Random Wavelet Series

• Random wavelet series are wavelet-based representations with stochastic coefficients that capture multifractal properties and local regularity.
• They utilize wavelet localization to connect scale-dependent coefficient statistics with function boundedness, continuity, and Besov space memberships.
• Various models—including independent, process-induced, and heavy-tailed frameworks—provide distinct insights for multifractal spectra and simulation accuracy.

Random wavelet series are random functions, fields, or distributions represented through wavelet expansions whose coefficients are random, randomized, or induced by an underlying stochastic process. In the narrow sense used in multifractal analysis, a Random Wavelet Series is a one-periodic random function with scale-dependent random wavelet coefficients (Céline et al., 1 Oct 2025). In a broader probabilistic sense, the term also covers deterministic wavelet series randomized by i.i.d. multipliers, stable and multistable wavelet expansions, and dependent wavelet coefficient arrays arising from long-memory processes (Esser et al., 2023, Medina et al., 2019, Clausel et al., 2010). Across these formulations, wavelet localization makes coefficient moduli unusually informative for regularity, so boundedness, continuity, Besov membership, Hölder behavior, and multifractal spectra can often be read directly from scale-by-scale coefficient statistics (Esser et al., 2023, Horst et al., 2024).

1. Foundational constructions and coefficient models

A standard deterministic wavelet expansion on $\mathbb T^d$ or $\mathbb R^d$ is written in the non-$L^2$-normalized form

$$f=\sum_j f_j,\qquad f_j=\sum_{i,k} c^i_{j,k}\,\psi^{(i)}_{j,k},$$

with the scale envelope

$\omega_j=\sup_{i,k}|c^i_{j,k}|.$

This parameter $\omega_j$ is central in the regularity analysis of randomized wavelet expansions (Esser et al., 2023). In the one-periodic independent-coefficient model, one instead writes

$$f=\sum_{j\in\mathbb N_0}\sum_{k=0}^{2^j-1} c_{j,k}\psi_{j,k},$$

and studies the scale law of

$X_{j,k}=\frac{-\log_2|c_{j,k}|}{j},$

whose distribution prescribes the abundance of coefficients of a given size at each scale (Céline et al., 1 Oct 2025).

On $\mathbb R^d$, a widely used Besov-prior model takes coefficients of the form

$$a_{j,t,m} = 2^{j\alpha}(j+1)^\theta \Big(1+\frac{\|m\|_\infty}{2^j}\Big)^\gamma \,\xi_{j,t,m} \quad\text{for }j\ge 1,$$

with a separate coarse-scale exponent at $\mathbb R^d$0, and with i.i.d. template variables $\mathbb R^d$1 (Horst et al., 2024). Sparse variants replace $\mathbb R^d$2 by $\mathbb R^d$3, where $\mathbb R^d$4 are Bernoulli variables with scale-location dependent success probabilities (Horst et al., 2024).

Three recurrent paradigms organize the subject.

Paradigm	Coefficient mechanism	Representative result
Randomization of a fixed series	$\mathbb R^d$5	Unbounded multipliers can destroy continuity and local boundedness (Esser et al., 2023)
Independent random coefficients	$\mathbb R^d$6 i.i.d. across positions at fixed scale	Exact $\mathbb R^d$7-spectrum from wavelet coefficient statistics (Céline et al., 1 Oct 2025)
Process-induced coefficient field	$\mathbb R^d$8 obtained by wavelet transform of a stochastic process	Limits in Wiener chaos and generalized self-similar processes (Clausel et al., 2010)

This variety matters because the phrase “random wavelet series” is not tied to a single probabilistic architecture. Some theories emphasize synthesis from prescribed random coefficients, whereas others analyze wavelet coefficients generated by an ambient random process. A plausible implication is that the most robust structural distinction is not constructive versus analytic, but whether regularity is controlled directly by coefficient moduli or by more global interactions between coefficients.

2. Regularity, continuity, and Besov behavior

For deterministic wavelet series, a sharp continuity threshold is expressed through $\mathbb R^d$9. If

$L^2$0

then the wavelet series converges normally,

$L^2$1

hence $L^2$2 converges uniformly to a bounded function, and if the wavelets are continuous then $L^2$3 is continuous (Esser et al., 2023). Conversely, if $L^2$4 is nonnegative and $L^2$5, there exists a wavelet series with $L^2$6 for all $L^2$7 that is nowhere locally bounded (Esser et al., 2023). In this deterministic framework, $L^2$8-summability of $L^2$9 is therefore the exact threshold for continuity.

Randomization changes this picture in a way that is opposite to the familiar Fourier phenomenon. If

$$f=\sum_j f_j,\qquad f_j=\sum_{i,k} c^i_{j,k}\,\psi^{(i)}_{j,k},$$0

with i.i.d. multipliers $$f=\sum_j f_j,\qquad f_j=\sum_{i,k} c^i_{j,k}\,\psi^{(i)}_{j,k},$$1 having unbounded support in the literal sense

$$f=\sum_j f_j,\qquad f_j=\sum_{i,k} c^i_{j,k}\,\psi^{(i)}_{j,k},$$2

then unbounded randomization can destroy boundedness and continuity “very violently” (Esser et al., 2023). The paper proves that for almost every $$f=\sum_j f_j,\qquad f_j=\sum_{i,k} c^i_{j,k}\,\psi^{(i)}_{j,k},$$3 in the prevalent sense, the randomized wavelet series associated with $$f=\sum_j f_j,\qquad f_j=\sum_{i,k} c^i_{j,k}\,\psi^{(i)}_{j,k},$$4 by unbounded i.i.d. multipliers is almost surely nowhere locally bounded (Esser et al., 2023). The decisive distinction is bounded versus unbounded multipliers, not Gaussian versus Rademacher. Bounded multipliers preserve the coefficient-modulus criteria behind $$f=\sum_j f_j,\qquad f_j=\sum_{i,k} c^i_{j,k}\,\psi^{(i)}_{j,k},$$5, Hölder, Sobolev, and Besov regularity, whereas unbounded multipliers create arbitrarily large local coefficients that wavelet localization turns into genuine local singularities (Esser et al., 2023).

Gaussian randomization yields a finer threshold. Since there are $$f=\sum_j f_j,\qquad f_j=\sum_{i,k} c^i_{j,k}\,\psi^{(i)}_{j,k},$$6 coefficients at scale $$f=\sum_j f_j,\qquad f_j=\sum_{i,k} c^i_{j,k}\,\psi^{(i)}_{j,k},$$7, one has almost surely

$$f=\sum_j f_j,\qquad f_j=\sum_{i,k} c^i_{j,k}\,\psi^{(i)}_{j,k},$$8

so

$$f=\sum_j f_j,\qquad f_j=\sum_{i,k} c^i_{j,k}\,\psi^{(i)}_{j,k},$$9

is sufficient for continuity of the Gaussian-randomized series (Esser et al., 2023). For $\omega_j=\sup_{i,k}|c^i_{j,k}|.$0, if the multipliers have exponential tail of order $\omega_j=\sup_{i,k}|c^i_{j,k}|.$1, then almost surely

$\omega_j=\sup_{i,k}|c^i_{j,k}|.$2

so the Hölder exponent is preserved but the modulus acquires a logarithmic correction (Esser et al., 2023). In the language of the uniform Hölder exponent,

$\omega_j=\sup_{i,k}|c^i_{j,k}|.$3

This rules out any general smoothing effect of unbounded randomization in wavelet coordinates (Esser et al., 2023).

Besov regularity admits a complementary coefficient-envelope characterization. For the non-sparse Besov prior on $\omega_j=\sup_{i,k}|c^i_{j,k}|.$4, almost sure membership in $\omega_j=\sup_{i,k}|c^i_{j,k}|.$5 is governed by Property A: $\omega_j=\sup_{i,k}|c^i_{j,k}|.$6 or

$\omega_j=\sup_{i,k}|c^i_{j,k}|.$7

with endpoint modifications for $\omega_j=\sup_{i,k}|c^i_{j,k}|.$8 and $\omega_j=\sup_{i,k}|c^i_{j,k}|.$9 (Horst et al., 2024). Under mild moment assumptions on the template variable $\omega_j$0, this criterion is sufficient; under the nondegeneracy condition $\omega_j$1, it is also necessary at the coefficient level, and essentially necessary at the function level once the wavelet characterization assumptions are imposed (Horst et al., 2024). The same deterministic threshold also controls finiteness of moments and, under stronger assumptions, finiteness of exponential moments of the Besov norm (Horst et al., 2024). This suggests that in many random wavelet models the probabilistic input primarily affects integrability requirements, while the location of the regularity threshold remains deterministic.

3. Multifractal analysis and $\omega_j$2-spectrum theory

Classical multifractal analysis of wavelet series is often formulated through Hölder exponents and iso-Hölder sets. For random wavelet series built from Gibbs measures, with coefficients

$\omega_j$3

and independent random perturbations of these coefficients, the ordinary singularity spectrum is inherited from the Gibbs measure through an affine change of variables (Jin, 2010). The main geometric refinement concerns the graph and range restricted to iso-Hölder sets. If

$\omega_j$4

then almost surely, for $\omega_j$5,

$\omega_j$6

for the graph and range singularity spectra (Jin, 2010). The proof uses Gibbs measures on subshifts that avoid the zero set of the mother wavelet, a technical device needed to make the potential-theoretic lower bounds work (Jin, 2010).

A major extension replaces Hölder exponents by $\omega_j$7-exponents, which remain meaningful for functions that are only locally in $\omega_j$8. For $\omega_j$9, the $$f=\sum_{j\in\mathbb N_0}\sum_{k=0}^{2^j-1} c_{j,k}\psi_{j,k},$$0-exponent is

$$f=\sum_{j\in\mathbb N_0}\sum_{k=0}^{2^j-1} c_{j,k}\psi_{j,k},$$1

and the corresponding $$f=\sum_{j\in\mathbb N_0}\sum_{k=0}^{2^j-1} c_{j,k}\psi_{j,k},$$2-spectrum is

$$f=\sum_{j\in\mathbb N_0}\sum_{k=0}^{2^j-1} c_{j,k}\psi_{j,k},$$3

(Céline et al., 1 Oct 2025). The key control is through the distribution of wavelet coefficients across scales, encoded by the wavelet density $$f=\sum_{j\in\mathbb N_0}\sum_{k=0}^{2^j-1} c_{j,k}\psi_{j,k},$$4 or profile $$f=\sum_{j\in\mathbb N_0}\sum_{k=0}^{2^j-1} c_{j,k}\psi_{j,k},$$5. If $$f=\sum_{j\in\mathbb N_0}\sum_{k=0}^{2^j-1} c_{j,k}\psi_{j,k},$$6, then for every $$f=\sum_{j\in\mathbb N_0}\sum_{k=0}^{2^j-1} c_{j,k}\psi_{j,k},$$7,

$$f=\sum_{j\in\mathbb N_0}\sum_{k=0}^{2^j-1} c_{j,k}\psi_{j,k},$$8

(Céline et al., 1 Oct 2025). For Random Wavelet Series in the narrow independent-coefficient sense, this upper bound is sharp almost surely: for all $$f=\sum_{j\in\mathbb N_0}\sum_{k=0}^{2^j-1} c_{j,k}\psi_{j,k},$$9, the support of the $X_{j,k}=\frac{-\log_2|c_{j,k}|}{j},$0-spectrum is $X_{j,k}=\frac{-\log_2|c_{j,k}|}{j},$1, and the $X_{j,k}=\frac{-\log_2|c_{j,k}|}{j},$2-large deviation wavelet formalism holds (Céline et al., 1 Oct 2025). The same bound is realized by a prevalent set of functions in the deterministic spaces $X_{j,k}=\frac{-\log_2|c_{j,k}|}{j},$3 with prescribed wavelet profile (Céline et al., 1 Oct 2025). A plausible implication is that, within coefficient-constrained classes, the large-deviation formula describes not merely exceptional models but typical behavior in both probabilistic and prevalence senses.

Not all random wavelet series satisfy the usual multifractal formalisms. Lacunary wavelet series on Cantor sets show that a desynchronization between dyadic wavelet scales and the scales of the Cantor construction can destroy both the Legendre formalism and the leader large deviation formalism (Esser et al., 2022). In the duplicated model on $X_{j,k}=\frac{-\log_2|c_{j,k}|}{j},$4, nonzero coefficients have the fixed size $X_{j,k}=\frac{-\log_2|c_{j,k}|}{j},$5 but are only allowed on dyadic intervals lying inside $X_{j,k}=\frac{-\log_2|c_{j,k}|}{j},$6, and are activated by Bernoulli variables of parameter $X_{j,k}=\frac{-\log_2|c_{j,k}|}{j},$7 (Esser et al., 2022). For $X_{j,k}=\frac{-\log_2|c_{j,k}|}{j},$8, the multifractal spectrum is piecewise affine with a phase transition at $X_{j,k}=\frac{-\log_2|c_{j,k}|}{j},$9,

$\mathbb R^d$0

whereas the leader large deviation spectrum keeps the classical line $\mathbb R^d$1 over its support (Esser et al., 2022). This provides a concrete counterexample to the widespread expectation that wavelet-leader statistics always reproduce the true singularity spectrum.

4. Dependent coefficient arrays, long memory, and Wiener chaos

A second branch of the subject studies random wavelet coefficients produced by analyzing stochastic processes rather than prescribing coefficients directly. For a centered stationary Gaussian sequence $\mathbb R^d$2 with long memory,

$\mathbb R^d$3

and a nonlinear subordinated process $\mathbb R^d$4, wavelet coefficients are defined by

$\mathbb R^d$5

(Clausel et al., 2010). Expanding $\mathbb R^d$6 in Hermite polynomials,

$\mathbb R^d$7

with Hermite rank

$\mathbb R^d$8

one finds that, after normalization by $\mathbb R^d$9, the random coefficient field converges in finite-dimensional distributions to a field in the $$a_{j,t,m} = 2^{j\alpha}(j+1)^\theta \Big(1+\frac{\|m\|_\infty}{2^j}\Big)^\gamma \,\xi_{j,t,m} \quad\text{for }j\ge 1,$$0-th Wiener chaos (Clausel et al., 2010). More precisely,

$$a_{j,t,m} = 2^{j\alpha}(j+1)^\theta \Big(1+\frac{\|m\|_\infty}{2^j}\Big)^\gamma \,\xi_{j,t,m} \quad\text{for }j\ge 1,$$1

and the limit coefficients are themselves the wavelet coefficients of a generalized self-similar Hermite process $$a_{j,t,m} = 2^{j\alpha}(j+1)^\theta \Big(1+\frac{\|m\|_\infty}{2^j}\Big)^\gamma \,\xi_{j,t,m} \quad\text{for }j\ge 1,$$2 (Clausel et al., 2010). The limiting coefficient field is therefore neither white-noise-like nor independent across locations and scales; it is a structured, chaos-valued, dependent random coefficient array.

The corresponding wavelet scalogram theory shows that even second-order statistics of such coefficients can have nontrivial chaotic limits. For

$$a_{j,t,m} = 2^{j\alpha}(j+1)^\theta \Big(1+\frac{\|m\|_\infty}{2^j}\Big)^\gamma \,\xi_{j,t,m} \quad\text{for }j\ge 1,$$3

the asymptotic law of the centered scalogram depends on the Hermite expansion of $$a_{j,t,m} = 2^{j\alpha}(j+1)^\theta \Big(1+\frac{\|m\|_\infty}{2^j}\Big)^\gamma \,\xi_{j,t,m} \quad\text{for }j\ge 1,$$4 and on the balance between the number of coefficients $$a_{j,t,m} = 2^{j\alpha}(j+1)^\theta \Big(1+\frac{\|m\|_\infty}{2^j}\Big)^\gamma \,\xi_{j,t,m} \quad\text{for }j\ge 1,$$5 and the scale parameter $$a_{j,t,m} = 2^{j\alpha}(j+1)^\theta \Big(1+\frac{\|m\|_\infty}{2^j}\Big)^\gamma \,\xi_{j,t,m} \quad\text{for }j\ge 1,$$6 (Clausel et al., 2012). The limit can be Gaussian, Rosenblatt, or a higher-order Hermite distribution. In particular, when $$a_{j,t,m} = 2^{j\alpha}(j+1)^\theta \Big(1+\frac{\|m\|_\infty}{2^j}\Big)^\gamma \,\xi_{j,t,m} \quad\text{for }j\ge 1,$$7 with $$a_{j,t,m} = 2^{j\alpha}(j+1)^\theta \Big(1+\frac{\|m\|_\infty}{2^j}\Big)^\gamma \,\xi_{j,t,m} \quad\text{for }j\ge 1,$$8, cross terms of the form $$a_{j,t,m} = 2^{j\alpha}(j+1)^\theta \Big(1+\frac{\|m\|_\infty}{2^j}\Big)^\gamma \,\xi_{j,t,m} \quad\text{for }j\ge 1,$$9 may dominate and yield a limit in chaos of order $\mathbb R^d$00 (Clausel et al., 2012). This is a sharp failure of any naive reduction principle based only on Hermite rank.

A related but analysis-oriented perspective studies statistics of random wavelet coefficients and their moduli rather than synthesis formulas. For processes with stationary increments, the wavelet transform $\mathbb R^d$01 yields coefficient fields whose ordinary cross-scale covariance is nearly diagonal, whereas nonlinear statistics such as $\mathbb R^d$02 and $\mathbb R^d$03 reveal non-Gaussian dependencies across scales (Morel et al., 2022). The scattering cross-spectrum

$\mathbb R^d$04

is scale-invariant for self-similar processes in the paper’s wide-sense formulation (Morel et al., 2022). This suggests that, for dependent coefficient fields, cross-scale envelope statistics may be as fundamental as the coefficient law itself.

5. Stable, multistable, anisotropic, and multiplicative series

Random wavelet series with heavy-tailed coefficients require different convergence mechanisms. For i.i.d. symmetric $\mathbb R^d$05-stable variables $\mathbb R^d$06, the fractional wavelet series

$\mathbb R^d$07

converges almost surely in $\mathbb R^d$08 under explicit conditions linking $\mathbb R^d$09, $\mathbb R^d$10, the dimension $\mathbb R^d$11, and the wavelet regularity (Medina et al., 2019). A modified series,

$\mathbb R^d$12

is pointwise defined and has a measurable version in a higher-regularity regime (Medina et al., 2019). In the Gaussian case $\mathbb R^d$13, the field is self-similar with exponent $\mathbb R^d$14, but for $\mathbb R^d$15 exact self-similarity is broken because the law depends on an $\mathbb R^d$16-norm of wavelet coefficients rather than a basis-invariant $\mathbb R^d$17-norm (Medina et al., 2019).

For harmonizable fractional stable sheets, the anisotropic expansion is indexed by multi-scales $\mathbb R^d$18 and locations $\mathbb R^d$19,

$\mathbb R^d$20

with stable coefficients $\mathbb R^d$21 defined by integration against a rotationally invariant stable random measure (Ayache et al., 2019). The series converges almost surely in every Hölder space $\mathbb R^d$22 with $\mathbb R^d$23, and this regularity feeds into a uniform Hausdorff-dimension theorem for inverse images of the associated $\mathbb R^d$24-valued sheet (Ayache et al., 2019). The construction relies on a Fourier-side wavelet expansion and a LePage representation to control the stable coefficients (Ayache et al., 2019).

Multistable models lead to yet another type of wavelet random series. For the random field

$\mathbb R^d$25

the Haar-wavelet expansion of the kernel yields

$\mathbb R^d$26

where $\mathbb R^d$27 are Haar-wavelet integrals of the multistable random measure (Ayache et al., 2020). The truncated series converges almost surely and uniformly on $\mathbb R^d$28, with rate

$\mathbb R^d$29

(Ayache et al., 2020). This strong convergence is used to define and simulate the multifractional multistable Riemann–Liouville process $\mathbb R^d$30 (Ayache et al., 2020).

A multiplicative variant arises for positive processes $\mathbb R^d$31, where $\mathbb R^d$32 itself has an additive random wavelet expansion

$\mathbb R^d$33

Exponentiation yields the multiplicative wavelet representation

$\mathbb R^d$34

with explicit truncation conditions guaranteeing simulation with given accuracy and reliability in both $\mathbb R^d$35 and $\mathbb R^d$36 when $\mathbb R^d$37 is strictly sub-Gaussian (Turchyn, 2014). This suggests that the additive wavelet series paradigm can be transported to nonlinear positive models without abandoning explicit probabilistic error control.

6. Simulation, inference, and the boundaries of the concept

Several works exploit random wavelet series as constructive simulation devices. For stationary strictly sub-Gaussian $\mathbb R^d$38 with spectral density $\mathbb R^d$39, one has a random wavelet expansion

$\mathbb R^d$40

with spectral formulas for $\mathbb R^d$41 and $\mathbb R^d$42 (Turchyn, 2019). Truncating this series yields a finite simulator $\mathbb R^d$43, which in turn gives plug-in simulators for nonlinear processes

$\mathbb R^d$44

together with explicit lower bounds on truncation depths ensuring prescribed accuracy $\mathbb R^d$45 and reliability $\mathbb R^d$46 in $\mathbb R^d$47 (Turchyn, 2019). This is a direct example of wavelet-based stochastic synthesis guided by rigorous error analysis.

Wavelet-domain stochastic modeling can, however, extend beyond the formal class of random wavelet series. In the superstatistical interpolation model, each Gaussian component $\mathbb R^d$48 has a multiwavelet expansion with correlated Gaussian coefficients and sparse covariance in the transformed basis, but the full non-Gaussian process is assembled pointwise as $\mathbb R^d$49, using a slowly varying log-normal latent process $\mathbb R^d$50 (Lübke et al., 2022). The authors explicitly emphasize that this is not a standard random wavelet series with independent random coefficients; rather, it is a hierarchical random field over $\mathbb R^d$51 whose Gaussian components are synthesized in wavelet space (Lübke et al., 2022).

The same distinction appears on the deterministic side. A threshold autoregressive model with time-varying threshold represented by

$\mathbb R^d$52

uses a wavelet series inside a stochastic time-series model, but the coefficients $\mathbb R^d$53 are deterministic unknown parameters estimated by conditional least squares rather than random variables (Davis et al., 18 May 2026). The paper explicitly states that it does not study random wavelet series in the formal probabilistic sense (Davis et al., 18 May 2026). This is a useful boundary case: wavelet series can be central to stochastic modeling without the model itself being a random wavelet series.

Taken together, these developments show that “random wavelet series” names both a concrete class of probabilistic expansions and a broader methodological zone in which wavelet coefficients, wavelet moduli, and wavelet-domain envelopes become the natural coordinates for regularity theory, multifractal analysis, and simulation. The strongest unifying principle is that wavelet localization converts coefficient statistics into spatially localized probabilistic geometry. In the independent-coefficient setting this yields sharp $\mathbb R^d$54-spectrum and Besov criteria (Céline et al., 1 Oct 2025, Horst et al., 2024); in randomized deterministic series it makes unbounded multipliers destructive rather than smoothing (Esser et al., 2023); and in stable, long-memory, or multistable settings it produces random coefficient fields whose dependence and heavy tails remain visible at every scale (Medina et al., 2019, Clausel et al., 2010).