{"id":"cd172024-1060-4207-a388-e2678383f040","arxiv_id":"2411.18155","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Random wavelet series with multiplicative random coefficients lie in a given Besov space almost surely exactly when the coefficient decay parameters satisfy three inequalities, with matching moment bounds and a separate sparse-coefficient version.","lead":"This paper identifies the exact parameter conditions under which a random wavelet series is almost surely smooth enough to belong to a prescribed Besov space, and when its norm has finite moments or exponential moments. The complete characterization covers all Besov exponents p and q on R^d, including sparse random coefficients, which matters for designing priors in Bayesian inverse problems.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.8's sufficiency for Bernoulli-sparse priors is false in the p=∞, q<∞ boundary: Property A\" allows s+d/2+α=0, but the ℓq norm of constant scale weights diverges, and a deterministic counterexample satisfies A\" while the Besov norm is infinite.","rationale":"The reader identified the moment assumptions on the template variable as the weakest point and accepted the paper with moderate confidence. A closer reading of the sufficiency proof for Besov–Bernoulli priors reveals a sharper and more serious issue: Theorem 3.8 claims finiteness of E[||a||^r] and almost sure membership in B^s_{p,q} under Property A\", but Property A\"(b) is too permissive for p=∞ and q<∞. The proof's Step 2 uses the inequality ||(1)_{j∈N}||_{ℓq} < ∞, which holds only for q=∞. In the boundary case s+d/2+α=0, q<∞, the estimate fails and, as shown by the explicit deterministic example with X≡1, λ≡1, β=γ=-1, the Besov sequence norm is infinite. This is not a matter of missing concentration or moment assumptions; it is an internal inconsistency between Property A\" and Property A' (which correctly separates q<∞ from q=∞). The main results for the non-sparse Besov prior (series (1.1)) appear unaffected, and the fix for the Bernoulli-sparse case is localized: make Property A\"(b) q-dependent exactly as in Property A'. Because the error invalidates a stated theorem and the claimed complete characterization for series (1.2) in a boundary regime, acceptance should be conditional on this correction rather than unconditional.","tokens_in":49806,"tokens_out":25863,"duration_ms":223480,"concrete_test":"Run the deterministic case d=1, p=∞, q=1, α=0, β=γ=-1, µ=ν=0, s=-1/2, X≡1 through Theorem 3.8. Verify that Property A\" is satisfied but ||a||_{b^{-1/2}_{∞,1}} = Σ_{j,t} sup_m (1+|m|/2^j)^{-1} = ∞, so the theorem's conclusion E[||a||^r]<∞ fails. Then correct Property A\"(b) to require s+d/2+α<0 for q<∞ (and s+d/2+α≤0 only for q=∞), and recheck Theorem 3.8 Step 2; if the corrected statement restores the estimate, the fix is confirmed.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The sufficiency result for Besov–Bernoulli priors overreaches at the p=∞, q<∞ boundary. Property A\" (Definition 2.8(b)) states that for p=∞ it is enough to have γ≤0, β≤0, and s+d/2+α≤0, with no distinction between q<∞ and q=∞. But in Theorem 3.8, Step 2 bounds ~η2 ≤ R ||(2^{j(s+d/2+α)})_{j∈N,t∈T_j}||_{ℓq} and claims this is ≲R when s+d/2+α≤0. That implication is false for q<∞: the ℓq norm of the constant sequence 1 over infinitely many scales is infinite. A concrete counterexample: d=1, α=0, β=γ=-1, µ=ν=0, p=∞, q=1, s=-1/2, with X≡1. Then λj,t,m=1 and ξj,t,m=1 a.s., so aj,t,m=(1+|m|/2^j)^{-1}. Property A\" is satisfied, since s+d/2+α=0≤0. But the b^{-1/2}_{∞,1} norm is Σ_{j∈N0,t∈T_j} sup_m (1+|m|/2^j)^{-1} = ∞, so E[||a||^r]=∞ and f∉B^{-1/2}_{∞,1}. This directly contradicts Theorem 3.8 and invalidates the claimed sufficiency direction for Bernoulli-sparse series in this boundary case. The correct condition should mirror Property A' (Definition 2.8(c)/(d)): for p=∞, q<∞ one needs s+d/2+α<0, while s+d/2+α≤0 is only sufficient for q=∞.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies almost-sure Besov regularity, moment finiteness, and exponential moment finiteness for random wavelet series on R^d whose coefficients are products of i.i.d. random variables and deterministic scale/shift decays. For the dense prior (1.1), Theorems 3.3--3.5 and 4.1--4.2 give, under mild moment assumptions on the template variable X, necessary and sufficient conditions (Property A) for f ∈ B^s_{p,q}(R^d) a.s., for E[||f||^r] < ∞, and for E[exp(c||f||^r)] < ∞. For the sparse Bernoulli prior (1.2), Theorems 3.8 and 4.3--4.4 claim analogous but only partially matching conditions (Property A'' versus Property A'). Section 5 discusses sharpness of the moment assumptions, including the p = ∞ endpoint. The proofs are detailed and mostly self-contained, relying on wavelet characterizations, Borel--Cantelli arguments, Paley--Zygmund inequalities, and auxiliary maximal lemmas.","tokens_in":50194,"tokens_out":7106,"duration_ms":62959,"significance":"If the main results were correct, the paper would be a valuable reference for Bayesian inverse problems and for random wavelet series, since it treats the full range p,q ∈ (0,∞], distinguishes p = ∞ and q = ∞ cases carefully, and gives two-sided characterizations rather than only sufficient conditions. The dense-prior theorems appear sound and are supported by self-contained proofs, and the sharpness discussion in Section 5 is a genuine strength. However, the sufficiency statement for the Bernoulli prior, Theorem 3.8, contains a load-bearing error at the p = ∞, q < ∞ boundary, so the claimed complete characterization for sparse priors is not established as stated.","major_comments":[{"comment":"Theorem 3.8 is false in the stated form for p = ∞, q < ∞. In Step 2 of the proof, the authors bound ~η2 ≤ R ||(2^{j(s+d/2+α)})_{j∈N,t∈T_j}||_{ℓq} and claim this is ≲ R because s+d/2+α ≤ 0 by Property A''. For q < ∞ and s+d/2+α = 0, the sequence is identically 1 over infinitely many scales, so its ℓq norm is infinite. A concrete counterexample is d=1, α=0, β=γ=−1, µ=ν=0, p=∞, q=1, s=−1/2, X≡1. Then Property A'' holds (γ≤0, β≤0, s+d/2+α=0), and the coefficients are a_{j,t,m}=(1+|m|/2^j)^{-1}. The b^{-1/2}_{∞,1} norm is Σ_{j≥1} sup_m (1+|m|/2^j)^{-1} = ∞, so f∉B^{-1/2}_{∞,1}(R) and E[||a||^r]=∞, contradicting Theorem 3.8. The sufficiency condition for p=∞ must distinguish q: for q<∞ one needs s+d/2+α<0, while equality is admissible only for q=∞, matching Property A'(c)/(d). This is a load-bearing issue because it invalidates the claimed sufficiency direction for the Bernoulli-sparse prior in a whole parameter regime.","section":"§3.8 and Definition 2.8(b)"}],"minor_comments":[{"comment":"The statement assumes µ,ν ∈ (−∞,0], but the proof contains the line 'since µ ≥ 0 and δ+δε+ε ≥ 0'. This should read 'since µ,ν ≤ 0 (so ̺_{j,τ} ≤ 1) and δ+δε−ε ≥ 0'; the proof also writes 'δ+δε+ε' where 'δ+δε−ε' is meant. As written, the sign conventions are internally inconsistent, though the underlying argument can be repaired.","section":"Lemma A.8"},{"comment":"There is a typo in the first paragraph: 'He present a brief outline' should be 'We present a brief outline'.","section":"Section 2.2"},{"comment":"The sentence 'Let further (ξ_{j,t,m})... by a family of of random variables' contains a duplicated 'of'.","section":"Lemma 3.2"},{"comment":"The statement begins 'Assume that = (a_{j,t,m})...' and is missing the symbol 'a' before the equals sign.","section":"Proposition 5.2"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Take a close look at Theorem 3.8 before you trust the sparse-prior half of this paper. The stress-test counterexample is valid. For d=1, α=0, β=γ=-1, µ=ν=0, p=∞, q=1, s=-1/2, and X≡1, Property A\" holds (since s+d/2+α=0), but the b^s_{∞,1} norm of the coefficient sequence is Σ_j 1 = ∞, so f is not in the claimed Besov space. The proof of Theorem 3.8 Step 2 treats the ℓq norm of the constant sequence 2^{j(s+d/2+α)} as finite when s+d/2+α≤0; that is only true for q=∞. The fix is straightforward: Property A\" (b) should require strict inequality for q<∞, matching Property A' (c)/(d).\n\nWhat the paper does well is the non-sparse part. The characterization of almost sure Besov regularity, moment finiteness, and MGF finiteness for series of the form (1.1) on R^d, with general i.i.d. templates, p≠q, and the full range p,q∈(0,∞], looks correct and is a genuine advance over earlier torus/domain/Gaussian results. The proofs are detailed and self-contained modulo standard wavelet theorems, and Section 5 is honest about sharpness of moment assumptions. That half deserves publication.\n\nBeyond the boundary flaw, a minor sign typo appears in Lemma A.8 ('since µ ≥ 0' should read µ ≤ 0), but the inequality still goes through. The reader's report is fair on novelty and soundness of the main results but missed the counterexample; the acceptance is premature as the paper stands.\n\nThe paper will be valuable to researchers working on Besov priors for Bayesian inverse problems and random wavelet series. I would send it to a serious referee, but with the sparse-prior boundary case explicitly flagged. With a corrected Property A\" and adjusted proof, it should be publishable; without that fix, the sparse-prior theorems are false as stated.","headline":"Strong new results on non-sparse Besov priors, but the Bernoulli-sparse sufficiency theorem has a genuine boundary counterexample at p=∞, q<∞ that must be fixed before acceptance.","tokens_in":50741,"tokens_out":5018,"would_cite":true,"duration_ms":45644,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["42B35","42C40","46F25","60G60","60H50"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that a random wavelet series of the form (1.1) almost surely belongs to the Besov space $B^s_{p,q}(\\mathbb{R}^d)$ if and only if the parameters satisfy $\\beta<-d/p$ and $s+d/2+\\alpha<0$, with the boundary case…","keywords":["random wavelet series","Besov regularity","Besov prior","Karhunen-Loève-type expansion","wavelet sequence space","moment generating function","Bernoulli sparsity","Bayesian inverse problems"],"falsifier":"Take a template $X$ with $P(X\\neq 0)>0$ but $E|X|^p=\\infty$ (for example, a tail of order $t^{-p}/\\log(t)^2$), choose parameters satisfying Property A, and simulate the Besov sequence norm of the coefficient array. The theorem predicts the norm diverges almost surely, while a weaker moment bound would predict possible convergence.","tokens_in":49605,"feed_emoji":"🎲","tokens_out":6339,"duration_ms":53703,"temperature":0.7,"pith_summary":"This paper studies random wavelet series on $\\mathbb{R}^d$ whose coefficients are independent copies of a template random variable multiplied by deterministic scale and shift factors. It proves that, under mild moment assumptions on the template, the series almost surely belongs to a Besov space $B^s_{p,q}(\\mathbb{R}^d)$ if and only if the two parameter inequalities $\\beta<-d/p$ and $s+d/2+\\alpha<0$ hold, with the boundary case $s+d/2+\\alpha=0$ requiring $\\theta<-1/q$ (and with the obvious replacements at $p=\\infty$ or $q=\\infty$). The same inequalities characterize finiteness of the $r$-th moment of the Besov norm and, for well-concentrated template variables, finiteness of the exponential moment $E[\\exp(c\\|f\\|^r)]$. A parallel condition governs the Bernoulli-sparse version of the series. A sympathetic reader would care because this turns a probabilistic regularity question into a checkable parameter test and closes gaps left by earlier treatments that required Gaussian or bounded coefficients.","feed_headline":"Two inequalities decide Besov regularity of random wavelet series","feed_subtitle":"The same sharp threshold controls finiteness of the norm's moments and of its exponential moments.","key_machinery":"The workhorse is the wavelet characterization of Besov spaces, which identifies $\\|f\\|_{B^s_{p,q}}$ (up to equivalence) with the weighted mixed $\\ell^p/\\ell^q$ norm of the wavelet coefficients. For the sufficiency direction, Lemma 3.1 controls the spatial sum appearing in that norm by a single random quantity $\\Xi$, defined as the supremum over dyadic shells of averages of $|\\xi|^p$; Property A makes the deterministic weights summable, while the moment condition on $X$ makes $\\Xi$ almost surely finite. For the necessity direction, truncated variables, the Paley-Zygmund inequality, and Borel-Cantelli arguments force each individual inequality: if any parameter crosses the threshold, independent blocks of coefficients with positive probability are large enough that the Besov norm diverges almost surely. A separate argument (Theorem 4.2) handles the subtle distinction between the coefficient sequence belonging to $b^s_{p,q}$ and the synthesized function belonging to $B^s_{p,q}$.","core_discovery":"The central claim is a complete if-and-only-if description of Besov regularity for the prior (1.1): with i.i.d. coefficients having a sufficiently nice template law, the random tempered distribution $f$ belongs to $B^s_{p,q}(\\mathbb{R}^d)$ almost surely exactly when Property A holds, namely $\\gamma<-d/p$, $\\beta<-d/p$, and either $s+d/2+\\alpha<0$ or $s+d/2+\\alpha=0$ with $\\theta<-1/q$; when $p=\\infty$ the strict inequalities on $\\gamma,\\beta$ become non-strict, and when $q=\\infty$ the strict inequality on $\\theta$ becomes non-strict. The same conditions are necessary and sufficient for $E[\\|f\\|^r_{B^s_{p,q}}]<\\infty$ and, under an exponential-moment condition on $|X|^{\\max\\{r,p\\}}$, for $E[\\exp(c\\|f\\|^r)]<\\infty$ for some $c>0$. For the Bernoulli-sparse series (1.2) an analogous condition, Property A$'$, is established. The paper also proves that the moment assumptions on the template cannot simply be dropped: essential boundedness is needed in the $p=\\infty$ borderline cases, and near-sharpness statements show only mild weakening is possible for $p<\\infty$.","pith_inferences":["The characterization is a sharp phase transition: crossing any single parameter over its threshold switches the series from almost-surely smooth to almost-surely not smooth, which should be visible in numerical samples of the prior.","Because the exponential-moment finiteness is tied to the same inequalities, Bayesian inverse problems using these priors inherit well-posedness exactly in the regime identified here, a consequence the paper motivates but does not develop.","The Bernoulli-sparse threshold suggests a design rule for sparse priors: to keep a target smoothness while thinning coefficients, one must compensate with $\\alpha$ or $\\mu$; one could test this by estimating Besov norms of truncated samples."],"forward_implications":["For the Besov prior (1.1), almost-sure membership in $B^s_{p,q}(\\mathbb{R}^d)$ is decided by the two inequalities $\\beta<-d/p$ and $s+d/2+\\alpha<0$, together with the stated boundary and endpoint modifications.","Under the same conditions, and with a well-concentrated template, $E[\\|f\\|^r_{B^s_{p,q}}]$ and $E[\\exp(c\\|f\\|^r_{B^s_{p,q}})]$ are finite for some $c>0$, so the norm's tail is sub-exponential in the regular regime.","The Bernoulli-sparse series (1.2) has an analogous characterization, Property A$'$, in which the sparsity parameter $\\mu$ shifts the smoothness threshold by $\\mu/p$.","The moment assumptions on the template are essentially optimal: for $p=\\infty$, boundedness cannot be dropped in the borderline cases, and for $p<\\infty$ the conditions can only be weakened mildly.","All results transfer to weighted Besov spaces with polynomial weights by a parameter shift explained in Remark 2.7."],"supporting_citations":[{"why":"Supplies the wavelet characterization of Besov spaces that turns the function-space question into a sequence-space question.","marker":"[21]"},{"why":"Supplies the existence of compactly supported smooth wavelets with vanishing moments used to build the basis.","marker":"[12]"},{"why":"Provides the motivating Besov-prior result on the torus that the paper extends to $\\mathbb{R}^d$ with general $p,q$.","marker":"[18]"},{"why":"Gives the Bayesian inverse problem context where finiteness of exponential moments matters.","marker":"[8]"},{"why":"Earlier study of random wavelet series regularity to which the present paper's conditions are compared.","marker":"[3]"},{"why":"The prior sparse wavelet-series result whose Bernoulli plus Gaussian coefficients are generalized here.","marker":"[7]"}],"fun_headline_variants":["Necessary and sufficient Besov regularity for random wavelet series","Sharp threshold for Besov regularity of random wavelet series","Random wavelet series: Besov regularity fully characterized","Exact if-and-only-if for Besov regularity of random wavelet series","Complete characterization of Besov regularity for random wavelet series"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole characterization rests on the template random variable $X$ having enough integrability: a finite moment $E|X|^{p(1+\\varepsilon)}$ when $p<\\infty$, and essential boundedness when $p=\\infty$, together with $P(X\\neq 0)>0$ for the necessity direction.","fun_headline_variants_meta":{"raw":{"variants":["Necessary and sufficient Besov regularity for random wavelet series","Sharp threshold for Besov regularity of random wavelet series","Random wavelet series: Besov regularity fully characterized","Exact if-and-only-if for Besov regularity of random wavelet series","Complete characterization of Besov regularity for random wavelet series"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001117,"raw_usage":{"total_tokens":4645,"prompt_tokens":938,"completion_tokens":3707,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":554,"completion_tokens_details":{"reasoning_tokens":3626}},"tokens_in":554,"tokens_out":3707,"duration_ms":25214,"temperature":1.0,"reasoning_tokens":3626,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T11:27:32.049355+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a template $X$ with $P(X\\neq 0)>0$ but $E|X|^p=\\infty$ (for example, a tail of order $t^{-p}/\\log(t)^2$), choose parameters satisfying Property A, and simulate the Besov sequence norm of the coefficient array. The theorem predicts the norm diverges almost surely, while a weaker moment bound would predict possible convergence.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the wavelet characterization of Besov spaces that turns the function-space question into a sequence-space question."},{"cited_title":"Grohs, A","cited_arxiv_id":null,"evidence_quote":"Supplies the existence of compactly supported smooth wavelets with vanishing moments used to build the basis."},{"cited_title":"Lassas, E","cited_arxiv_id":null,"evidence_quote":"Provides the motivating Besov-prior result on the torus that the paper extends to $\\mathbb{R}^d$ with general $p,q$."},{"cited_title":"Dashti, S","cited_arxiv_id":null,"evidence_quote":"Gives the Bayesian inverse problem context where finiteness of exponential moments matters."},{"cited_title":"Aubry and S","cited_arxiv_id":null,"evidence_quote":"Earlier study of random wavelet series regularity to which the present paper's conditions are compared."},{"cited_title":"Cioica, S","cited_arxiv_id":null,"evidence_quote":"The prior sparse wavelet-series result whose Bernoulli plus Gaussian coefficients are generalized here."}],"review_version":1}