Local limit theorem for a perturbed sample paths of induced order statistics
- Authors: Bitter I.I.1
-
Affiliations:
- Laboratory of Stochastic Analysis and its Applications, HSE University
- Issue: No 113 (2025)
- Pages: 6-20
- Section: Systems analysis
- URL: https://journals.rcsi.science/1819-2440/article/view/289704
- ID: 289704
Cite item
Abstract
In this paper we derive a local limit theorem for a perturbed sample paths of normalized sums of induced order statistics obtained from a sequence of independent identically distributed random vectors under weak regularity conditions on the coefficients. The situation under consideration is a typical example of the problem of estimating the rate of convergence of discrete-time Markov processes to diffusions, when the corresponding trends and diffusion coefficients of the Markov chain and the diffusion limit coincide only asymptotically. Under the conditions described above, the classical result of Konakov and Mammen (2000) on the rate of weak convergence of triangular arrays of discrete Markov processes to a diffusion process with coefficients that coincide with the coefficients of the chains turns out to be inapplicable. Our approach is based on the study of the uniform distance between the transition densities of the underlying inhomogeneous Markov chain and the limiting gaussian diffusion process. In particular, the convergence rate estimate derived from the well-known classical limit theorem and the parametrix-type stability bounds.
About the authors
Ilya Igorevich Bitter
Laboratory of Stochastic Analysis and its Applications, HSE University
Email: ilya.bitter@yandex.ru
Moscow
References
- BITTER I., KONAKOV V. L1 and L ∞ stability of transitiondensities of perturbed diffusions // Random Oper. Stoch. Eq. –2021. – Vol. 29(4). – P. 287–308.
- BHATTACHARYA R., RAO R. Normal approximations andasymptotic expansions. – Wiley and sons, 1976.
- BHATTACHARYA P.K. Convergence of Sample Paths ofNormalized Sums of Induced Order Statistics // Ann. Statist. –September, 1974. – Vol. 2(5). – P. 1034–1039.
- CRAMER H., LEADBETTER M.R. Stationary and relatedstochastic processes. – Dover Publications, Inc., Mineola, NY,2004.
- CS ˙ ORG ´ O M., R ´ EV ´ ESZ P. Strong Approximations of thequantile process // The Annals of Statistics. – 1978. – Vol. 6,No. 4. – P. 882–894.
- CS ˙ ORG ´ O M. Quantile Processes with Statistical Applications //Society for IndustrialL and Applied Mathematics. – 1983.
- DAVID H.A. Concomitants of order statistics // Bull. Internat.Statist. Inst. – 1973. – Vol. 45. – P. 295–300.
- DAVID H.A., GALAMBOS J. The asymptotic theory ofconcomitants of order statistics // J. Appl. Probab. – 1974. –Vol. 11. – P. 762–770.
- DAVYDOV YU., EGOROV V. Functional limit theorems forinduced order statistics // Mathematical Methods of Statistics. –January, 2000. – Vol. 9(3). – P. 297–313.
- DELARUE F., MENOZZI S. Density estimates for a randomnoise propagating through a chain of differential equations // J.Funct. Anal. – 2010. – Vol. 259(6). – P. 1577–1630.
- GALAMBOS J. The Asymptotic Theory of Extreme OrderStatistics. – Krieger, Malabar, FL, 1987.
- KONAKOV V., MAMMEN E. Local Limit Theorems andStrong Approximations for Robbins-Monro Procedures //arXiv:2304.10673. – 2023.
- KONAKOV V., MAMMEN E. Local limit theorems fortransition densities of Markov chains converging to diffusions //Probability Theory and Related Fields. – 2000. – Vol. 117. –P. 551–587.
- KONAKOV V., KOZHINA A., MENOZZI S. Stability ofdensities for perturbed diffusions and Markov chains // ESAIM:Probability and Statistics. – 2017. – Vol. 21. – P. 88–112.
- MENOZZI S., PESCE A., ZHANG X. Density and gradientestimates for non-degenerate Brownian SDEs with unboundedmeasurable drift // J. Diff. Eq. –2021. – Vol. 272. – P. 330–369.
- SEN P.K. A note on invariance principles for induced orderstatistics // The Annals of Probab. – 1976. – Vol. 4. –P. 474–479.
- KOZHINA A. Weak error for the Euler scheme approximationof degenerate diffusions with nonsmooth coefficients // Fundam.Prikl. Mat. – 2018. – Vol. 22, Iss. 3. – P. 91–118.
- SKOROHOD A.V. Studies in the theory of randomprocesses. – Addison-Wesley, Reading, Massachussetts, 1965.[English translation of Skorohod A.V. Issledovaniya po teoriisluchaynykh protsessov. – Kiev University Press, 1961.]
- STROOCK D.W., VARADHAN S.R. Multidimensionaldiffusion processes. – Springer, Berlin, Heidelberg, New York,1979.
- VAN DER VAAR A.W. Asymptotic Statistics. – CambridgeUniversity Press, 1998.
- YANG S.S. General distribution theory of the concomitants oforder statistics // The Annals of Statist. – 1977. – Vol. 5. –P. 996–1002.
Supplementary files
