Confidence Bands for Local Non-Stationarity Detection ^†

Abstract

This article develops an asymptotic theory for the sample (time-averaged) variance of the locally stationary wavelet packet (LSWP) processes. Our focus is the periodogram and the sample variance, and in particular its average. Under a near-epoch-dependence (NED) condition on the periodogram, and mild moment and summability assumptions, we establish a central limit theorem and a consistent estimator for the average process variance. These results yield asymptotic stationarity bands for the average variance, providing a simple, theoretically grounded benchmark for second-order stationarity in packet-domain models.

1. Introduction

For a stationary time series, second-order properties such as the autocovariance function and spectrum are time-invariant. In many applications (financial volatility, geophysics, biomedical signals), these properties evolve over time: series can appear approximately stationary on short intervals but exhibit clear changes on longer horizons. The locally stationary literature formalizes this idea by allowing a smoothly varying local spectrum; foundational work includes Priestley’s evolutionary-spectrum formulation [1] and later treatments of evolutionary spectra [2,3], as well as subsequent general frameworks and reviews [4,5]. Wavelet-based models provide an alternative, multiscale notion of local second-order structure. The locally stationary wavelet (LSW) model of [6] represents a nonstationary process by a superposition of non-decimated wavelet atoms with slowly varying amplitudes, leading to the evolutionary wavelet spectrum (EWS) and associated (bias-corrected) wavelet periodograms. Wavelet packet extensions enrich the time–frequency tiling and can better capture oscillatory features; the locally stationary wavelet packet (LSWP) framework of [7] develops basis selection and model fitting for this class. This paper is a theory note aimed at a single inferential building block: a central limit theorem (CLT) for the average sample variance of LSWP processes. The main results are: (i) an $L_2$ near-epoch-dependence (NED) control for the time-varying periodogram and the average sample variance; (ii) a CLT for the average sample variance; and (iii) consistent estimators of the corresponding average (marginal) variance and its standard error, yielding its asymptotic stationarity bands. Section 2 introduces the relevant definitions and assumptions. Section 3 states the NED bounds and the CLT and describes the resulting stationarity bands. Proofs and technical lemmas are collected in Appendix A.

2. Notation, Definitions and Assumptions

We consider a locally stationary wavelet packet representation from [7]. For a nontrivial theory we recall the necessary concepts from that paper, and define some further tools and notation. Non-decimated wavelet packets are locally supported oscillatory functions ${\psi }_{j,i,k}(t)$, where the index $j=1,2,\dots ,{\log }_2(T)$ controls the support width and the scale of oscillation, the index $i=0,1,\dots ,2^j$ controls the oscillation frequency within a fixed scale, and k denotes translation. In what follows, $\mathcal{B}$ denotes a redundant collection of dyadic increasing partitions of the frequency interval $(0,1/2]$, and each $b\in \mathcal{B}$ is a disjoint cover of that interval. Summation over packet indices is therefore taken over $p,p^{\prime },q\in b$. A basis b can be selected from $\mathcal{B}$ as shown in [7].

Definition 1 (Non-decimated wavelet packets).

Let ${\left\{{\psi }_{p,k}(t)\right\}}_{p\in b,k\in \mathbb{Z}}$ be a non-decimated wavelet packet family (a redundant packet frame), indexed by the flattened packet index $p=(j_p,i_p)$ and translation k. Let ${\mathcal{L}}_p$ denote the support length of packet p, with ${\mathcal{L}}_p\asymp 2^{j_p}$.

Definition 2 (Cross-correlation wavelet packets).

For $p,p^{\prime }\in b$ the cross-correlation wavelet packet is defined by

$${\mathrm{\Psi }}_{p,p^{\prime }}(\tau )=\sum _k{\psi }_{p,k}{\psi }_{p^{\prime },k-\tau },$$

(1)

where ${\psi }_{p,k}$ are the non-decimated wavelet packets from Definition 1. When $p^{\prime }=p$, ${\mathrm{\Psi }}_{p,p}(\tau )={\mathrm{\Psi }}_p(\tau )$ is the corresponding autocorrelation wavelet packet. Let ${\mathcal{L}}_{p,p^{\prime }}$ denote the support length of ${\mathrm{\Psi }}_{p,p^{\prime }}$, with ${\mathcal{L}}_{p,p^{\prime }}\asymp 2^{j_p\wedge j_{p^{\prime }}}$.

Definition 3 (Locally stationary wavelet packet process).

A locally stationary wavelet packet (LSWP) process is a triangular array $\left\{X_{t;T}\right\}$ with the mean-square representation

$$X_{t;T}=\sum _{p\in b}\sum _k{\omega }_{p,k;T}{\psi }_{p,k}(t){\epsilon }_{p,k},$$

(2)

where b is a partition element of the redundant dictionary $\mathcal{B}$. The amplitudes satisfy

$$\underset{k}{\sup }|{\omega }_{p,k;T}-W_p(k/T)|\le {\displaystyle \frac{C_p}{T}},{\parallel W_p^2\parallel }_{TV}\le L_p,L_p\le K{2}^{-j_p},$$

for finite constants $C_p$, $L_p$ and K, and ${||x||}_{TV}$ is the total variation norm of x.

More details on this setup and on the packet indexing conventions are given in [7]. The assumptions below are stated at the level needed for the theory that follows. They emphasize (i) moment conditions on the innovations, (ii) summability and tail decay of packet-correlation kernels, and (iii) a multiscale growth regime ensuring uniformity over the packet indices used in the cumulative statistics.

Assumption 1 (Innovations: moments and weak dependence).

The array ${\left\{{ϵ}_{q,m}\right\}}_{q,m}$ satisfies $\mathbb{E}{ϵ}_{q,m}=0$, $\mathbb{E}{ϵ}_{q,m}^2=1$, and ${\sup }_{q,m}\mathbb{E}{|{ϵ}_{q,m}|}^{4+\delta }<\infty$ for some $\delta >0$. Moreover, for each fixed q the sequence ${\left\{{ϵ}_{q,m}\right\}}_m$ is α-mixing (see [8]), with coefficients ${\alpha }_{ϵ}(r)$ such that ${\sum }_{r\ge 1}{\alpha }_{ϵ}{(r)}^{\delta /(2+\delta )}<\infty$.

Assumption 2 (Weighted tail summability of packet correlations).

There exists $\epsilon >0$ such that, uniformly over all relevant indices $(q,p^{\prime })$,

$$\sum _{b\in \mathbb{Z}}{(1+|b|)}^{1+2\epsilon }{\mathrm{\Psi }}_{q,p^{\prime }}^2(b)<\infty ,\sum _{b\in \mathbb{Z}}{(1+|b|)}^{1+2\epsilon }{\mathrm{\Psi }}_{q,p^{\prime }}^4(b)<\infty .$$

Assumption 3 (Uniform bounds on time-varying amplitudes).

There exist finite constants $C_2,C_4$ such that ${\sup }_{q,m,T}{\omega }_{q,m;T}^2\le C_2$ and ${\sup }_{q,m,T}{\omega }_{q,m;T}^4\le C_4$. Whenever sums over packet indices are used, the corresponding scale-energy is summable so that the resulting bounds are finite.

Assumption 4 (Multiscale growth regime).

Inner sums defining the LSWP process and the local second-order structure are taken over $p\in b$, while the cumulative statistics use a finite sample truncation $1\le p\le P_T$ satisfying ${\log }_2(T)\le P_T<T$. All bounds involving cumulative statistics are uniform in $(k,T)$ and over $p\le P_T$.

Remark 1 (Local spectra and local covariance for LSWP processes).

For $z\in [0,1)$ and $k=\left[zT\right]$, and for increasing T, we have $\left[zT\right]/T\to z$. This is the rescaled time. We then define the evolutionary wavelet packet spectrum (EWPS) by

$$S_p(z)={|W_p(z)|}^2,$$

(3)

where the single index p is a shorthand for the packet pair $(j_p,i_p)$; see [7]. The marginal EWPS is ${\overline{S}}_p={\int }_0^1{S}_p(z)dz$. The time-localized covariance is

$$C(z,\tau )=\sum _{p\in b}{S}_p(z){\mathrm{\Psi }}_p(\tau ),$$

(4)

where ${\mathrm{\Psi }}_p(\tau )$ is the autocorrelation packet from Definition 2. Since ${\mathrm{\Psi }}_p(0)=1$ for all p, the time-localized variance is

$${\sigma }^2(z)=C(z,0)=\sum _{p\in b}{S}_p(z).$$

(5)

The corresponding marginal variance is

$${\sigma }^2={\int }_0^{1}C(z,0)dz=\sum _{p\in b}{\int }_0^1{S}_p(z)dz.$$

(6)

Definition 4 (Packet periodogram and bias-corrected periodogram).

Given a packet family $\left\{{\psi }_{p,k}\right\}$, define the raw packet periodogram by $I_{p,k;T}={\left|{\sum }_t{\psi }_p(t-k)X_{t;T}\right|}^2$. Let $\mathbf{A}=(A_{p,p^{\prime }})$ with $A_{p,p^{\prime }}={\sum }_{\tau }{\mathrm{\Psi }}_{p,p^{\prime }}^2(\tau )$, and let ${\mathbf{A}}^{-1}=(A_{p,p^{\prime }}^{-1})$ denote its inverse on the relevant packet set. The bias-corrected (asymptotically unbiased) packet periodogram is

$$L_{p,k;T}={\displaystyle \sum _{p^{\prime }=1}^{P_T}}{A}_{p,p^{\prime }}^{-1}{I}_{p^{\prime },k;T}.$$

(7)

The bias-corrected packet periodogram satisfies $\mathbb{E}{L}_{p,k;T}=S_p(z)+O(T^{-1})$ uniformly in k.

Definition 5 (Locally stationary sample variance).

The locally stationary sample variance (LSSV) is the sequence

$$s_{k;T}^2={\displaystyle \sum _{p=1}^{P_T}}{L}_{p,k;T}.$$

We center it by the deterministic local variance curve and define

$$U_{k;T}=s_{k;T}^2-{\sigma }^2(k/T),b_T^2=\mathbb{E}{\left({\displaystyle \sum _{k=0}^{T-1}}{U}_{k;T}\right)}^2.$$

3. ${\mathit{L}}_{\mathbf{2}}$ NED Properties of LSWP Statistics

This section studies the dependence structure of the bias-corrected packet periodogram and of the locally stationary sample variance. We use the $L_2$ near-epoch-dependence framework of [8], with the innovation array as the underlying weakly dependent process.

Definition 6.

For an array ${\left\{{\left\{{\xi }_{k;T}\right\}}_{k=-\infty }^{\infty }\right\}}_{T=1}^{\infty }$ and ${\mathcal{G}}_{k-d;T}^{k+d}=\sigma ({\xi }_{k-d;T},\dots ,{\xi }_{k+d;T})$, an integrable array ${\left\{\left\{X_{k;T}\right\}\right\}}_T$ is said to be $L_2$ near-epoch-dependent if

$${∥X_{k;T}-\mathbb{E}(X_{k;T}\mid {\mathcal{G}}_{k-d;T}^{k+d})∥}_2\le a_T^{(1)}{a}_{d;T}^{(2)}$$

(8)

uniformly in k, where $a_{d;T}^{(2)}\to 0$ as $d\to \infty$. If $a_{d;T}^{(2)}=O(d^{\lambda -ϵ})$ for some $ϵ>0$, the array is said to be $L_2$ NED of size λ (see [8] for details).

For the asymptotic arguments below we also define the following:

Definition 7.

Let

$${\kappa }_i^{ϵ}={\mathrm{cum}}_i({ϵ}_{q,m}),{\mathcal{G}}_{k-d;T}^{k+d}=\sigma ({\overline{ϵ}}_{k-d;T},\dots ,{\overline{ϵ}}_{k+d;T}),$$

where ${\overline{ϵ}}_{m;T}={({ϵ}_{1,m},\dots ,{ϵ}_{P_T,m})}^{\prime }$, and let ${\tilde{S}}_p={\sup }_z{S}_p(z)$. Define

$${\delta }_T=\underset{1\le p\le P_T}{\sup }{\displaystyle \frac{C_p}{{\tilde{S}}_p}},with{\delta }_T/T\in {\ell }_{\infty }.$$

Under Assumption 1, the innovation array is $\alpha$-mixing. The next theorem gives the required $L_2$ NED bound for the bias-corrected packet periodogram.

Theorem 1.

Let $X_{t;T}$ be specified as in Definition 3 and satisfy Assumptions 1–4. Let ${\left\{L_{p,k;T}\right\}}_k$ be the bias-corrected packet periodogram sequence for packet p from Definition 4. Then for each $p\in b$, there exist constants ${\overline{b}}_{p;T}=1$ and ${\nu }_{p,d;T}=O(d^{-1/2-\epsilon })$ such that, uniformly in k,

$${∥L_{p,k;T}-\mathbb{E}\left(L_{p,k;T}\mid {\mathcal{G}}_{k-d;T}^{k+d}\right)∥}_2\le {\overline{b}}_{p;T}{\nu }_{p,d;T}.$$

(9)

In particular, ${\left\{L_{p,k;T}\right\}}_k$ is $L_2$ NED of size $-1/2-\epsilon$.

The LSSV sequence inherits this dependence structure by summing over the finitely many packet indices included in the finite sample truncation $P_T$.

Proposition 1.

Let $X_{t;T}$ be specified as in Definition 3 and satisfy Assumptions 1–4. Consider the sequences ${\left\{L_{p,k;T}\right\}}_k$ and ${\left\{s_{k;T}^2\right\}}_k$ from Definitions 4 and 5. If, for every $p\le P_T$, ${\left\{L_{p,k;T}\right\}}_k$ satisfies an $L_2$ NED bound of size $-1/2-\epsilon$ with constants $({\overline{b}}_{p;T},{\nu }_{p,d;T})$, then ${\left\{s_{k;T}^2\right\}}_k$ also satisfies an $L_2$ NED bound of size $-1/2-\epsilon$.

Weak Convergence

We derive a central limit theorem for the centered local sample variance. Since, for all k,

$$\mathbb{E}{s}_{k;T}^2={\sigma }^2(k/T)+O(T^{-1}),$$

centering by ${\sigma }^2(k/T)$ is asymptotically equivalent to centering by the exact mean.

Theorem 2.

Let $X_{t;T}$ be specified as in Definition 3 and satisfy Assumptions 1–4. Consider ${\left\{U_{k;T}\right\}}_k$, ${\left\{s_{k;T}^2\right\}}_k$ and $b_T^2$ from Definition 5. Then

$$b_T^{-1}{\displaystyle \sum _{k=0}^{T-1}}{U}_{k;T}⇝\mathcal{N}(0,1).$$

(10)

Moreover, the $O(T^{-1})$ remainder in the centering can be omitted because

$$\sqrt{T}\underset{k}{\sup }\left|\mathbb{E}{s}_{k;T}^2-{\sigma }^2(k/T)\right|=O(T^{-1/2})\to 0.$$

To construct confidence bands to assess departures from stationarity using the average sample variance, we require (i) a consistent estimator of the average variance level and (ii) a consistent estimator of the scaling constant $b_T/T$ appearing in (10).

Corollary 1.

Under the conditions of Theorem 2,

$${\displaystyle \frac{1}{T}}{\displaystyle \sum _{k=0}^{T-1}}{s}_{k;T}^2-{\displaystyle \frac{1}{T}}{\displaystyle \sum _{k=0}^{T-1}}{\sigma }^2(k/T)\stackrel{p}{\to }0.$$

Proposition 2.

Assume the conditions of Theorem 2 and, in addition, that ${\left\{U_{k;T}\right\}}_k$ is locally short memory so that ${\sup }_T{\sum }_{h\ge 0}{\sup }_k|\mathrm{Cov}(U_{k;T},U_{k+h;T})|<\infty$. Let w be bounded with $w(0)=1$ and $w(x)=0$ for $x>1$, and let $L_T\to \infty$ with $L_T/T\to 0$. Define

$${\widehat{b}}_T^2={\displaystyle \sum _{k=0}^{T-1}}{U}_{k;T}^2+2{\displaystyle \sum _{h=1}^{L_T}}w(h/L_T){\displaystyle \sum _{k=h}^{T-1}}{U}_{k;T}{U}_{k-h;T},{\widehat{b}}_T/T={\displaystyle \frac{\sqrt{{\widehat{b}}_T^2}}{T}}.$$

If ${\lim \inf }_{T\to \infty }{b}_T^2/T>0$, then ${\widehat{b}}_T/T/(b_T/T)\stackrel{p}{\to }1$.

Let ${\overline{s}}_T^2=T^{-1}{\sum }_{k=0}^{T-1}{s}_{k;T}^2$ and let $z_{1-\alpha /2}$ denote the $(1-\alpha /2)$ quantile of the standard normal distribution. Then, for confidence level $(1-\alpha )$,

$${\overline{s}}_T^2\pm z_{1-\alpha /2}{\widehat{b}}_T/T$$

are the endpoints of an asymptotic stationarity band for the local variance sequence ${\left\{s_{k;T}^2\right\}}_{k=0}^{T-1}$.