OscillatoryProcess

The Envelope of Oscillatory Harmonizable Processes: A Theoretical Exposition

The class of oscillatory harmonizable processes represents a significant extension of Priestley's oscillatory stationary processes, generalizing their spectral structure to accommodate non-stationary characteristics while retaining harmonizability. This report rigorously examines the mathematical foundations of these processes, focusing on the non-uniqueness problem associated with their envelope representations and the conditions under which uniqueness can be established. By synthesizing operator-theoretic methods, spectral representations, and covariance analysis, we delineate the structural prerequisites for resolving the envelope ambiguity inherent in earlier formulations.

Foundations of Oscillatory Harmonizable Processes

Spectral Representations and Covariance Structures

A stochastic process $X(t)$ is termed weakly harmonizable if it admits a spectral representation:

$$ X(t) = \int_{\mathbb{R}} e^{it\lambda} , dZ(\lambda), $$

where $Z(\cdot)$ is a stochastic measure with orthogonal increments, and its covariance $r(s, t)$ is expressible as a double integral over a spectral bimeasure $F(\cdot, \cdot)$:

$$ r(s, t) = \iint_{\mathbb{R}^2} e^{i(s\lambda - t\lambda')} , dF(\lambda, \lambda') [6][14]. $$

This framework generalizes stationary processes, where $F$ concentrates on the diagonal $\lambda = \lambda'$. Priestley's oscillatory processes further modulate such representations via a time-dependent amplitude function $A(t, \lambda)$, yielding:

$$ X(t) = \int_{\mathbb{R}} A(t, \lambda)e^{it\lambda} , dZ(\lambda) [8][12]. $$

The covariance of such processes becomes:

$$ r(s, t) = \int_{\mathbb{R}} A(s, \lambda)A(t, \lambda)e^{i\lambda(s - t)} , dG(\lambda), $$

where $G$ is the spectral measure of the stationary component [12][14].

Extension to Oscillatory Harmonizable Processes

The oscillatory harmonizable class relaxes the stationarity assumption, allowing $Z(\cdot)$ to exhibit non-orthogonal increments. Formally, an oscillatory weakly harmonizable process has covariance:

$$ r(s, t) = \iint_{\mathbb{R}^2} A(s, \lambda)A(t, \lambda')e^{i(s\lambda - t\lambda')} , dF(\lambda, \lambda'), $$

where $F$ is a bimeasure of bounded Fréchet variation [14][6]. The amplitude function $A(t,\lambda)$ is defined via a signed measure $H(t,\cdot)$:

$$ A(t, \lambda) = \int_{\mathbb{R}} e^{itx} , dH(t, x), $$

ensuring a maximum at $\lambda=0$ independent of $t$ [14][9]. This construction subsumes Priestley's oscillatory processes when $F$ concentrates on the diagonal.

The Envelope Non-Uniqueness Problem

Hilbert Transforms and Quadrature Processes

For a weakly harmonizable process $X(t)$, the Hilbert transform $\tilde{X}(t)$ is defined as:

$$ \tilde{X}(t) = \text{Im } 2 \int_{\mathbb{R}} e^{it\lambda} , dZ(\lambda) [16][18]. $$

The envelope $R(t)$, given by:

$$ R(t) = \sqrt{X^2(t) + \tilde{X}^2(t)}, $$

serves as a time-dependent amplitude measure. However, for oscillatory harmonizable processes $Y(t) = \int_{\mathbb{R}} A(t, \lambda)e^{it\lambda} , dZ(\lambda)$, the corresponding quadrature process $\tilde{Y}(t)$ and envelope inherit non-uniqueness from the non-stationary spectral structure [18][11].

Hasofer's Non-Uniqueness Example

Hasofer demonstrated that distinct amplitude functions $A_1(t, \lambda)$ and $A_2(t, \lambda)$ can yield identical covariance structures $r(s, t)$, leading to multiple envelope representations for the same process [14][4]. This ambiguity stems from the over-parametrization inherent in the spectral bimeasure $F(\lambda, \lambda')$ when decoupled from diagonal constraints [6][14].

Uniqueness Under Time-Varying Filters

Operator-Theoretic Formulation

Consider a time-varying filter $h(t_0, t, u)$ initiated at $t_0$, whose Fourier transform $A(t_0, t, \lambda)$ satisfies:

$$ A(t_0, t, \lambda) = \int_{\mathbb{R}} e^{i\lambda u}h(t_0, t, u) , du, $$

with $A(t_0, t, \lambda)$ maximized at $\lambda=0$. Filtering a weakly harmonizable process $X(t)$ through $h$ produces an oscillatory harmonizable process:

$$ Y(t_0, t) = \int_{\mathbb{R}} h(t_0, t, u)X(u) , du [10][17]. $$

As $t_0 \to -\infty$, if $h(t_0, t, u)$ converges to a time-invariant kernel $h(t-u)$, then $Y(t) = \lim_{t_0 \to -\infty} Y(t_0, t)$ becomes:

$$ Y(t) = \int_{\mathbb{R}} h(t-u)X(u) , du, $$

retaining weak harmonizability [6][14].

Invertibility and Uniqueness

Assuming the filter $h$ is invertible, the mapping $Y = KX$ implies $X = K^{-1}Y$. For two representations $Y = KX$ and $Y = K'X'$, compatibility requires:

$$ X' = K'^{-1}KX, $$

ensuring the Hilbert transforms $\tilde{X}'$ and $\tilde{X}$ satisfy:

$$ \tilde{X}' = K'^{-1}K\tilde{X} [16][18]. $$

Consequently, the quadrature processes $\tilde{Y}$ and $\tilde{Y}'$ coincide, enforcing envelope uniqueness [14][11].

Theorem: Uniqueness of the Envelope

Theorem. Let $X(t)$ be a weakly harmonizable process with spectral measure $Z(\cdot)$ having no origin jumps. If a time-varying filter $h(t_0, t, u)$ converges to a time-invariant $h(t-u)$ as $t_0 \to -\infty$, and the resulting oscillatory harmonizable process $Y(t) = \int h(t-u)X(u) , du$ is well-defined, then the envelope $R(t) = \sqrt{Y^2(t) + \tilde{Y}^2(t)}$ is unique [14][11].

Proof. The spectral representations of $Y(t)$ under distinct filters $h$ and $h'$ must satisfy:

$$ \iint_{\mathbb{R}^2} \left( T_1(s, \lambda)T_1(t, \lambda') - T_2(s, \lambda)T_2(t, \lambda') \right) e^{i(s\lambda - t\lambda')} , dF(\lambda, \lambda') = 0, $$

for all $s, t$, where $T_1$ and $T_2$ correspond to the operators $K$ and $K'$. This forces $T_1 = T_2$, ensuring $\tilde{Y}$ is unique [14][6].

Conclusion

The non-uniqueness of envelope representations for oscillatory harmonizable processes arises from the flexibility in spectral bimeasure parametrization. By constraining amplitude functions through time-invariant filters in the asymptotic limit, the Hilbert transform-induced quadrature process becomes uniquely determined. This theoretical resolution underscores the centrality of operator commutativity and spectral convergence in stabilizing envelope definitions, providing a rigorous foundation for further analytical explorations within the harmonizable paradigm.