{"id":"8f585877-8f93-49b6-9c70-41f774a438ed","arxiv_id":"2505.13206","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A new CRM family, built by mixing stable and generalized gamma processes over their index of variation, reaches rapid variation with a closed-form Laplace exponent and yields almost extremely sparse Caron-Fox graphs with E ~ N log N.","lead":"This paper introduces a new family of random measures, the mixed generalized gamma processes, whose Laplace exponent remains tractable in the rapid variation regime, up to index of variation 1. Used as a graph model, it produces networks with near-linear edge growth and heavy-tailed degree distributions, matching features of large real-world networks.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The paper's verification of the Caron et al. tail condition (Lemma S1) is misstated as printed, and Prop. 4.1 and Cor. 4.2 give inconsistent edge constants; as a result the quantitative sparsity claim is not currently self-contained, though both issues appear repairable.","rationale":"Prop. 3.1 gives rho_mSt(w) ~ w^{-2}/log^2(1/w) at zero for alpha=1, tau=0, so the layered verification in Lemma S1 is the natural place to look. The printed ell(1/x) containing x^{-1} cannot be slowly varying, so the theorem's hypothesis is not literally verified; however, the error appears typographical, because integrating the correct leading order reproduces x^{-1} log^{-2}(1/x), and the rest of the argument (definition of C, Lemma S2, degree distribution) is internally consistent with that corrected tail. Similarly, the disagreement between Prop. 4.1 and Cor. 4.2 is a factor W^2; the Poisson representation of E[E_t] yields t^2 W/2 and makes Cor. 4.2 correct, so Prop. 4.1 is again typographical. These are not fatal to the qualitative claim that the model is almost extremely sparse with degree tail 1/(j(j-1)), and the closed-form Laplace exponent (Eq. 7) is an independent, parameter-free derivation. But the two printed errors concern exactly the parts of the argument that connect the new CRM to graph sparsity, so the conditional verdict should stand: accept only after the tail lemma and Prop. 4.1 constant are corrected. The reader's weakest-assumption points to the same Lemma S1 issue; my read therefore agrees.","tokens_in":27684,"tokens_out":31863,"duration_ms":315098,"concrete_test":"Recompute the two disputed quantities from first principles: (i) integrate rho_mGG(w) ~ eta c w^{-2}/log^2(1/w) from x to infinity and check whether the result is eta c x^{-1}/log^2(1/x) (Caron's condition) or the x^{-2} form implied by the printed Lemma S1; (ii) compute E[E_t] ~ (t^2/2) integral psi_mGG(2w) rho_mGG(w) dw from the Poisson process representation and compare with Prop. 4.1's t^2/(2W) and with Cor. 4.2's implied W t^2/2. Both are direct analytic checks that distinguish typo from substantive error.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that the mGG Caron-Fox graphs satisfy N_t^(e) ~ (W/(4C)) N_t log N_t is imported from Caron et al. (2023, Prop. 11), and the gate for that import is the tail condition on the mGG Levy intensity. The supplementary verification (Lemma S1) is incoherent as printed: it asserts bar-rho_mGG(x;1,0,beta,c,eta) ~ x^{-1} ell(1/x), and then defines ell(1/x) = eta c (1-tau)^{-1} x^{-1} / log^2(1/x). With the extra x^{-1}, ell is not slowly varying, and the displayed behavior is bar-rho ~ x^{-2}/log^2(1/x), contradicting Prop. 3.1. Direct integration of the Prop. 3.1 behavior rho(w) ~ eta c w^{-2}/log^2(1/w) gives bar-rho(x) ~ eta c x^{-1}/log^2(1/x), so the condition can be satisfied, but the manuscript does not state it correctly. Since Prop. 4.1, Cor. 4.2, and Prop. 4.3 all inherit from the same theorem, the quantitative sparsity claim is unsupported until this is fixed. Independently, Section 4 contains a constant inconsistency: Prop. 4.1 writes N_t^(e) ~ t^2/(2W), whereas Cor. 4.2 together with N_t ~ t^2 C/log t implies N_t^(e) ~ W t^2/2; the Poisson-process calculation of E[E_t] gives t^2 W/2, so Prop. 4.1 appears to have a reciprocal typo.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a new family of completely random measures (CRMs), the mixed stable and mixed generalized gamma CRMs, constructed by mixing the Lévy intensity of stable/generalized gamma processes over the stability index. The main theoretical contributions are a closed-form Laplace exponent ψ_mSt(t) = (t^α − t^τ)/((α−τ) log t), identification of the index of variation α ∈ (0,1] (including the rapid-variation case α = 1), asymptotic forms of the Lévy intensity, moment formulas, a size-biased series representation, and a Riemann-sum approximation of the total mass. The paper then applies the model in the Caron–Fox random graph framework with α = 1, τ = 0, β > 0, claiming almost extremely sparse graphs with N_t^{(e)} ~ (W/(4C)) N_t log N_t and an asymptotic degree distribution P(degree = j | degree ≥ 2) → 1/(j(j−1)). The paper also presents MCMC posterior inference and experiments on synthetic and three real-world networks. The central claims depend on importing asymptotic results from Caron et al. (2023), with the required tail condition verified in the supplementary material as Lemma S1, and on explicit constants in Proposition 4.1.","tokens_in":28102,"tokens_out":17943,"duration_ms":152787,"significance":"If the technical issues described below are corrected, this is a valuable contribution. The proposed CRM class fills a genuine gap: existing tractable CRMs have index of variation α < 1, and the mSt/mGG construction provides the first tractable example with α = 1 while retaining explicit Laplace exponents, moments, and simulation algorithms. The resulting graph model produces a simple family of almost extremely sparse networks with near-linear edge growth and a power-law degree distribution with exponent 2, and the paper backs this with reproducible code (GitHub), MCMC diagnostics (including multivariate Gelman–Rubin), and posterior predictive checks on real networks. These strengths are tangible: the Laplace-exponent formula and the Riemann-sum approximation of the total mass are clean, parameter-free derivations, and the real-data experiments are carefully presented. However, the quantitative sparsity claims, which are the paper's headline results, currently rest on a misstated verification of the Caron et al. (2023) tail condition and an inconsistent constant in Proposition 4.1; both are repairable but must be fixed before the results are reliable.","major_comments":[{"comment":"Lemma S1 states that the tail Lévy intensity satisfies \\bar ρ_mGG(x) ~ x^{-1} \\tilde ℓ(1/x) with \\tilde ℓ(1/x) = ηc/(1−τ) x^{-1}/log^2(1/x). With the factor x^{-1} inside \\tilde ℓ, the function \\tilde ℓ is not slowly varying and the displayed product behaves as ηc/(1−τ) x^{-2}/log^2(1/x). This contradicts the direct tail integration of the Proposition 3.1 behavior ρ_mGG(w) ~ ηc w^{-2}/log^2(1/w), which gives \\bar ρ_mGG(x) ~ ηc x^{-1}/log^2(1/x). The correct slowly varying function should be \\tilde ℓ(1/x) = ηc/(1−τ) / log^2(1/x) (without the extra x^{-1}). Because Lemma S1 is the verification of the condition of Caron et al. (2023, Proposition 11) on which Proposition 4.1, Corollary 4.2, and Proposition 4.3 all depend, the quantitative sparsity results are not supported as printed. The surrounding text also contains notational confusions between x and w (e.g., 'ρmGG(w; 1,τ,β,c,η ) =w−2L(1/x)' with L(1/x) evaluated at x→0 but depending on w), which should be cleaned up in the revision.","section":"Supplementary Material, Section S1.2, Lemma S1"},{"comment":"Proposition 4.1 states N_t^{(e)} ~ t^2/(2W), but the direct Poisson-process calculation of the expected number of edges gives E[N_t^{(e)}] = (1/2)t^2 ∫∫(1−e^{-2ww'}) ρ(dw)ρ(dw') = (1/2)t^2 W, where W is defined in the proposition. Moreover, combining N_t ~ t^2 C/log t with Corollary 4.2 gives N_t^{(e)} ~ (W/(4C)) × (t^2 C/log t) × 2 log t = W t^2/2, i.e., N_t^{(e)} ~ W t^2/2. Therefore the displayed asymptotic in Eq. (20) has an inverted W; it should read N_t^{(e)} ~ W t^2/2. This is not just a typographical nuisance because it makes Proposition 4.1 inconsistent with Corollary 4.2; the constant needs to be corrected so the quantitative sparsity statement is coherent.","section":"Section 4.2, Proposition 4.1, Eq. (20)"}],"minor_comments":[{"comment":"The formula for C is typeset ambiguously: 'C = 2(ηc)^2 β^{-1}-1+logβ/(logβ)^2' should be displayed as C = 2(ηc)^2 (β^{-1} − 1 + log β)/(log β)^2, matching the closed form obtained from the mean in Proposition 3.2.","section":"Section 4.2, Proposition 4.1"},{"comment":"In the Flickr row, the number of edges is given as '155,55,041', which is malformed; this should be a standard thousands-separated number such as 15,555,041 (the value should be checked against the original data source).","section":"Section 5.2, Table 2"},{"comment":"The caption states 'parameters α = 0, τ = 1', which violates the model constraints 0 ≤ τ < α ≤ 1; this is presumably a typo for α = 1, τ = 0 and should be corrected.","section":"Supplementary Material, Figure S10 caption"},{"comment":"The comparison model is described as the 'Generalized Gamma CRM (■) with parameters τ = 1 and σ = 0.5' in the caption, while the text says 'here with α = 0.5'; the notation should be unified to avoid confusion about which parameter is the stability index.","section":"Section 5, Figure 3 caption and text"},{"comment":"The proof of Proposition 3.5 would be clearer if the Riemann-sum limit were stated for the shifted argument, i.e., (1/n) Σ_{i=1}^n ((ct+β)^{s_i} − β^{s_i}) → ψ_mSt(β+ct) − ψ_mSt(β), rather than the current (S30) which displays only (1/n) Σ t^{s_i}; the step to the Laplace transform of the total mass is then immediate.","section":"Supplementary Material, Section S1.1.5, Eq. (S30)"}],"recommendation":"major_revision","confidential_remarks":"The paper's central claims are defensible and the construction is genuinely novel, but the current version contains two load-bearing technical errors: the supplementary verification of the Caron et al. tail condition is misstated (extra x^{-1} in the slowly varying function), and the constant in Proposition 4.1 is inverted. Both appear easily fixable within the manuscript's scope, which is why I recommend major revision rather than rejection. The external dependence on Caron et al. (2023, Proposition 11) is acceptable, but the revision should make the verification fully self-contained and free of these errors."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a real new object, and it should go to review. Mixing stable (or generalized gamma) Lévy intensities over the index of variation gives a CRM whose Laplace exponent is (t^α − t^τ)/((α−τ) log t), closing the α=1 gap in Table 1 with a tractable formula. The size-biased representation, the Lambert-W inversion, and the moments are worked out cleanly. Applied to Caron–Fox graphs, the qualitative story—near-linear edge growth, degree-1 node dominance, power-law tail exponent 2 among degree≥2 nodes—follows robustly from the theory. The code is public, which also helps.\n\nTwo things on the page need to be fixed before the quantitative claims are self-contained. First, Prop 4.1 states N_t^(e) ~ t^2/(2W); Corollary 4.2, together with N_t ~ t^2 C/log t, gives N_t^(e) ~ W t^2/2, and a direct Poisson-process calculation of E[E_t] gives t^2 W/2. So Prop 4.1 has a reciprocal typo. Second, Lemma S1 is misstated as printed: it asserts bar-rho_mGG(x) ~ x^{-1} ell(1/x) and then defines ell(1/x) = ηc/(1−τ) x^{-1}/log^2(1/x), which is not slowly varying. Direct integration of Prop 3.1 gives bar-rho ~ ηc x^{-1}/log^2(1/x), so the Caron et al. condition can be met, but the manuscript does not say it correctly. Both are local and repairable; neither destroys the construction. The β>0 condition for finite mean m is stated properly.\n\nLess central: the real-data section is thin. It shows posterior predictive degree distributions against empirical ones, but has no quantitative baselines on those datasets, so the claim of practical advantages is not really demonstrated. The synthetic MCMC checks are more convincing. Also, the sparsity results are theorems inherited from Caron et al. (2023), not predictions fitted to data; that is normal in this area, and the external dependency is a published result, so I do not count it against the paper.\n\nThe paper is for researchers in Bayesian nonparametrics and sparse graph models who need a tractable CRM at α=1. It deserves a serious referee: the construction is new, the proofs are mostly solid modulo typos, and the code is available. A referee should ask for the constant fix, the Lemma S1 correction, and real-data baselines, but those are revision items, not rejection items. I would send it out.","headline":"A genuinely new tractable CRM with index of variation 1, worth reviewing, but two quantitative statements on the page need correcting first.","tokens_in":28670,"tokens_out":3357,"would_cite":true,"duration_ms":34407,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60G57","60G55","05C80","62F15"],"pacs":[],"model":"deepseek-v4-flash","headline":"A new family of completely random measures reaches the rapid-variation regime α=1 and yields almost extremely sparse graphs.","keywords":["completely random measures","rapid variation","index of variation","sparse networks","generalized gamma process","stable process","size-biased representation","Bayesian nonparametrics"],"falsifier":"Compute the tail integral $\\bar\\rho_{\\mathrm{mGG}}(x)=\\int_x^\\infty \\rho_{\\mathrm{mGG}}(w;1,0,\\beta,c,\\eta)\\,dw$ numerically for small $x$, for example $x=10^{-6}$ to $10^{-3}$, and compare it with $x^{-1}$ times a slowly varying function. The paper's Proposition 3.1 implies $\\bar\\rho(x)\\sim x^{-1}\\log^{-2}(1/x)$; if the numerical tail instead behaves as $x^{-2}\\log^{-2}(1/x)$, or if $\\bar\\rho(x)\\,x\\,\\log^2(1/x)$ does not tend to a finite positive constant, then the constants $C$ and $W$ in Corollary 4.2 do not have their stated values and the extreme-sparsity conclusion loses its proof. A Monte Carlo check would simulate growing graphs and test whether $N_t^{(e)}/(N_t\\log N_t)$ stabilises at $W/4C$.","tokens_in":27475,"feed_emoji":"🕸️","tokens_out":12201,"duration_ms":119417,"temperature":0.7,"pith_summary":"The paper introduces a new family of completely random measures (CRMs), the mixed Stable and mixed Generalized Gamma CRMs, whose index of variation can reach $\\alpha=1$, the rapid-variation regime. Its central object is a closed-form Laplace exponent, $\\psi_{\\mathrm{mSt}}(t;\\alpha,\\tau) = (t^\\alpha-t^\\tau)/((\\alpha-\\tau)\\log t)$, obtained by averaging stable Laplace exponents over the stability index. In the CRM-based sparse graph construction, the paper proves that with $\\alpha=1$, $\\tau=0$, the number of edges satisfies $N_t^{(e)} \\sim (W/4C)\\, N_t \\log N_t$, so the graph is almost extremely sparse: edges outnumber nodes but stay far below $N_t^{1+\\epsilon}$. The degree distribution among nodes of degree at least two converges to $1/(j(j-1))$, a power-law tail with exponent 2. This matters because large real networks show nearly linear edge growth, which requires the previously missing $\\alpha=1$ case; the paper also gives simulation and posterior inference algorithms based on the mixture structure.","feed_headline":"Edges grow almost linearly in a new sparse-network model","feed_subtitle":"Mixing stable CRMs over their index of variation yields near-linear edge growth and a 1/(j(j−1)) degree tail.","key_machinery":"The load-bearing identity is the Laplace exponent of the mixed Stable CRM, $\\psi_{\\mathrm{mSt}}(t;\\alpha,\\tau)=\\frac{1}{\\alpha-\\tau}\\int_\\tau^\\alpha t^s\\,ds=\\frac{t^\\alpha-t^\\tau}{(\\alpha-\\tau)\\log t}$, with $\\alpha\\in(0,1]$ and $\\tau\\in[0,\\alpha)$. Because the integral of $t^s$ against $s$ produces a logarithm, the exponent is rapidly varying at $t\\to\\infty$ when $\\alpha=1$; the full model tilts and scales this base, giving $\\psi_{\\mathrm{mGG}}(t)=\\eta(\\psi_{\\mathrm{mSt}}(\\beta+ct;\\alpha,\\tau)-\\psi_{\\mathrm{mSt}}(\\beta;\\alpha,\\tau))$. A size-biased representation expresses each weight through an inverse Laplace exponent and a latent $S_j\\in(\\tau,\\alpha)$, enabling exact simulation and the MCMC scheme. The graph asymptotics flow from the tail behaviour of the Lévy intensity, $\\rho_{\\mathrm{mGG}}(w)\\sim \\eta c^\\alpha \\rho_{\\mathrm{mSt}}(w)$, which produces the slowly varying logarithm corrections used in the sparsity theorems.","core_discovery":"The discovery is that one can build a completely random measure with rapid variation — index of variation $\\alpha\\in(0,1]$ — by mixing the Lévy intensities of stable or generalized gamma processes over their stability parameter. Although the mixture intensity has no closed form, its Laplace exponent does: $\\psi_{\\mathrm{mSt}}(t;\\alpha,\\tau)=(t^\\alpha-t^\\tau)/((\\alpha-\\tau)\\log t)$, and the full mixed Generalized Gamma CRM has $\\psi_{\\mathrm{mGG}}(t;\\alpha,\\tau,\\beta,c,\\eta)=\\eta(\\psi_{\\mathrm{mSt}}(\\beta+ct;\\alpha,\\tau)-\\psi_{\\mathrm{mSt}}(\\beta;\\alpha,\\tau))$. With $\\alpha=1$, $\\tau=0$, the exponent $\\psi_{\\mathrm{mSt}}(t;1,0)=(t-1)/\\log t$ is rapidly varying, so the associated sparse graph sequence is extremely sparse in the sense $N_t^{(e)}/N_t \\to \\infty$ yet $N_t^{(e)}/N_t^{1+\\epsilon} \\to 0$; explicitly $N_t^{(e)} \\sim (W/4C)\\, N_t \\log N_t$. Among nodes of degree at least 2, the asymptotic degree distribution is $1/(j(j-1))$. The construction includes stable and generalized gamma CRMs as limiting cases, and its size-biased representation makes simulation and MCMC inference practical.","pith_inferences":["One testable extension is to apply the same mixing recipe to other CRM families, such as the stable beta process; the paper's asymptotic conditions suggest the resulting CRM should also be rapidly varying.","In edge-exchangeable graph models, the same CRM would give multigraphs whose edge count grows almost linearly in node count, extending the paper's results beyond the exchangeable-vertex setting.","The universal $1/(j(j-1))$ tail among nodes of degree at least two could serve as a model-checking diagnostic: subsamples of increasing size should show $N_{t,j}/\\tilde N_{t,2}$ stabilizing at $1/(j(j-1))$."],"forward_implications":["The rapid-variation CRM fills a gap in Bayesian nonparametric models: clustering and feature-allocation processes built on it should grow nearly linearly, $K_n \\sim n/\\log n$.","In the graph setting, the almost-extremely-sparse regime $N_t^{(e)} \\sim (W/4C)\\, N_t \\log N_t$ matches the near-linear edge growth observed in real large networks, which standard $\\alpha<1$ CRMs cannot produce.","The asymptotic degree distribution among nodes of degree at least two is $1/(j(j-1))$, giving a power law with exponent 2.","The size-biased sampler has truncation error $R_n \\sim \\eta/\\log n$, and the Riemann-sum total-mass sampler converges at rate $1/n$, making simulation feasible.","The posterior MCMC algorithm recovers sociability parameters and model parameters on synthetic and real subgraphs, and the predictive degree distributions fit the three real-world datasets tested."],"supporting_citations":[{"why":"Supplies the CRM-based sparse graph construction used in Section 4, including the latent multigraph representation and posterior sampling scheme.","marker":"(Caron and Fox, 2017)"},{"why":"Provides Proposition 11, Theorem 2 and Corollary 1 that transfer tail Lévy-intensity asymptotics into node and edge sparsity and degree-distribution results.","marker":"(Caron et al., 2023)"},{"why":"Gives the size-biased representation of CRMs on which the simulation algorithms and Proposition 3.3 are built.","marker":"(Perman et al., 1992)"},{"why":"Supplies the truncation-error proof strategy and series representation used for Proposition 3.4.","marker":"(Lee et al., 2023)"},{"why":"Provides the Karamata, Tauberian and conjugate-pair results used to derive intensity asymptotics and inverse Laplace exponents.","marker":"(Bingham et al., 1987)"},{"why":"Gives the algorithm for sampling exponentially tilted stable random variables used in the total-mass Riemann-sum approximation.","marker":"(Devroye, 2009)"},{"why":"Supplies efficient sampling algorithms for exponentially tilted stable distributions, also used in the total-mass approximation.","marker":"(Hofert, 2011)"}],"fun_headline_variants":["Mixing CRMs yields near-linear edge growth in sparse networks","Rapidly varying CRMs give sparse networks with linear edge growth","A tractable CRM with rapid variation for extremely sparse networks","Stable-gamma mixtures produce sparse networks with near-linear edges","New CRM class achieves near-linear edge growth for sparse graphs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument depends on the precise rate at which the jumps of the mixed Generalized Gamma CRM thin out near zero: the tail must decay like $x^{-1}$ divided by a slowly varying function. The displayed verification in the supplementary material contains an extra factor of $x^{-1}$ in the slowly varying part, which would change the tail to $x^{-2}$ and break the constants in the edge-count formula; the sparsity results also require the tilting parameter $\\beta$ to be positive so that the total sociability mass has finite mean.","fun_headline_variants_meta":{"raw":{"variants":["Mixing CRMs yields near-linear edge growth in sparse networks","Rapidly varying CRMs give sparse networks with linear edge growth","A tractable CRM with rapid variation for extremely sparse networks","Stable-gamma mixtures produce sparse networks with near-linear edges","New CRM class achieves near-linear edge growth for sparse graphs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000457,"raw_usage":{"total_tokens":2348,"prompt_tokens":1058,"completion_tokens":1290,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":674,"completion_tokens_details":{"reasoning_tokens":1205}},"tokens_in":674,"tokens_out":1290,"duration_ms":10730,"temperature":1.0,"reasoning_tokens":1205,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T20:19:37.473457+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the tail integral $\\bar\\rho_{\\mathrm{mGG}}(x)=\\int_x^\\infty \\rho_{\\mathrm{mGG}}(w;1,0,\\beta,c,\\eta)\\,dw$ numerically for small $x$, for example $x=10^{-6}$ to $10^{-3}$, and compare it with $x^{-1}$ times a slowly varying function. The paper's Proposition 3.1 implies $\\bar\\rho(x)\\sim x^{-1}\\log^{-2}(1/x)$; if the numerical tail instead behaves as $x^{-2}\\log^{-2}(1/x)$, or if $\\bar\\rho(x)\\,x\\,\\log^2(1/x)$ does not tend to a finite positive constant, then the constants $C$ and $W$ in Corollary 4.2 do not have their stated values and the extreme-sparsity conclusion loses its proof. A Monte Carlo check would simulate growing graphs and test whether $N_t^{(e)}/(N_t\\log N_t)$ stabilises at $W/4C$.","supporting_citations":[{"cited_title":"and Fox, E","cited_arxiv_id":null,"evidence_quote":"Supplies the CRM-based sparse graph construction used in Section 4, including the latent multigraph representation and posterior sampling scheme."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides Proposition 11, Theorem 2 and Corollary 1 that transfer tail Lévy-intensity asymptotics into node and edge sparsity and degree-distribution results."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the truncation-error proof strategy and series representation used for Proposition 3.4."},{"cited_title":"H., Goldie, C","cited_arxiv_id":null,"evidence_quote":"Provides the Karamata, Tauberian and conjugate-pair results used to derive intensity asymptotics and inverse Laplace exponents."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the algorithm for sampling exponentially tilted stable random variables used in the total-mass Riemann-sum approximation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies efficient sampling algorithms for exponentially tilted stable distributions, also used in the total-mass approximation."}],"review_version":1}