[Paper Review] A central limit theorem for the sample autocorrelations of a Lévy driven continuous time moving average process

Serge Cohen, Alexander Lindner|arXiv (Cornell University)|Jun 14, 2012

Financial Risk and Volatility Modeling13 references4 citations

TL;DR

This paper establishes a central limit theorem for sample autocorrelations of Lévy-driven continuous time moving average processes, showing that asymptotic normality holds with a variance correction term not present in discrete-time models. The key contribution is that the asymptotic variance of sample autocorrelations includes an extra term dependent on the fourth moment of the Lévy process, invalidating naive discrete-time approximations and enabling asymptotically normal estimation of the Hurst exponent in fractional Lévy processes.

ABSTRACT

In this article we consider Lévy driven continuous time moving average processes observed on a lattice, which are stationary time series. We show asymptotic normality of the sample mean, the sample autocovariances and the sample autocorrelations. A comparison with the classical setting of discrete moving average time series shows that in the last case a correction term should be added to the classical Bartlett formula that yields the asymptotic variance. An application to the asymptotic normality of the estimator of the Hurst exponent of fractional Lévy processes is also deduced from these results.

Motivation & Objective

• To establish asymptotic normality of sample mean, autocovariances, and autocorrelations for Lévy-driven continuous time moving average processes.
• To identify the discrepancy between continuous-time and discrete-time moving average models in the asymptotic variance of sample autocorrelations.
• To derive a corrected version of Bartlett’s formula that includes a fourth moment correction term for continuous-time processes.
• To apply the results to the asymptotic normality of the Hurst exponent estimator for fractional Lévy processes.
• To demonstrate that the sampled process $(X_{n riangle})_{n\in\mathbb{Z}}$ cannot generally be treated as a discrete-time moving average process with i.i.d. noise.

Proposed method

• Model the process as $X_t = \mu + \int_{\mathbb{R}} f(t-s) \, dL_s$, where $L$ is a zero-mean Lévy process with finite variance and $f \in L^2(\mathbb{R})$.
• Analyze the asymptotic distribution of the sample mean $\overline{X}_{n;\Delta}$, sample autocovariance $\widehat{\gamma}_{n;\Delta}(\Delta h)$, and sample autocorrelation $\widehat{\rho}_{n;\Delta}(\Delta h)$ as $n \to \infty$.
• Use a spectral representation and functional central limit theorem techniques to derive the asymptotic variance, incorporating the fourth moment of $L$.
• Establish conditions under which $\sqrt{n}(\widehat{\rho}_{n;\Delta}(\Delta h) - \rho(h))$ converges to a normal distribution with a corrected variance formula.
• Apply the results to fractional Lévy processes by analyzing the kernel $f_d(s) = (s_+)^d - (s_-)^d$ and verifying integrability and moment conditions.
• Use the empirical autocorrelation of the increment process $Z_t = X_t - X_{t-1}$ to construct a consistent and asymptotically normal estimator of the Hurst exponent $d$.

Experimental results

Research questions

• RQ1 Does the sample autocorrelation of a Lévy-driven continuous time moving average process satisfy a central limit theorem?
• RQ2 How does the asymptotic variance of the sample autocorrelation differ from the classical Bartlett formula in discrete-time models?
• RQ3 What role does the fourth moment of the driving Lévy process play in the asymptotic distribution of sample autocorrelations?
• RQ4 Can the Hurst exponent of a fractional Lévy process be consistently and asymptotically normally estimated from discrete sampling?
• RQ5 Is the sampled process $X_{n\Delta}$ equivalent to a discrete-time moving average process with i.i.d. noise in terms of asymptotic distributional properties?

Key findings

• The sample autocorrelations of the continuous-time process are asymptotically normal, but their asymptotic variance includes an extra term proportional to the fourth cumulant of the Lévy process, which is absent in discrete-time models.
• The correction term invalidates the naive assumption that sampled continuous-time processes can be treated as discrete-time moving averages with i.i.d. noise.
• For $d < 1/4$, the estimator $\widehat{d}$ defined via the sample autocorrelation of $X_t$ is asymptotically normal with $\sqrt{n}(\widehat{d} - d) \to N(0, \sigma^2)$.
• For $d \in (0, 1/2)$, the estimator $\tilde{d} = \phi^{-1}(\rho_n^*(1))$ based on the increment process $Z_t = X_t - X_{t-1}$ is also asymptotically normal.
• The kernel functions $f_d(s)$ and their differences $\tilde{f}_d(s)$ are in $L^2(\mathbb{R}) \cap L^4(\mathbb{R})$, ensuring the validity of the central limit theorem.
• The function $g_0(u)$, which appears in the asymptotic variance formula, is bounded and in $L^2([0,1])$, confirming the applicability of the functional central limit theorem.