Note on asymptotic behavior of spatial sign autocovariance matrices

• Department of Accounting and Finance, University of Turku, Turku, Finland [mtvout@utu.it]

• Department of Mathematics and Systems Analysis, Aalto University, Espoo, Finland [pauliina.ilmonen@aalto.fi]

• Department of Information and Service Management, Aalto University, Espoo, Finland [lauri.viitasaari@aalto.fi]

• Department of Mathematics and Statistics, University of Turku, Turku, Finland [niko.lietzen@utu.fi]

Keywords: Spatial signs – Autocovariance matrix – Gaussian subordinated process

1 Introduction

In the communication (see Voutilainen et al. [2023]), we examine the asymptotic properties of the spatial sign autocovariance matrix estimator of a subordinated Gaussian process with a known location parameter. We note that Gaussian subordinated processes provide a large class of processes covering, for example, stationary processes with arbitrary marginal distributions (Viitasaari and Ilmonen [2020]). The spatial sign covariance matrix, a term coined in Visuri et al. [2000], is a cornerstone in modern multivariate robust statistics. The classical definition can straightforwardly be extended to take temporal dependencies into account. To the best of our knowledge, the asymptotic properties of the spatial sign autocovariance matrix have not previously been considered under non-trivial temporal dependency structures.

2 Description of the Main Results

Let $X={(X_t)}_{t\in \mathbb{N}}$ be a $d$-variate stationary Gaussian process and let $f_k:{ℝ}^d\to ℝ$, where $k\in \{1,\mathrm{\dots },n\}$, be measurable functions. We call the $n$-variate process $Z={(Z_t)}_{t\in \mathbb{N}}$, where $Z_t={[Z_t^{(1)},\mathrm{\dots },Z_t^{(n)}]}^{\top }$ with $Z_t^{(k)}=f_k(X_t)$ a Gaussian subordinated process. By the assumption that the location of $Z$ is known, the spatial sign autocovariance matrix with lag $\tau$ and the corresponding estimator are defined as

$$\gamma (\tau )=𝔼\left[\frac{Z_t}{\parallel Z_t\parallel }\frac{Z_{t+\tau }^{\top }}{\parallel Z_{t+\tau }\parallel }\right]\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}{\widehat{\gamma }}_T(\tau )=\frac{1}{T}\sum _{t=1}^{T-\tau }\frac{Z_t}{\parallel Z_t\parallel }\frac{Z_{t+\tau }^{\top }}{\parallel Z_{t+\tau }\parallel },$$

respectively. In addition, let $r_X:\mathbb{Z}\to ℝ$ be the autocovariance matrix function of $X$. Then, if $r_X(t)\to 0$ as $t\to \mathrm{\infty }$, we obtain the weak consistency

$$\underset{T\to \mathrm{\infty }}{lim}\parallel {\widehat{\gamma }}_T(\tau )-\gamma (\tau )\parallel \stackrel{\mathbb{P}}{=}0,$$

where $\parallel \cdot \parallel$ is an arbitrary matrix norm.

Furthermore, the convergence of $\sqrt{T}\mathrm{vec}({\widehat{\gamma }}_T(\tau )-\gamma (\tau ))$ is dictated by the joint convergence of

$$\frac{\sqrt{T}}{T-\tau }\sum _{t=1}^{T-\tau }(g_{i,j}({\tilde{X}}_t)-𝔼g_{i,j}({\tilde{X}}_t)),$$ (1)

where $i,j\in \{1,\mathrm{\dots },n\}$, ${\tilde{X}}_t={[X_t^{\top },X_{t+\tau }^{\top }]}^{\top }$ and $g_{i,j}:{ℝ}^{2d}\to ℝ$ are bounded functions expressible in terms of the functions $f_k$. The convergence of (1) depends on two factors. Namely, the rate of $r_X(t)\to 0$ and the Hermite ranks of $g_{i,j}({\tilde{X}}_1)$, which is the smallest non-zero degree present in the corresponding generalized Wiener-Hermite polynomial expansion. If the above-mentioned Hermite ranks are at least $q$ and

then as an application of the multivariate Breuer-Major theorem, we obtain a jointly normal limit for (1). It is worth to mention that due to the instability of Hermite ranks, this approach covers also long-range dependent processes $Z$.