arXiv subjects

Nan Rao

Publications and source records attributed to Nan Rao.

4 recordsLinked to original sources

arXiv2023

Fractional Brownian Motion: Local Modulus of Continuity with Refined Almost Sure Upper Bound and First Exit Time from One-sided Barrier

Based on an optimal rate wavelet series representation, we derive a local modulus of continuity result with a refined almost sure upper bound for fractional Brownian motion. \sloppy The obtained upper bound of the small fractional Brownian increments is of order $\mathcal O_{a.s.}\big(|h|^H\sqrt{\log\log |h|^{-1}}\big)$ as $|h|\to0$, and an upper bound of its $p$th moment is provided, for any $p>0$. This result fills the gap of the law of iterated logarithm for fractional Brownian motion, where the moments' information of the random multiplier in the upper bound is missing. With this enhanced upper bound and some new results on the distribution of the maximum of fractional Brownian motion, we obtain a new and refined asymptotic estimate of the upper-tail probability for a fractional Brownian motion to first exit from a positive-valued barrier over time $T$, as $T\to+\infty$.

math.PR↗

arXiv2020

Cluster Analysis on Locally Asymptotically Self-similar Processes with Known Number of Clusters

We conduct cluster analysis on a class of locally asymptotically self-similar stochastic processes, which includes multifractional Brownian motion as a representative. When the true number of clusters is supposed to be known, a new covariance-based dissimilarity measure is introduced, from which we obtain the approximately asymptotically consistent clustering algorithms. In simulation studies, clustering data sampled from multifractional Brownian motions with distinct functional Hurst parameters illustrates the approximated asymptotic consistency of the proposed algorithms. Clustering global financial markets' equity indexes returns and sovereign CDS spreads provides a successful real world application.

stat.ML↗

arXiv2019

Some Developments in Clustering Analysis on Stochastic Processes

We review some developments on clustering stochastic processes and come with the conclusion that asymptotically consistent clustering algorithms can be obtained when the processes are ergodic and the dissimilarity measure satisfies the triangle inequality. Examples are provided when the processes are distribution ergodic, covariance ergodic and locally asymptotically self-similar, respectively.

stat.ML↗

arXiv2019

Covariance-based Dissimilarity Measures Applied to Clustering Wide-sense Stationary Ergodic Processes

We introduce a new unsupervised learning problem: clustering wide-sense stationary ergodic stochastic processes. A covariance-based dissimilarity measure together with asymptotically consistent algorithms is designed for clustering offline and online datasets, respectively. We also suggest a formal criterion on the efficiency of dissimilarity measures, and discuss of some approach to improve the efficiency of our clustering algorithms, when they are applied to cluster particular type of processes, such as self-similar processes with wide-sense stationary ergodic increments. Clustering synthetic data and real-world data are provided as examples of applications.

stat.ML↗