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Two absolutely summable linear filters of a stationary time series compose by convolution: $c_r={\sum }_j{b}_j{a}_{r-j}$ and ${\sum }_r|c_r|\le ({\sum }_r|a_r|)({\sum }_j|b_j|)$. Absolute convergence in the space of square-integrable random variables justifies exchanging the sums. The frequency responses multiply, giving the spectral density transformation under a linear filter twice.

A linear filter of a stationary time series with real coefficients $(a_j{)}_{j\in \mathbb{Z}}$ satisfying ${\sum }_j|a_j|<\infty$ is $Y_t={\sum }_j{a}_j{X}_{t-j}$. Its filter generating function is the Laurent series $A(z)={\sum }_j{a}_j{z}^j$, absolutely convergent on $|z|=1$, and its filter transfer function is $H_A(\lambda )=A(e^{-i\lambda })$. For a one-sided filter this is a power series; a general two-sided filter need not converge on a larger annulus.

The filter theorem states that if $X$ is second-order stationary, then $Y$ exists in mean square, is second-order stationary, and has mean and covarianceIts spectral measure of a stationary time series is $d{F}_Y(\lambda )=|H_A(\lambda ){|}^{2}d{F}_X(\lambda )$. In particular, if $X$ has a time-series spectral density in the convention ${\gamma }_X(h)={\int }_{-\pi }^{\pi }{e}^{ih\lambda }{f}_X(\lambda )d\lambda$, thenTo prove existence, the $L^2$ norm of a tail of the filter is at most $\Vert X_0{\Vert }_2$ times the sum of the omitted $|a_j|$, which tends to zero. Hence finite partial sums converge in $L^2$. Expectations and covariances pass to the limit, and the covariance double sum is absolutely convergent because $|{\gamma }_X(h)|\le {\gamma }_X(0)$. Its formula depends only on $h$, proving stationarity. Insert the spectral representation of ${\gamma }_X(h-j+k)$ and interchange sums and integration; the bound ${\sum }_{j,k}|a_j{a}_k|F_X([-\pi ,\pi ])<\infty$ justifies the interchange. The resulting factor is ${\sum }_j{a}_j{e}^{-ij\lambda }{\sum }_k{a}_k{e}^{ik\lambda }=|A(e^{-i\lambda }){|}^2$. This proves the spectral statement, including the spectral density transformation under a linear filter without assuming that the input autocovariances are absolutely summable.

For an autoregressive moving-average model, use real polynomials normalized by ${\varphi }_0={\theta }_0=1$ and first work with the reduced noise-driven representation. If $\varphi$ has no zero on the unit circle, the rational function $C(z)=\theta (z)/\varphi (z)$ has a Laurent expansion in an annulus containing that circle, with absolutely summable coefficients. Filtering the white noise with it gives a stationary solution. Its time-series spectral density isConversely, for coprime $\varphi ,\theta$, a unit-circle zero of $\varphi$ would make this ratio nonintegrable, since the numerator does not vanish there; thus there is no finite-variance stationary noise-driven solution. A stationary solution can use a bilateral filter when autoregressive roots lie inside the unit circle. The stronger causal condition, meaning dependence only on present and past noise, requires every zero of $\varphi$ strictly outside the unit disk. Common factors must be handled before applying these conditions; a cancelled unit-root factor can also allow additional deterministic stationary components, which are outside the minimal noise-driven convention.

Identifiability means that different admissible parameters cannot produce the same stationary process law. In this zero-mean Gaussian setting, its law is determined by its covariance, equivalently its spectrum. Unrestricted ARMA coefficients are not identifiable from that law: if a moving-average factor has root $r$ inside the disk, then on $|z|=1$,Reflecting the root outside and multiplying the innovation variance by $|r{|}^{-2}$ preserves the spectrum, and therefore the Gaussian law. Common polynomial factors and arbitrary overall scaling give further redundancy. A regular canonical identifiable parameterization requires constant coefficients one, coprime polynomials of their stated minimal orders, all zeros of both $\varphi$ and $\theta$ outside the closed unit disk, and ${\sigma }^2>0$.

Here is a proof of canonical identifiability of Gaussian ARMA models. Suppose two such models have equal spectra and write their innovation standard deviations as $\sigma ,\tau >0$. The ratio $R(z)=\sigma \theta (z)\tilde{\varphi }(z)/[\tau \tilde{\theta }(z)\varphi (z)]$ and its reciprocal are analytic on a neighbourhood of the closed unit disk, with modulus one on its boundary. The maximum modulus principle applied to both makes $|R|=1$ throughout the disk, so $R$ is constant. Its value at zero is the positive number $\sigma /\tau$, forcing $R=1$ and $\sigma =\tau$. Therefore $\theta \tilde{\varphi }=\tilde{\theta }\varphi$; coprimality and the normalization at zero force equality of both polynomials. This proves the claimed parameter uniqueness within that canonical class.

In this class, $D(z)=\varphi (z)/\theta (z)$ is analytic on a disk of radius greater than one. Its Taylor series coefficients are absolutely summable, soThis is an invertible time-series representation, proved by composition of the stable filters. Causality makes ${\epsilon }_t$ orthogonal to past observations; invertibility makes past noises recoverable from past observations. Their closed past linear spans agree, so this noise is the linear innovation and the one-sided expansion $X_t={\sum }_{j\ge 0}{c}_j{\epsilon }_{t-j}$ is the Wold representation. Mere stationarity without the stated canonical restrictions does not imply invertibility of the originally written noise representation.

For this one-sided canonical representation, $C(z)=\theta (z)/\varphi (z)={\sum }_{j\ge 0}{c}_j{z}^j$. Comparing coefficients gives the ARMA coefficient recursionsThus for exactly the range asked for,The characteristic rates are reciprocals of the zeros of $\varphi$. Partial fractions therefore express the tail as sums of $P_r(k)r^k$, where $|r|<1$ and the degree of $P_r$ is at most one less than the corresponding root multiplicity. Hence the coefficients decay exponentially, with possible polynomial and damped oscillatory factors. If $p=0$, they instead vanish after $q$. For a noncausal or noninvertible original specification, the Wold coefficients refer to the canonical spectral factor; using the original driving noise as the Wold innovation without checking these restrictions would be incorrect.

The autocovariance is $\gamma (h)={\sigma }^2{\sum }_{j\ge 0}{c}_j{c}_{j+|h|}$. Analyticity of $C$ on a disk of radius $R>1$ gives $|c_j|\le M{R}^{-j}$ for a slightly smaller radius than its first pole, and consequently $|\gamma (h)|\le {\sigma }^2{M}^2{R}^{-|h|}/(1-R^{-2})$. Moreover, multiply the ARMA equation at time $t$ by $X_{t-h}$ and take expectations. For $h>q$, all noises ${\epsilon }_{t-s}$ with $0\le s\le q$ are later than $X_{t-h}$ and orthogonal to it. For $h>\max (p-1,q)$ this givesThe covariance tail therefore has the same characteristic exponential rates and multiplicity patterns as the filter coefficients, subject to possible cancellation of amplitudes. Pure moving averages have an exact covariance cutoff.

For the final process, factor the polynomials:The common factor has its zero at $3/2$, so it can be cancelled as a stable filter. Explicitly, let $Y_t=(1+B/3)X_t-{\epsilon }_t$; the original equation gives $Y_t=(2/3)Y_{t-1}$. The second moments of $Y_t$ are uniformly bounded because those of $X_t$ and ${\epsilon }_t$ are. Iterating gives $Y_t=(2/3{)}^m{Y}_{t-m}$, whose $L^2$ norm tends to zero as $m\to \infty$, so $Y_t=0$ in mean square. ThusSumming the covariance series givesThe process is a causal stationary autoregressive process of order one, with alternating-sign geometric correlations, and is invertible. The displayed ARMA(2,1) description is nonminimal because of its common factor; its minimal identifiable class is ARMA(1,0).

For a discrete-time weakly stationary process, the spectral density of a stationary process, when it exists, is a nonnegative function satisfyingIf ${\sum }_h|\gamma (h)|<\infty$, thenA general second-order stationary process has a spectral measure of a stationary time series, which need not have a spectral density: a centered random constant has an atom at zero. Thus the spectral density calculation below assumes that $f_X$ exists; the corresponding measure identity holds without that extra assumption.

Absolute summability of the filter coefficients gives mean-square convergence becauseConsequently the bilateral filtered process is well defined, with constant mean $m{\sum }_s{a}_s$. Its covariance isThe sum is absolutely convergent, since $|{\gamma }_X(j)|\le {\gamma }_X(0)$ and $\sum |a_s|<\infty$. It depends only on lag, proving weak stationarity of $Y$.

Insert the spectral representation and interchange summation and integration using absolute summability and the finite total spectral mass. For real coefficients,Therefore the spectral density transformation under a linear filter isFor a bilateral sequence, $A$ is guaranteed to converge on the unit circle. Its negative powers need not converge inside the disk: for example, $a_{-j}=2^{-j}$ makes the series diverge at $z=1/4$. Only the unit-circle values are used here, so the unnecessarily broad domain printed for $A$ does not affect the proof.

A real weakly stationary process has finite second moments, a time-independent mean $\mu$, and autocovariance function ${\gamma }_k=\mathrm{Cov}(X_t,X_{t+k})$ depending only on the lag. Use the angular-frequency conventionThe time-series spectral density is nonnegative and integrates to ${\gamma }_0$. The displayed Fourier series is ordinarily justified by ${\sum }_k|{\gamma }_k|<\infty$. More generally, weak stationarity gives a spectral measure of a stationary time series, not necessarily a spectral density. If a spectral density exists without this summability, its Fourier inversion can be understood using Fejér sums, which converge in $L^1$ and at its Lebesgue points. Thus spectral density existence should not be inferred from weak stationarity alone.

A white noise process $WN(0,{\sigma }^2)$ has mean zero and $\mathrm{Cov}({\epsilon }_t,{\epsilon }_s)={\sigma }^2{1}_{\{t=s\}}$. Neither independence nor a normal distribution is needed for this second-order definition. Its autocovariance function is zero at nonzero lags, soFor a real absolutely summable filter $(c_r)$, the series defining $Y_t$ has mean-square convergence: its tails have $L^2$ norm at most $\Vert X_0{\Vert }_2{\sum }_{\text{tail}}|c_r|$. Its mean is $\mu {\sum }_r{c}_r$, andThe sum is absolutely convergent, since $|{\gamma }_X(j)|\le {\gamma }_X(0)$ by the Cauchy-Schwarz inequality. It depends only on $k$, proving that the output is a weakly stationary process. Substitute the spectral integral for ${\gamma }_X$ and interchange the sums and integral, justified by $f_X\in L^1$ and ${\sum }_{r,s}|c_r{c}_s|<\infty$. This givesBecause the coefficients are real, the two factors are complex conjugates. Hence the spectral density transformation under a linear filter isFor complex coefficients the same formula holds with the conjugate-covariance convention.

Applying this result to the order-two moving-average model yieldsEquivalently, its nonzero autocovariances are ${\gamma }_0={\sigma }^2(1+{\theta }_1^2+{\theta }_2^2)$, ${\gamma }_1={\sigma }^2{\theta }_1(1+{\theta }_2)$ and ${\gamma }_2={\sigma }^2{\theta }_2$, together with ${\gamma }_{-k}={\gamma }_k$.

For the order-one moving-average process, invert its filter by a geometric series. The required coefficients areThey are absolutely summable for $|\theta |<1$. Direct cancellation giveswith mean-square convergence. Thus the inverse filter recovers exactly the original white noise process, including its variance; it is an invertible time-series representation.

The composition of absolutely summable time-series filters is $(1-B^{12})(1-B)$, with $B$ the backshift operator. Thusand its only nonzero coefficients are $c_0=1$, $c_1=-1$, $c_{12}=-1$, $c_{13}=1$. Multiplication of the two filter gains givesEquivalently, its spectral density transformation under a linear filter is $f_Z=16{\sin }^2(\lambda /2){\sin }^2(6\lambda )f_X$.

Use the two-sided spectral density of a stationary process conventionThus white noise of variance ${\sigma }^2$ has density ${\sigma }^2/(2\pi )$.

Absolute summability makes the linear filter of a stationary time series well defined in $L^2$: ${\sum }_r\Vert a_r{X}_{t-r}{\Vert }_2\le \Vert X_0{\Vert }_2{\sum }_r|a_r|<\infty$. Its mean is $m_Y=m_X{\sum }_r{a}_r$. For its autocovariance, continuity of the $L^2$ inner product givesThe sum is absolutely convergent because $|{\gamma }_X(j)|\le {\gamma }_X(0)$. It depends only on $k$, proving weak stationarity. Inserting the spectral inversion formula and exchanging the absolutely dominated sum with the integral yieldsHere the coefficients are real, so the two factors are complex conjugates. This is the spectral density transformation under a linear filter.

Applying a second linear filter of a stationary time series givesThe composition of absolutely summable time-series filters can also be calculated directly:Indeed ${\sum }_{\ell }|c_{\ell }|\le \Vert a{\Vert }_1\Vert b{\Vert }_1$; this bound also justifies the interchange of the two $L^2$ sums. The filter gain multiplies under this composition. The sketches for the two individual filters are:

The series defining the linear filter of a stationary time series converges in $L^2$, because the norm of a tail is at most $\Vert X_0{\Vert }_2$ times the corresponding tail sum of $|a_s|$. Its mean is the original mean times ${\sum }_s{a}_s$. Its covariance is time independent and equalsUnder the summability assumption in (i),We may therefore interchange all the sums. Setting $h=k-s+u$ in the Fourier series givesThe second factor is the conjugate of the first because the coefficients are real. HenceThis spectral density transformation under a linear filter also holds whenever the original spectral measure has a density, without covariance summability: insert its integral representation into the covariance double sum, justified by absolute summability of the filter coefficients.

Ideal spectrum. Write $P_i(f)=C/(f^2+f_c^2)$, with $C>0$. The overdamped harmonic-trap power spectrum is a Lorentzian line shape:Thus its log-log plot is horizontal at low frequency and approaches slope $-2$ at high frequency, changing around $f_c$. The left panel below compares two characteristic frequencies with the same illustrative $C$; a common numerator is explicitly assumed for this sketch. Their high-frequency tails coincide and their low-frequency plateaux differ as $f_c^{-2}$.

The right panel plots all nine original data points, including the very-low-frequency excess and the highest-frequency suppression. It also shows the ideal curve inferred only from the requested intermediate range.

Graphical corner estimate. Between roughly $1$ and $100\mathrm{Hz}$ the measured power spectral density is approximately the plateau $P_0=10^{-6}{\mathrm{V}}^2\mathrm{H}{\mathrm{z}}^{-1}$. At $10^4$ and $10^5\mathrm{Hz}$,so the descending asymptote is $C/f^2$ with $C\simeq 1{\mathrm{V}}^2\mathrm{Hz}$. The graphical corner frequency of a Lorentzian spectrum is where these asymptotes intersect:Independently, the measured value $P_m(1000\mathrm{Hz})=5\times 10^{-7}{\mathrm{V}}^2\mathrm{H}{\mathrm{z}}^{-1}=P_0/2$ is exactly the half-plateau condition $P_i(f_c)=P_0/2$. Both graphical readings agree. The inferred ideal curve is approximately

Low-frequency excess and recording duration. The two lowest-frequency measurements increase approximately as $1/f$, whereas the ideal trapped oscillator tends to a constant plateau. This resembles flicker noise or slow apparatus drift superposed on the motion. The transimpedance amplifier, detector or environment could contribute; the spectrum alone does not identify the source uniquely. It would be incorrect to attribute that whole excess to the ideal Brownian oscillator.

A finite record of duration $T$ has ordinary Fourier frequency resolution of order $1/T$. Resolving the lowest tabulated frequency $0.01\mathrm{Hz}$ therefore requires at leastThis is a formal lower bound corresponding to about one cycle; a reliable noise-spectrum estimate normally needs a substantially longer record and averaging.

Sampling. The largest reported frequency is $f_{\max }=10^6\mathrm{Hz}$. The Nyquist–Shannon sampling theorem therefore gives the Nyquist lower boundIn practice choose a rate above $2\mathrm{MS}{\mathrm{s}}^{-1}$, with an analogue anti-aliasing filter suppressing unmeasured content above half the sampling rate. Exact equality cannot recover an arbitrary pure sinusoid precisely at the boundary for every phase; it is the limiting lower-bound answer rather than a comfortable practical sampling design.

High-frequency departure and amplifier design. Extrapolating the fitted $f^{-2}$ tail to $10^6\mathrm{Hz}$ predicts approximately $10^{-12}{\mathrm{V}}^2\mathrm{H}{\mathrm{z}}^{-1}$, while the measured value is $10^{-15}{\mathrm{V}}^2\mathrm{H}{\mathrm{z}}^{-1}$. The observed drop from $10^5$ to $10^6\mathrm{Hz}$ has effective logarithmic slope $-5$, much steeper than the ideal $-2$.

A detector or amplifier bandwidth limit, or intentional filtering before digitization, can explain such suppression. For a linear filter the spectral density transformation under a linear filter is $P_{\mathrm{out}}=|H{|}^2{P}_{\mathrm{in}}$, so the last point corresponds approximately to an additional relative voltage gain $\sqrt{10^{-3}}=0.0316$, if attributed entirely to the readout transfer function. One additional point does not determine a unique filter order or identify the circuit responsible.

Rolling off an amplifier's gain beyond the useful signal band reduces integrated high-frequency noise, suppresses interference, and helps prevent aliasing in the analogue-to-digital converter. Limiting high-frequency loop gain can also improve feedback stability and suppress unwanted oscillation. The cutoff must retain the frequencies needed for the measurement; excessive filtering would distort the desired fluctuation spectrum itself.