, n d } j 1 ?Z a stationary sequence of martingale differences. The analysis of this martingale-difference sequence shall need results for generalized (d ? 1)-dimensional models (to be defined properly first). To complete the details of this strategy would require an induction argument
Convergence criteria for multiparameter stochastic processes and some applications, Ann. Math. Stat, vol.42, pp.1656-1670, 1971. ,
Invariance principles for self-similar set-indexed random fields, Trans. Amer. Math. Soc, vol.366, p.3256190, 2014. ,
Invariance principles for operator-scaling Gaussian random fields, Ann. Appl. Probab, vol.27, pp.1190-1234, 2017. ,
Operator scaling stable random fields, Stochastic Process. Appl, vol.117, p.2290879, 2007. ,
, Convergence of Probability Measures, p.1700749, 1999.
, Regular Variation. Encyclopedia of Mathematics and Its Applications 27, p.898871, 1987.
DOI : 10.1017/cbo9780511721434
On the central limit theorem for stationary mixing random fields, Ann. Probab, vol.10, 1982. ,
DOI : 10.1214/aop/1176993726
URL : https://doi.org/10.1214/aop/1176993726
Exponential inequalities and functional central limit theorems for a random fields, ESAIM Probab. Stat, vol.5, p.1875665, 2001. ,
DOI : 10.1051/ps:2001103
URL : https://www.esaim-ps.org/articles/ps/pdf/2001/01/psVol5-4.pdf
From infinite urn schemes to decompositions of self-similar Gaussian processes, Electron. J. Probab, vol.21, issue.43, p.3530320, 2016. ,
URL : https://hal.archives-ouvertes.fr/hal-01184411
A simple construction of the fractional Brownian motion, Stochastic Process. Appl, vol.109, p.2031768, 2004. ,
URL : https://hal.archives-ouvertes.fr/hal-00101987
Notes on the occupancy problem with infinitely many boxes: General asymptotics and power laws, Probab. Surv, vol.4, p.2318403, 2007. ,
Power law Pólya's urn and fractional Brownian motion, 2013. ,
Stochastic partial differential equations driven by multiparameter fractional white noise, Stochastic Processes, Physics and Geometry: New Interplays, II (Leipzig, vol.29, pp.327-337, 1999. ,
Foundations of Modern Probability, 1997. ,
On the fractional anisotropic Wiener field, Probab. Math. Statist, vol.16, pp.85-98, 1996. ,
Central limit theorems for certain infinite urn schemes, J. Math. Mech, vol.17, pp.373-401, 1967. ,
Fractional Brownian motion as a weak limit of Poisson shot noise processes -With applications to finance, Stochastic Process. Appl, vol.113, pp.333-351, 2004. ,
Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum, Dokl. Akad. Nauk SSSR, vol.26, p.3441, 1940. ,
Invariance principles for non-isotropic long memory random fields, Stat. Inference Stoch. Process, vol.10, pp.255-282, 2007. ,
A decomposition of the bifractional Brownian motion and some applications, Statist. Probab. Lett, vol.79, p.2499385, 2009. ,
Fractional Brownian motions, fractional noises and applications, SIAM Rev, vol.10, pp.422-437, 1968. ,
Dependent central limit theorems and invariance principles, Ann. Probab, vol.2, p.358933, 1974. ,
Scaling limits for cumulative input processes, Math. Oper. Res, vol.32, p.2363203, 2007. ,
Multiparameter fractional Brownian motion and quasi-linear stochastic partial differential equations, Stoch. Stoch. Rep, vol.71, p.1922562, 2001. ,
On fractional Brownian motion limits in one dimensional nearest-neighbor symmetric simple exclusion, ALEA Lat. Am. J. Probab. Math. Stat, vol.4, pp.245-255, 2008. ,
Long-Range Dependence and Self-Similarity, Cambridge Series in Statistical and Probabilistic Mathematics 45, p.3729426, 2017. ,
Combinatorial Stochastic Processes. Lecture Notes in Math. 1875, p.2245368, 2006. ,
Scaling transition for long-range dependent Gaussian random fields, Stochastic Process. Appl, vol.125, p.3322863, 2015. ,
Aggregation of autoregressive random fields and anisotropic long-range dependence, Bernoulli, vol.22, p.3498033, 2016. ,
Stochastic Processes and Long Range Dependence, p.3561100, 2016. ,
DOI : 10.1007/978-3-319-45575-4
Operator-scaling Gaussian random fields via aggregation, 2017. ,
An invariance principle for fractional Brownian sheets, J. Theoret. Probab, vol.27, pp.1124-1139, 2014. ,
DOI : 10.1007/s10959-013-0483-2
URL : http://arxiv.org/pdf/1109.5160
Sample path properties of anisotropic Gaussian random fields, A Minicourse on Stochastic Partial Differential Equations, p.2508776, 1962. ,
DOI : 10.1007/978-3-540-85994-9_5
URL : http://www.stt.msu.edu/~xiaoyimi/anigaussian3.pdf