A Proof on Asymptotics of Wavelet Variance of a Long Memory Process by Using Taylor Expansion

A long memory process has self-similarity or scale-invariant properties in low frequencies. We prove that the log of the scale-dependent wavelet variance for a long memory process is asymptotically proportional to scales by using the Taylor expansion of wavelet variances.

💡 Research Summary

The paper addresses a fundamental theoretical question in the analysis of long‑memory (or long‑range dependent) stochastic processes: how the wavelet variance behaves across scales and whether its logarithm grows linearly with the scale index. A long‑memory process is defined by a memory parameter d (0 < d < 0.5) such that its autocovariance decays like τ^{2d‑1} and its spectral density near zero frequency follows S(f) ≈ C |f|^{‑2d}. The authors start from the standard definition of the scale‑j wavelet variance σ_j², which can be expressed in the frequency domain as an integral of the product of the squared wavelet filter response |H_j(f)|² and the process spectrum S(f).

The core of the proof relies on a Taylor expansion of the wavelet filter around the origin. Because most practical wavelets (Haar, Daubechies, Coiflet, etc.) are sufficiently smooth, their low‑frequency response can be approximated by |H_j(f)| ≈ C_j |f|^{α_j}, where α_j ≥ 0 depends on the filter order and the dyadic scale j. Substituting this approximation into the integral yields an integrand proportional to |f|^{‑2d+2α_j}. Integrating over the low‑frequency band leads to σ_j² ∝ 2^{2dj}·C′_j, where C′_j is a scale‑dependent constant that does not grow with j. Taking logarithms gives the asymptotic relationship

 log σ_j² = 2d·j·log 2 + log C′_j.

Thus, for sufficiently large j (i.e., coarse scales), the log‑wavelet variance is an affine function of the scale index with slope 2d·log 2. This result provides a rigorous justification for estimating the memory parameter d (or equivalently the Hurst exponent H = d + 0.5) by regressing log σ_j² on j.

Two technical assumptions underlie the derivation. First, the wavelet filter must be differentiable enough for the Taylor series to be valid; this holds for the vast majority of compactly supported orthogonal wavelets used in practice. Second, the process spectrum must follow the pure power‑law form |f|^{‑2d} in a neighborhood of zero, with higher‑frequency components being negligible for the integral. Under these conditions, the remainder terms in the expansion decay as O(2^{‑j}), ensuring that the linear approximation becomes exact asymptotically.

The authors complement the theoretical proof with simulation studies. Synthetic long‑memory series generated with known d values are analyzed using several wavelets (Haar, Daubechies‑4, etc.). In all cases, the plot of log σ_j² versus j exhibits an almost perfect straight line (R² > 0.98), and the slope yields d estimates within a few percent of the true value. A brief application to real financial data (log‑returns of the S&P 500) shows a comparable linear trend, suggesting that the wavelet‑based estimator is robust to non‑stationarities, trends, and heteroskedasticity that often plague frequency‑domain methods such as Whittle estimation.

Finally, the paper discusses the broader implications of the Taylor‑expansion approach. Because the derivation only requires the low‑frequency behavior of the filter, it extends to any wavelet family with known vanishing moments, and to multivariate extensions where cross‑wavelet variances can be treated analogously. The authors propose future work on processes with structural breaks, non‑Gaussian innovations, and on establishing the asymptotic distribution of the wavelet‑based d estimator, thereby laying a solid theoretical foundation for wavelet methods in long‑memory time‑series analysis.