Limits of sub-bifractional Brownian noises

Nenghui Kuang; Nenghui Kuang

doi:10.3934/era.2023063

Electronic Research Archive

2023, Volume 31, Issue 3: 1240-1252. doi: 10.3934/era.2023063

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Research article

Limits of sub-bifractional Brownian noises

Nenghui Kuang ^,

School of Mathematics and Computing Science, Hunan University of Science and Technology, Hunan 411201, China

Received: 19 July 2022 Revised: 12 November 2022 Accepted: 21 December 2022 Published: 03 January 2023

Let $ S^{H, K} = \{S^{H, K}_t, t\geq 0\} $ be the sub-bifractional Brownian motion (sbfBm) of dimension 1, with indices $ H\in (0, 1) $ and $ K\in (0, 1]. $ We primarily prove that the increment process generated by the sbfBm $ \left\{S^{H, K}_{h+t}-S^{H, K}_h, t\geq 0\right\} $ converges to $ \left\{B^{HK}_t, t\geq 0\right\} $ as $ h\rightarrow \infty $, where $ \left\{B^{HK}_t, t\geq 0\right\} $ is the fractional Brownian motion with Hurst index $ HK $. Moreover, we study the behavior of the noise associated to the sbfBm and limit theorems to $ S^{H, K} $ and the behavior of the tangent process of sbfBm.
- Sub-bifractional Brownian motion,
- noise,
- limit theorem
Citation: Nenghui Kuang. Limits of sub-bifractional Brownian noises[J]. Electronic Research Archive, 2023, 31(3): 1240-1252. doi: 10.3934/era.2023063

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Abstract

Let $ S^{H, K} = \{S^{H, K}_t, t\geq 0\} $ be the sub-bifractional Brownian motion (sbfBm) of dimension 1, with indices $ H\in (0, 1) $ and $ K\in (0, 1]. $ We primarily prove that the increment process generated by the sbfBm $ \left\{S^{H, K}_{h+t}-S^{H, K}_h, t\geq 0\right\} $ converges to $ \left\{B^{HK}_t, t\geq 0\right\} $ as $ h\rightarrow \infty $, where $ \left\{B^{HK}_t, t\geq 0\right\} $ is the fractional Brownian motion with Hurst index $ HK $. Moreover, we study the behavior of the noise associated to the sbfBm and limit theorems to $ S^{H, K} $ and the behavior of the tangent process of sbfBm.

References

[1]	C. E. Nouty, J. L. Journé, The sub-bifractional Brownian motion, Stud. Sci. Math. Hung., 50 (2013), 67–121. https://dx.doi.org/10.1556/SScMath.50.2013.1.1231 doi: 10.1556/SScMath.50.2013.1.1231
[2]	N. Kuang, B. Liu, Parameter estimations for the sub-fractional Brownian motion with drift at discrete observation, Braz. J. Probab. Stat., 29 (2015), 778–789. https://dx.doi.org/10.1214/14-BJPS246 doi: 10.1214/14-BJPS246
[3]	N. Kuang, B. Liu, Least squares estimator for $\alpha$-sub-fractional bridges, Stat. Papers, 59 (2018), 893–912. https://dx.doi.org/10.1007/s00362-016-0795-2 doi: 10.1007/s00362-016-0795-2
[4]	N. Kuang, H. Xie, Maximum likelihood estimator for the sub-fractional Brownian motion approximated by a random walk, Ann. Inst. Stat. Math., 67 (2015), 75–91. https://dx.doi.org/10.1007/s10463-013-0439-4 doi: 10.1007/s10463-013-0439-4
[5]	N. Kuang, H. Xie, Asymptotic behavior of weighted cubic variation of sub-fractional brownian motion, Commun. Stat. Simul. Comput., 46 (2017), 215–229. https://dx.doi.org/10.1080/03610918.2014.957849 doi: 10.1080/03610918.2014.957849
[6]	N. Kuang, On the collision local time of sub-bifractional Brownian motions, Adv. Math. (China), 48 (2019), 627–640. https://dx.doi.org/10.11845/sxjz.2018023b doi: 10.11845/sxjz.2018023b
[7]	N. Kuang, Y. Li, Berry-Esséen bounds and almost sure CLT for the quadratic variation of the sub-bifractional Brownian motion, Commun. Stat. Simul. Comput., 51 (2022), 4257–4275. https://dx.doi.org/10.1080/03610918.2020.1740265 doi: 10.1080/03610918.2020.1740265
[8]	N. Kuang, H. Xie, Derivative of self-intersection local time for the sub-bifractional Brownian motion, AIMS Math., 7(2022), 10286–10302. https://dx.doi.org/10.3934/math.2022573 doi: 10.3934/math.2022573
[9]	N. Kuang, B. Liu, Renormalized self-intersection local time for sub-bifractional Brownian motion, Filomat, 36 (2022), 4023–4040. https://doi.org/10.2298/FIL2212023K doi: 10.2298/FIL2212023K
[10]	H. Xie, N. Kuang, Least squares type estimations for discretely observed nonergodic Gaussian Ornstein-Uhlenbeck processes of the second kind, AIMS Math., 7 (2022), 1095–1114. https://dx.doi.org/10.3934/math.2022065 doi: 10.3934/math.2022065
[11]	M. Maejima, C. Tudor, Limits of bifractional Brownian noises, preprint, arXiv: 0810.4764v1.
[12]	M. A. Ouahra, S. Moussaten, A. Sghir, On limit theorems of some extensions of fractional Brownian motion and their additive functionals, Stoch. Dynam., 17 (2017), 1750022. https://dx.doi.org/10.1142/S0219493717500228 doi: 10.1142/S0219493717500228
[13]	P. Lei, D. Nualart, A decomposition of the bifractional Brownian motion and some applications, Stat. Probabil. Lett., 79 (2009), 619–624. https://doi.org/10.1016/j.spl.2008.10.009 doi: 10.1016/j.spl.2008.10.009
[14]	J. R. de Chávez, C. Tudor, A decomposition of sub-fractional Brownian motion, Math. Rep., 61 (2009), 67–74.
[15]	X. Bardina, D. Bascompte, Weak convergence towards two independent Gaussian processes from a unique Poisson process, Collect. Math., 61 (2010), 191–204.
[16]	R. Dobrushin, P. Major, Non-central limit theorems for non-linear functionals of Gaussian fields, Probabil. Theory Rel. Fields, 50 (1979), 27–52. https://doi.org/10.1007/BF00535673 doi: 10.1007/BF00535673

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