, Boston (2002), pp. 171-193. | Zbl

[19] T. Morita, Local limit theorem and distribution of periodic orbits of Lasota-Yorke transformations with infinite Markov partition. J. Math. Soc. Jpn 46 (1994) 309-343. | Zbl

[20] M. Peligrad and S. Utev, Central limit theorem for linear processes. Ann. Probab. 25 (1997) 443-456. | Zbl

[21] E. Rio, About the Lindeberg method for strongly mixing sequences. ESAIM: PS 1 (1995) 35-61. | Zbl | Numdam

[22] M. Rosenblatt, A central limit theorem and a strong mixing condition. Proc. Natl. Acad. Sci. USA 42 (1956) 43-47. | Zbl

[23] Y.A. Rozanov and V.A. Volkonskii, Some limit theorems for random functions I. Theory Probab. Appl. 4 (1959) 178-197. | Zbl

[24] C.J. Stone, On local and ratio limit theorems. Proc. Fifth Berkeley Symp. Math. Statist. Probab. Univ. Californie (1966), pp. 217-224. | Zbl

[25] S.A. Utev, Central limit theorem for dependent random variables. Probab. Theory Math. Statist. 2 (1990) 519-528. | Zbl

[26] S.A. Utev, Sums of random variables with ϕ-mixing. Siberian Adv. Math. 1 (1991) 124-155. | Zbl

[27] C.S. Withers, Central limit theorems for dependent variables. I. Z. Wahrsch. Verw. Gebiete 57 (1981) 509-534 (corrigendum in Z. Wahrsch. Verw. Gebiete 63 (1983) 555). | Zbl

Cité par Sources :