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Chunhao Cai

Publications and source records attributed to Chunhao Cai.

At least 19 recordsLinked to original sources
arXiv2026

Spectral Simplicity and Joint Eigenvalue Densities for a Non-Gaussian Brownian Time Change

We study a Brownian time change on the unit square whose speed measure is constructed from a Dirichlet eigenfunction expansion with independent non-Gaussian coefficients. For $0<\gamma<\sqrt2$, the measure is obtained by a second-moment martingale argument. Finite coefficient translations induce coherent exponential tilts of the speed measure, and conditioning on the complementary coefficients gives positive Lebesgue densities on every finite-dimensional orbit. Unitary transport along these orbits gives a common-domain analytic family and explicit first-order cluster derivatives. A first-order splitting argument proves almost-sure simplicity, while the local eigenfunction-square identity and a Vandermonde argument give joint densities for all finite vectors of ordered positive eigenvalues.

math.PR↗
arXiv2026

Response Calculus for Spectral Simplicity and Joint Eigenvalue Densities

We develop a perturbative response calculus for spectral problems obtained by changing the speed measure of a fixed symmetric energy form in a Gaussian environment. If Cameron--Martin translation acts by $\mu_{x+f}=e^{\kappa f}\mu_x$, unitary transport identifies the varying $L^2$ spaces and produces a common-domain analytic family. For a positive eigenvalue $\Lambda$, the first-order operator is $-\kappa\Lambda$ times the compression of multiplication by $f$ to the $\Lambda$-eigenspace; its eigenvalues are the derivatives of the analytic branches issuing from $\Lambda$. A countable separation condition and finite-dimensional Gaussian disintegration then give almost-sure simplicity; a response-transversality condition and the inverse function theorem give joint densities for all finite vectors of positive ordered eigenvalues. We verify these hypotheses for every $0<\gamma<2$ in two models: Dirichlet Liouville Brownian motion on an arbitrary bounded connected planar domain, and the Liouville--Cauchy operator on the circle. In the first model the whole spectrum is almost surely simple; in the second the constants form the deterministic zero mode and the positive spectrum is almost surely simple. Transversality follows from a local eigenfunction-square identity in the Brownian case and its nonlocal jump-form analogue in the Cauchy case.

math.PR↗
arXiv2026

Truncated Signature Information for Mixed Fractional Brownian Paths

We study finite expected-signature information for mixed-fBm paths with Hurst indices above $1/4$. Up to level three, the only parameter-dependent expected features are the variance transform $q_\theta$ and the time-ordered transform $R_\theta$. We prove the scale tradeoff $2K$ level-two scales versus $K$ selected level-two/three scales, together with separation and local inverse bounds.

math.PR↗
arXiv2026

A GPH-Filtered Hannan--Rissanen Information Criterion for ARFIMA Order Selection

We propose a simple two-stage order selector for finite-order ARFIMA models. First, a preliminary log-periodogram estimate of the memory parameter is used to fractionally filter the data. Second, a Hannan--Rissanen residual construction is applied to the filtered series, and the autoregressive and moving-average orders are selected by a generalized information criterion over a growing candidate rectangle. The search bounds are allowed to satisfy \(P_n,Q_n\to\infty\), whereas the true orders remain fixed and finite. The penalty is allowed to be larger than the ordinary BIC penalty so that it dominates the error introduced by preliminary long-memory estimation and by the Hannan--Rissanen residual approximation. We prove a uniform residual-variance approximation over the growing rectangle and combine it with a population separation argument between the true finite ARMA representation and underfitted alternatives. The resulting generalized information-criterion selector is consistent.

math.ST↗
arXiv2026

Second-order PACF asymptotics and discrimination between fractional Gaussian noise and $\FARIMA(0,d,0)$

Fractional Gaussian noise and $\FARIMA(0,d,0)$ have the same long-memory pole $|\theta|^{-2d}$ and hence the same leading PACF law $\alpha(n)\sim d/n$. We show that this agreement breaks at the first non-universal order. For $0<d<1/2$, the pure fGn PACF satisfies $$ \alpha_{\fGn}(n)=\frac d n+\frac{C_{\fGn}(d)}{n^2}+o(n^{-2}), \qquad C_{\fGn}(d)<d^2, $$ The proof uses the Bingham--Inoue--Kasahara representation, a phase-coefficient expansion for fGn, and a Hankel-operator perturbation argument. Thus the fGn spectral envelope is invisible at first order but visible in second-order finite prediction, explaining why short-memory order selection can differ when fGn data are fitted by FARIMA-type models.

math.ST↗
arXiv2026

Boundary Inference for Mixed Fractional Models under High-Frequency Observation Critical LAN and Score Tests at $H=3/4$

We study boundary inference at $H=3/4$ for mixed fractional Brownian motion and mixed fractional Ornstein--Uhlenbeck models under high-frequency observation. This boundary is economically important because it separates the critical and supercritical regimes of mixed fractional dynamics. We make three contributions. First, we identify the exact critical first-order scaling and show that, after removing the explicit linear component in the $H$-score, the transformed $(\sigma,H)$ block is already non-degenerate. Second, we establish critical score central limit theorems (CLT) and derive local asymptotic normality (LAN) with fully explicit leading information constants for both models. Third, we construct boundary-calibrated one-sided score tests for detecting entry into the supercritical region $H>3/4$ and discuss feasible implementation through restricted nuisance estimation. Monte Carlo evidence shows that the feasible statistic has the correct directional power but conservative null calibration. Finally, an intraday illustration on one-minute SPY data finds no persistent evidence in favor of $H>3/4$.

math.ST↗
arXiv2026

Local Asymptotic Normality for Mixed Fractional Brownian Motion Under High-Frequency Observation

In this paper we will consider the LAN property for both the Hurst parameter $H>3/4$ and the variance of the fractional Brownian motion plus an independent standard Brownian motion (called mixed fractional Brownian motion) with high-frequency observation. We will first remove the $H$-score linear term and orthogonalize the remainder through two non-diagonal transformations, then we can construct the CLT for the quadratic form base on $\| \cdot \|_{\mathrm{op}}/\|\cdot\|_F\to0$. At last we obtain a diagonal Gaussian LAN expansion with an explicit information matrix. Beyond the case of $H>3/4$, we also present that the $\| \cdot \|_{\mathrm{op}}/\|\cdot\|_F\to0$ method is also useful for the case of $H<3/4$ and the proof will be concise compared with the Whittle translation method. We consider that this method can be applied to this type of problem, including the fractional Ornstein-Uhlenbeck model and mixed fractional O-U process.

math.PR↗
arXiv2025

Local Asymptotic Normality for Mixed Fractional Brownian Motion with $0<H<3/4$

This paper establishes the Local Asymptotic Normality (LAN) property for the mixed fractional Brownian motion under high-frequency observations with Hurst index $H \in (0, 3/4)$. The simultaneous estimation of the volatility and the Hurst index encounters a degeneracy problem in the Fisher information matrix.

math.ST↗
arXiv2025

Drift estimation for a partially observed mixed fractional Ornstein--Uhlenbeck process

We consider estimation of the drift parameter $\vartheta>0$ in a \emph{partially observed} Ornstein--Uhlenbeck type model driven by a mixed fractional Brownian noise. Our framework extends the partially observed model of \cite{BrousteKleptsyna2010} to the \emph{mixed} case. We construct the canonical innovation representation, derive the associated Kalman filter and Riccati equations, and analyse the asymptotic behaviour of the filtering error covariance. Within the Ibragimov--Khasminskii LAN framework we prove that the MLE of $\vartheta$, based on continuous observation of the partially observed system on $[0,T]$, is consistent and asymptotically normal with rate $\sqrt{T}$ and the Fisher Information is the same as in \cite{BrousteKleptsyna2010} or the standard Brownian motion case.

math.ST↗
arXiv2025

A mixed fractional CIR model: positivity and an implicit Euler scheme

We consider a Cox--Ingersoll--Ross (CIR) type short rate model driven by a mixed fractional Brownian motion. Let $M=B+B^H$ be a one-dimensional mixed fractional Brownian motion with Hurst index $H>1/2$, and let $\mathbf{M}=(M,\mathbb{M}^{\mathrm{It\hat{o}}})$ denote its canonical It\^o rough path lift. We study the rough differential equation \begin{equation}\label{eqn1} \dd r_t = k(\theta-r_t)\,\dd t + \sigma\sqrt{r_t}\,\dd\mathbf{M}_t,\qquad r_0>0, \end{equation} and prove that, under the Feller condition $2k\theta>\sigma^2$, the unique rough path solution is almost surely strictly positive for all times. The proof relies on an It\^o type formula for rough paths, together with refined pathwise estimates for the mixed fractional Brownian motion, including L\'evy's modulus of continuity for the Brownian part and a law of the iterated logarithm for the fractional component. As a consequence, the positivity property of the classical CIR model extends to this non-Markovian rough path setting. We also establish the convergence of an implicit Euler scheme for the associated singular equation obtained by a square-root transformation.

math.PR↗
arXiv2025

Parameter estimation for fractional autoregressive process with seasonal structure

This paper introduces a new kind of seasonal fractional autoregressive process (SFAR) driven by fractional Gaussian noise (fGn). The new model includes a standard seasonal AR model and fGn. {The estimation of the parameters of this new model has to solve two problems: nonstationarity from the seasonal structure and long memory from fGn. We innovatively solve these by getting a stationary subsequence, making a stationary additive sequence, and then obtaining their spectral density. Then, we use one-step procedure for Generalized Least Squares Estimator (GLSE) and the Geweke Porter-Hudak (GPH) method to get better results. We prove that both the initial and one-step estimators are consistent and asymptotically normal. Finally, we use Monte Carlo simulations with finite-sized samples to demonstrate the performance of these estimators. Moreover, through empirical analysis, it is shown that the SFAR model can simulate some real world phenomena better than general models.

stat.AP↗
arXiv2025

Parameter estimation for fractional autoregressive process with periodic structure

This paper introduces a new periodic fractional autoregressive process (PFAR) driven by fractional Gaussian noise (fGn) to model time series of precipitation evapotranspiration. Compared with the similar model in [\emph{Water Resources Research}, \textbf{20} (1984) 1898--1908], the new model incorporates a periodic structure via specialized varying coefficients and captures long memory and rough voltality through fGn for $0<H<1$, rather than via fractional differencing. In this work, Generalized Least Squares Estimation (GLSE) and the GPH method are employed to construct an initial estimator for the joint estimation of model parameters. A One-Step procedure is then used to obtain a more asymptotically efficient estimator. The paper proves that both estimators are consistent and asymptotically normal, and their performance is demonstrated via Monte Carlo simulations with finite-size samples. Simulation studies suggest that, while both estimation methods can accurately estimate the model parameters, the One-Step estimator outperforms the initial estimator.

math.ST↗
arXiv2024

RNN-BSDE method for high-dimensional fractional backward stochastic differential equations with Wick-It\^o integrals

Fractional Brownian motions(fBMs) are not semimartingales so the classical theory of It\^o integral can't apply to fBMs. Wick integration as one of the applications of Malliavin calculus to stochastic analysis is a fine definition for fBMs. We consider the fractional forward backward stochastic differential equations(fFBSDEs) driven by a fBM that have the Hurst parameter in (1/2,1) where $\int_{0}^{t} f_s \, dB_s^H$ is in the sense of a Wick integral, and relate our fFBSDEs to the system of partial differential equations by using an analogue of the It\^o formula for Wick integrals. And we develop a deep learning algorithm referred to as the RNN-BSDE method based on recurrent neural networks which is exactly designed for solving high-dimensional fractional BSDEs and their corresponding partial differential equations.

math.PR↗
arXiv2024

Scaling limit of heavy tailed nearly unstable cumulative INAR($\infty$) processes and rough fractional diffusions

In this paper, we investigate the scaling limit of heavy-tailed nearly unstable cumulative INAR($\infty$) processes. These processes exhibit a power-law tail of the form $n^{-(1+\alpha)}$ for $\alpha \in (\frac{1}{2}, 1)$, and the $\ell^1$ norm of the kernel vector converges to 1. We demonstrate that the discrete-time scaling limit retains a long-memory property and can be viewed as an integrated fractional Cox-Ingersoll-Ross process. Moreover, we present an efficient method for simulating the fractional Cox-Ingersoll-Ross process. The simulation and Goodness-of-Fit Test code are available at https://github.com/gagawjbytw/INAR-rough-Heston.

math.PR↗
arXiv2022

Uncertainty-based Network for Few-shot Image Classification

The transductive inference is an effective technique in the few-shot learning task, where query sets update prototypes to improve themselves. However, these methods optimize the model by considering only the classification scores of the query instances as confidence while ignoring the uncertainty of these classification scores. In this paper, we propose a novel method called Uncertainty-Based Network, which models the uncertainty of classification results with the help of mutual information. Specifically, we first data augment and classify the query instance and calculate the mutual information of these classification scores. Then, mutual information is used as uncertainty to assign weights to classification scores, and the iterative update strategy based on classification scores and uncertainties assigns the optimal weights to query instances in prototype optimization. Extensive results on four benchmarks show that Uncertainty-Based Network achieves comparable performance in classification accuracy compared to state-of-the-art method.

cs.CV↗
arXiv2021

Learning Class-level Prototypes for Few-shot Learning

Few-shot learning aims to recognize new categories using very few labeled samples. Although few-shot learning has witnessed promising development in recent years, most existing methods adopt an average operation to calculate prototypes, thus limited by the outlier samples. In this work, we propose a simple yet effective framework for few-shot classification, which can learn to generate preferable prototypes from few support data, with the help of an episodic prototype generator module. The generated prototype is meant to be close to a certain \textit{\targetproto{}} and is less influenced by outlier samples. Extensive experiments demonstrate the effectiveness of this module, and our approach gets a significant raise over baseline models, and get a competitive result compared to previous methods on \textit{mini}ImageNet, \textit{tiered}ImageNet, and cross-domain (\textit{mini}ImageNet $\rightarrow$ CUB-200-2011) datasets.

cs.CV↗