## The Annals of Applied Probability

### A uniform law for convergence to the local times of linear fractional stable motions

James A. Duffy

#### Abstract

We provide a uniform law for the weak convergence of additive functionals of partial sum processes to the local times of linear fractional stable motions, in a setting sufficiently general for statistical applications. Our results are fundamental to the analysis of the global properties of nonparametric estimators of nonlinear statistical models that involve such processes as covariates.

#### Article information

Source
Ann. Appl. Probab., Volume 26, Number 1 (2016), 45-72.

Dates
Revised: October 2014
First available in Project Euclid: 5 January 2016

https://projecteuclid.org/euclid.aoap/1452003234

Digital Object Identifier
doi:10.1214/14-AAP1085

Mathematical Reviews number (MathSciNet)
MR3449313

Zentralblatt MATH identifier
1334.60044

#### Citation

Duffy, James A. A uniform law for convergence to the local times of linear fractional stable motions. Ann. Appl. Probab. 26 (2016), no. 1, 45--72. doi:10.1214/14-AAP1085. https://projecteuclid.org/euclid.aoap/1452003234

#### References

• Astrauskas, A. (1983). Limit theorems for sums of linearly generated random variables. Lith. Math. J. 23 127–134.
• Avram, F. and Taqqu, M. S. (1992). Weak convergence of sums of moving averages in the $\alpha$-stable domain of attraction. Ann. Probab. 20 483–503.
• Bercu, B. and Touati, A. (2008). Exponential inequalities for self-normalized martingales with applications. Ann. Appl. Probab. 18 1848–1869.
• Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge Univ. Press, Cambridge.
• Borodin, A. N. (1981). The asymptotic behavior of local times of recurrent random walks with finite variance. Theory Probab. Appl. 26 758–772.
• Borodin, A. N. (1982). On the asymptotic behavior of local times of recurrent random walks with infinite variance. Theory Probab. Appl. 29 318–333.
• Borodin, A. N. and Ibragimov, I. A. (1995). Limit theorems for functionals of random walks. Proc. Steklov Inst. Math. 195 1–259.
• Chan, N. and Wang, Q. (2014). Uniform convergence for nonparametric estimators with nonstationary data. Econometric Theory 30 1110–1133.
• Davidson, J. (1994). Stochastic Limit Theory: An Introduction for Econometricians. Oxford Univ. Press, New York.
• Davidson, J. and de Jong, R. M. (2000). The functional central limit theorem and weak convergence to stochastic integrals. II. Fractionally integrated processes. Econometric Theory 16 643–666.
• Duffy, J. A. (2015). Uniform convergence rates over maximal domains in structural nonparametric cointegrating regression. Unpublished manuscript, Univ. Oxford.
• Gao, J., Kanaya, S., Li, D. and Tjøstheim, D. (2015). Uniform consistency for nonparametric estimators in null recurrent time series. Econometric Theory 31 911–952.
• Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Application (Probability and Mathematical Statistics). Academic Press, New York.
• Hannan, E. J. (1979). The central limit theorem for time series regression. Stochastic Process. Appl. 9 281–289.
• Ibragimov, I. A. and Linnik, Yu. V. (1971). Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff Publishing, Groningen.
• Jeganathan, P. (2004). Convergence of functionals of sums of r.v.s to local times of fractional stable motions. Ann. Probab. 32 1771–1795.
• Jeganathan, P. (2008). Limit theorems for functionals of sums that converge to fractional Brownian and stable motions. Cowles Foundation Discussion Paper No. 1649, Yale Univ., New Haven, CT.
• Karlsen, H. A., Myklebust, T. and Tjøstheim, D. (2007). Nonparametric estimation in a nonlinear cointegration type model. Ann. Statist. 35 252–299.
• Karlsen, H. A. and Tjøstheim, D. (2001). Nonparametric estimation in null recurrent time series. Ann. Statist. 29 372–416.
• Kasahara, Y. and Maejima, M. (1988). Weighted sums of i.i.d. random variables attracted to integrals of stable processes. Probab. Theory Related Fields 78 75–96.
• Kasparis, I., Andreou, E. and Phillips, P. C. B. (2012). Nonparametric predictive regression. Cowles Foundation Discussion Paper No. 1878, Yale Univ., New Haven, CT.
• Kasparis, I. and Phillips, P. C. B. (2012). Dynamic misspecification in nonparametric cointegrating regression. J. Econometrics 168 270–284.
• Liu, W., Chan, N. and Wang, Q. (2014). Uniform approximation to local time with applications in non-linear cointegrating regression. Unpublished manuscript, Univ. Sydney.
• Park, J. Y. and Phillips, P. C. B. (2001). Nonlinear regressions with integrated time series. Econometrica 69 117–161.
• Perkins, E. (1982). Weak invariance principles for local time. Probab. Theory Related Fields 60 437–451.
• Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes: Stochastic Models With Infinite Variance. Chapman & Hall, New York.
• Skorohod, A. V. (1956). Limit theorems for stochastic processes. Theory Probab. Appl. 1 261–290.
• Tyran-Kamińska, M. (2010). Functional limit theorems for linear processes in the domain of attraction of stable laws. Statist. Probab. Lett. 80 975–981.
• van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge Univ. Press, Cambridge.
• van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes: With Applications to Statistics. Springer, New York.
• Wang, Q. and Phillips, P. C. B. (2009a). Asymptotic theory for local time density estimation and nonparametric cointegrating regression. Econometric Theory 25 710–738.
• Wang, Q. and Phillips, P. C. B. (2009b). Structural nonparametric cointegrating regression. Econometrica 77 1901–1948.
• Wang, Q. and Phillips, P. C. B. (2011). Asymptotic theory for zero energy functionals with nonparametric regression applications. Econometric Theory 27 235–259.
• Wang, Q. and Phillips, P. C. B. (2012). A specification test for nonlinear nonstationary models. Ann. Statist. 40 727–758.
• Wang, Q. and Phillips, P. C. B. (2015). Nonparametric cointegrating regression with endogeneity and long memory. Econometric Theory. To appear. DOI:10.1017/S0266466614000917.