bibkey: “whitt2007martingale” authors: “Ward Whitt” year: 2007 title: “Proofs of the martingale FCLT” doi: “10.1214/07-PS122” url: “https://arxiv.org/abs/0712.1929v2” claim: “Theorem 2.1(ii) gives a multidimensional Brownian limit from predictable quadratic-covariation convergence and negligible martingale and bracket jumps.” strata_touched: [] license: “citation-only” triage: “anchor”

Martingale limits and a posterior rank process

The primary text is Whitt, Probability Surveys 4 (2007), 268–302, arXiv:0712.1929v2. Theorem 2.1(ii), printed pages 270–271, assumes locally square-integrable vector martingales starting at zero. Their predictable quadratic covariations must converge to a deterministic covariance matrix times time. Expected maximal bracket jumps and expected squared maximal martingale jumps must vanish. The conclusion is weak convergence in Skorohod space to Brownian motion with that covariance matrix. A diagonal matrix gives independent Brownian coordinates. These are classical results.

Chapter 34 of the fluctuation volume uses this result only after replacing one entire posterior rank block by a calibrated independent Bernoulli vector. The two sides have limiting variance clocks v-plus and v-minus times time, zero cross brackets, jumps at most q to the power minus one quarter, and bracket jumps at most one over four square roots of q. The proof also supplies an elementary fourth-moment interpolation argument for tightness. Neither argument treats the exact fixed-size posterior partial sums as a martingale.

A closely related primary source is Leo Pasquazzi, Functional Central Limit Theorems for Conditional Poisson sampling Designs, arXiv:1905.01021v2. Its Theorem 5 states the joint independence mechanism: a deterministic conditional process limit is independent of a convergent data-measurable component. Its sampling FCLTs study Horvitz–Thompson and Hájek processes; the bounded-inclusion-probability setting and the later small-probability extension carry their own covariance, Lindeberg and function-class conditions. The latter extension’s assumption B0 has a positive limiting sample fraction. Those design theorems do not automatically apply to the sparse, data-selected rank strips here. Chapter 34 proves its conditional path convergence and joint independence directly on a separable continuous-path space.

The conditioning representation itself is established rejective-sampling structure; see Arratia–Goldstein–Langholz (2005). The quantitative local probability input is attributed separately to Siripraparat–Neammanee. Classical dyadic maximal estimates, Brownian stationary increments and conditional Gaussian covariances remain auxiliary methods. The new repo-derived statement is the joint actual stationary pair/path capacity process: one growing union label vector, two variance clocks, a common observed rank origin, uniform mean control and exact minimax centers. The lattice equality condition for origin-independent increments follows from that process and logistic variance symmetry. Independence from the absolute fine value at the origin is explicitly false. The inspected sources do not supply this entire model-specific bridge; this bounded comparison is not a global novelty claim.

Threshold groups and several separated rank boundaries

Chapter 36 of the posterior threshold volume applies the martingale theorem to a forward threshold-strip process under the calibrated independent Bernoulli law. All labels at the same observed score jump together. Small individual label increments therefore do not verify the theorem’s jump hypothesis. The proof first establishes uniform convergence of a monotone variance clock to a continuous function. This bounds the largest score-group variance. The independent Bernoulli fourth-moment bound then controls the sum of all grouped fourth moments, implying the required expected maximum squared-jump bound. A deterministic time change reduces the limiting clock to linear time, as required by Whitt’s stated theorem.

The threshold tail is reconstructed from a terminal tail variable and a forward strip martingale with the correct strict/non-strict endpoint conventions. This supplies a right-continuous finite process even when the nonlattice score law has atoms. It does not reverse a step process and assume that its path remains right-continuous. Nonlattice fixed-width localization is attributed to Stone, and the one-union posterior comparison and weighted exact-center bound use the local input described in Siripraparat–Neammanee.

The new repo-derived statement couples the entire compact threshold-tail field to finitely many separated critical-capacity processes in the actual stationary pair/path experiments. The common coarse origin is proved using actual interval-count variance and exact means. Joint Gaussian convergence establishes independence between the threshold field and the primitive boundary processes before their common random translation. Their absolute fine values retain dependence through that shared origin; their relative increment processes have an origin-independent product law. The covariance identities concern the limiting random variables only.

Monotone-clock convergence, martingale functional limits, Brownian tail covariance and stationary increments are classical auxiliary facts. Neither the martingale theorem nor the conditional-design independence mechanism alone provides this actual-model joint law, its grouped-atom verification, or its common exact posterior centers. This attribution concerns the checked proof ingredients and does not certify global originality. The result does not assert a fine-scale process over continuously varying thresholds, a shrinking-threshold tangent limit, or convergence of actual moments.

A Gaussian endpoint inside a collapsing score cluster

The same primary paper also states the classical tightness criterion in Theorem 3.2, printed page 279. Equations (14)–(15) define the ordinary and partition oscillation moduli. Tightness requires the partition modulus to vanish in probability as its mesh parameter tends to zero. The partition uses half-open intervals and controls the spacing of its internal points; the stated half-line version has a separate last-interval convention. An ordered triple collapsing at an interior point encounters at most one partition point. Hence one of its adjacent increments is bounded by a within-partition oscillation. Two adjacent increments that both stay nonzero with positive probability violate this necessary condition. This criterion and its elementary three-point consequence are classical.

Chapter 39 of the posterior threshold volume constructs one fixed nonlattice amplitude with exceptionally accurate rational approximations to its logarithmic jump ratio. On a legal subsequence, an entire rational line of count pairs fits inside the microscopic score window. Uniform Stirling estimates and a Riemann sum, with relative tail control, calculate its comparison mass. Actual one/two-row estimates, compensation and exact untilting transfer the mixed occupancy. One posterior-vector comparison and variance-weighted center estimate then give a Gaussian endpoint at scale sqrt(q/(Q sqrt(lambda))). Two disjoint portions of the same count line produce nondegenerate joint Gaussian increments at three deterministic score locations tending to zero. Their oscillation violates J1 tightness even at that endpoint normalization.

The arithmetic approximation, Poisson asymptotics, array Gaussian limits and topology criterion are established methods. The fixed-amplitude construction, its actual pair/path transport, sharp posterior scale and collapsed-cluster obstruction are the repo-derived result. An endpoint Gaussian alone neither proves tightness nor permits replacing the cluster by one jump. Equal-score labels remain grouped throughout.

For comparison, Borovkov–Borovkov’s A refined version of the integro-local Stone theorem, DOI 10.1016/j.spl.2016.12.004, arXiv:1607.05879v2, imposes the strong nonlattice characteristic-function condition (2), page 3, in Theorem 1. Its Remark 1, page 4, already explains a classical atom obstruction to arbitrarily small intervals. The fixed two-jump compound-Poisson law here is discrete and does not satisfy that strong condition. Its ordinary nonlattice status is insufficient to import the refined shrinking-window estimate. That theorem does not provide the present actual-data, fixed-cardinality posterior cluster or its process conclusion.

These comparisons delimit the inspected inputs and do not certify global originality. Chapter 39 asserts no actual moment convergence, universal amplitude law, or conclusion in another path topology. Its conditional posterior calculations are on the uniform-support-prior space; the fixed-support conclusions are unconditional and use equivariance.

Resolving the cluster by its compensation scale

Chapter 40 of the posterior threshold volume uses the same fixed near-arithmetic amplitude and legal subsequence as Chapter 39. Its exact compensated score step gives a smaller threshold scale, asymptotic to epsilon(1-alpha)sqrt(lambda). On the rational count line, this change of coordinates is exact. The uncompensated central score differs from the correct center by an unbounded number of these smaller scale units, despite a negligible difference at the outer-window scale.

The model-specific result is a compact exact-posterior profile jointly with its full cluster endpoint and the previous threshold/capacity field. Its limiting variance clock is a Gaussian distribution function. The endpoint-centered profile has a Brownian-bridge limit independent of the endpoint; the whole new Brownian motion is independent of the previous primitive field before its random translations. These statements concern one common observation and label vector, not separately sampled marginal limits.

Whitt’s Theorem 2.1(ii), printed pages 270–271, is used on a forward strip martingale under the calibrated independent Bernoulli law, after a deterministic change of time makes its limiting predictable variance linear. Uniform convergence of the monotone variance clock controls the maximum variance of a whole equal-score group. The Bernoulli fourth-moment bound controls the expected maximum squared jump. Initial and terminal tail sums are retained as independent auxiliary coordinates before the full-vector comparison transfers the endpoint and entire profile to the exact posterior. The weighted estimate described in Siripraparat–Neammanee’s note transfers every prefix center uniformly. It is not obtained by multiplying total variation by the number of labels.

The martingale theorem, Gaussian time changes, Brownian-bridge covariance identity, and independence of jointly Gaussian orthogonal coordinates are classical. The repo-derived content is the compensated arithmetic coordinate, actual mixed-row clock, exact posterior process transfer and joint separation from the existing scales. The full joint proof verifies Gaussianity before using small overlap covariances. Its deterministic conditional limit is adjoined to the common data origin using bounded continuous origin tests and bounded Lipschitz process tests, as in the conditional-limit mechanism attributed above to Pasquazzi.

This attribution does not assert global originality. The conclusion is restricted to the fixed Chapter 39 amplitude, each fixed beta and fixed compact parameter sets. It asserts neither actual moment convergence nor a universal microscopic limit for nonlattice amplitudes. The conditional posterior statements are under the uniform support prior; fixed-support conclusions use the unconditional permutation-equivariant law.

The critical arithmetic mesh and Gaussian staircase

Chapter 41 of the posterior threshold volume keeps the fixed amplitude of Chapters 39–40 and changes the legal count intensity to floor(theta Q squared), for fixed positive theta. Its compensated score step gives exact integer coordinates on the count line. The limiting group variances are discrete Gaussian weights, and the complete fixed-size posterior profile is a Gaussian staircase. The endpoint has the same sqrt(q/lambda) normalization as the isolated-atom example, while two nondegenerate group increments still obstruct J1 tightness in the original collapsing score window. An endpoint scale therefore does not determine a one-jump path law.

This is a different actual-model limit from the continuous profile in Chapter 40. Each complete count group now has positive limiting variance. Whitt’s continuous Brownian martingale limit theorem cannot be invoked by claiming its maximum-group-jump condition. Instead, on a fixed compact interval the jump positions are exactly the same finite set of integers at every sample size. The proof retains a left-tail variable, the finite whole-group vector and a right-tail endpoint variable. A bounded-label joint CLT and the continuous fixed-step reconstruction yield the compact J1 conclusion, including integer endpoints. Relative summable Poisson tails establish the full endpoint variance before this finite-vector argument is used.

The normalized weights are the real one-dimensional discrete Gaussian in Agostini and Améndola, Discrete Gaussian Distributions via Theta Functions, SIAM Journal on Applied Algebra and Geometry (2019), DOI 10.1137/18M1164937, arXiv:1801.02373v2, equations (2.2)–(2.3) and Definition 2.3, page 3. In their exponential convention the parameters are u = 0 and B = kappa/(2 pi theta) > 0. Only this explicitly positive, convergent series is used; no arbitrary prescribed-moment existence assertion is needed.

Independent Gaussian sums, discrete Gaussian weights, fixed-step path reconstruction and the Gaussian bridge projection are classical. So are Whitt’s necessary partition-modulus criterion, the exact conditional Bernoulli representation and the local probability bound described in Siripraparat–Neammanee’s note. The repo-derived content is the critical arithmetic sampling sequence, its actual mixed occupancy, exact posterior staircase, separation from the previous field, and an outer-path obstruction despite the matching endpoint normalization. One complete posterior union and its weighted center estimate preserve the common realization throughout.

The variance and covariance expressions describe the limiting Gaussian objects. They assert no actual moment convergence. The result fixes theta, beta, the amplitude and compact intervals; it does not supply a simultaneous law for varying theta or all nonlattice amplitudes. Conditional posterior calculations remain under the uniform support prior, and fixed-support conclusions use unconditional equivariance and one common direction event. The attribution delimits the classical tools and does not certify global originality.

Variable arithmetic phase and a criterion for the one-jump path

Chapter 44 of the posterior threshold volume keeps the same fixed ultra-Liouville amplitude and allows integer intensities satisfying log(Q) = o(lambda) and lambda = o(Q to the fourth power). The arithmetic phase theta = lambda/Q squared need not converge. The normalizing variance is q/lambda times the positive discrete Gaussian series already identified in the preceding section. That series and its small-phase and large-phase asymptotics are classical auxiliary facts, not additional new theorems of the chapter.

The repo-derived conclusion is a single actual-posterior endpoint normalization for the entire permitted intensity range, together with the necessary and sufficient condition theta tending to zero for J1 tightness of the outer-window path. In that regime, the limit is one standard Gaussian jump. This assertion is stronger than identifying the endpoint law, and it does not say that the actual window contains only the central count group. Noncentral groups can contain many labels while their share of normalized variance vanishes.

The new subquadratic estimate bounds the whole noncentral Poisson line relative to its central atom, including half-mean and off-cutoff tails. Actual one/two-row comparisons and calibration turn it into a vanishing relative conditional variance. The classical L2 martingale maximal inequality controls the ordered complete-group prefixes under the independent calibrated law. One full posterior-vector comparison and a separate simultaneous bound on the exact posterior centers then control the entire actual path. No maximal inequality is asserted directly for the fixed-size posterior, and total variation is not multiplied by an unbounded label count.

For the converse, every phase sequence not tending to zero has a subsequence with a positive finite or infinite phase limit. Two disjoint count-line halves then have independent, nondegenerate joint Gaussian limits while their score positions collapse to zero. The classical interior-triple consequence of Whitt’s Theorem 3.2, equations (14)–(15), excludes J1 tightness. Extended-real subsequence extraction combines the subcritical atom, critical discrete Gaussian, and supercritical integral regimes without replacing a changing phase by a fixed one.

A related primary source is Emmanuel Breuillard, Distributions diophantiennes et théorème limite local sur Rd, Probability Theory and Related Fields 132 (2005), 39–73, DOI 10.1007/s00440-004-0388-1. Proposition 3.1, printed page 43, characterizes its Diophantine assumption by a polynomial lower bound on one minus the characteristic-function modulus at large frequencies. Theorem 3.1, printed page 46, gives a local Edgeworth expansion with that assumption and explicit moment and test regularity conditions. At frequencies 2 pi Q/h, the fixed unit compound-Poisson signal increment here has modulus defect at most a constant times Q squared exp(-2 Q to the fourth power). Thus this Diophantine assumption also fails. Ordinary nonlattice status does not authorize importing that quantitative smooth-test expansion.

The source boundaries for the fixed-width local theorem, strong nonlattice refinements, discrete Gaussian series, and conditional Bernoulli representation remain those recorded above. These ingredients do not supply the changing actual-row occupancy, compensation separation, posterior centers, and necessary-and-sufficient path conclusion together. This is a bounded attribution assessment, not a global originality certificate. The theorem asserts no useful finite onset, actual moment convergence, other topology, varying amplitude, or growing-window limit.

Approximation error visible in the original score window

Chapter 45 of the microscopic window-phase volume uses the same fixed ultra-Liouville amplitude but samples at a later intensity, of order log of the inverse exact approximation tail, hence of order Q to the fifth power. The tail eta and the support size q combine into a phase chi = eta sqrt(q), which stays in a fixed positive compact interval for the specified rounded intensity construction. Bounded rounding errors do not imply that chi converges or that an arbitrary chosen phase is attainable.

The arithmetic integer gap still excludes off-line count pairs. In contrast to the earlier collapsing regimes, only a finite resolved segment of the count line now lies inside the microscopic score window. The positive score step is governed by the approximation error rather than compensation. Its mesh in the original window coordinate tends to zero. A uniform two-Poisson Stirling estimate, exact untilting, actual one/two-row comparisons and calibration produce a truncated Gaussian variance clock on that original interval.

Whitt’s classical Theorem 2.1(ii) again supplies the auxiliary functional limit after inverse-clock time change and Brownian continuation. Uniform monotone clocks control the largest whole-group variance; the independent Bernoulli fourth-moment bound controls whole-group jumps. The full-window posterior-vector comparison and a separate simultaneous exact-center bound then transfer the complete path. The endpoint variance is the mass of the truncated clock, not the variance of the entire count line. The endpoint and bridge are projections of the same limiting Brownian motion.

The repo-derived result is continuous-path subsequential Gaussian convergence at the original microscopic scale, and C-tightness along the entire rounded intensity sequence. Every phase-convergent subsequence has the stated clock; deterministic phase normalization gives a standard Gaussian endpoint along the full sequence. This does not extend the Chapter 44 criterion past its explicit intensity range. It instead separates two questions: whether the score window isolates one count line, and whether the relevant part of that line collapses to a single window location.

The Gaussian time change, martingale criterion, Stirling expansion, compact-subsequence argument and Gaussian bridge projection are classical auxiliary tools. The existing source boundaries for quantitative nonlattice local limits remain in force: the fixed increment distribution still fails the strong nonlattice and polynomial Diophantine assumptions discussed above. The present restricted sampling sequence and explicit count-line calculation do not establish fine-scale smoothing at every intensity.

This bounded attribution is not a global originality certificate. The result fixes the amplitude and beta and uses the stated intensity construction. It makes no actual-moment, useful finite-onset, arbitrary phase-attainment, all-amplitude, growing-window or efficiency claim. Conditional posterior arguments remain under the uniform support prior; fixed-support laws and the one common direction event transfer the whole process and its projections together.

Finite posterior meshes and compact-interval endpoints

Chapter 46 of the window-phase volume uses the same fixed Liouville amplitude at an intensity larger than the reopening scale by three times log(Q) divided by the exponential growth rate of q. The exact compensated spacing in the original score-window coordinate now stays bounded above and away from zero. Only finitely many count-line groups can enter that window. All of them have the same limiting normalized variance, including groups currently outside the window that are retained in one enlarged posterior vector.

The finite vector central limit theorem, conditional-Bernoulli representation, absolute Bernoulli local estimate and endpoint-fixing time changes are classical. The repository-derived content connects them to the recalculated actual pair/path model: uniform bounded-index point masses, actual mixed occupancy, calibration, a complementary variance reserve, one whole-vector comparison and simultaneous exact centers. A uniformly Lipschitz family of changing staircase maps gives a moving Gaussian-law approximation even when the path laws are not tight. This is not convergence to one fixed limiting process.

The topology is explicitly the ordinary J1 topology on the closed interval from minus one to one. Interior jumps approaching either endpoint are obstructed, while a jump exactly at the right endpoint is permitted. The original score window excludes its exact left endpoint. The proof derives both endpoint restrictions directly from the endpoint-fixing time-change definition and the one-sided limits of a cadlag path. It does not use Whitt’s half-line last-interval exception as a compact right-endpoint tightness theorem. Nor does it apply the Brownian martingale FCLT to the finite mesh: its group jumps have nonvanishing limiting variances.

At a reciprocal-integer limiting spacing, the two possible endpoint contacts share the same center and spacing phases. Their inclusion and interior/exterior status must be checked together. A phase limit alone does not determine a unique staircase law. The theorem does not claim that every classified boundary pattern, or a nontight resonant pattern, is attained by the legal rounded intensities. A sufficiently large fixed positive intensity shift does give a proved legal regime containing only the central group; its rounding bound is explicit.

A related primary predecessor is D. Ferger and D. Vogel, “Weak convergence of the empirical process and the rescaled empirical distribution function in the Skorokhod product space,” arXiv:1506.04324v1, Theorem 2.1 and Condition C.1. Its fixed iid distribution, specified one-sided derivatives and local/global scaling yield an independent transformed Brownian bridge and two-sided Poisson process. Its convention permits an atom at the localization point, with the left limit used on the negative side. Those facts are relevant precedents for local process limits and endpoint conventions. The source works on the real-line Skorokhod space; restriction at a limiting jump does not automatically give closed-interval J1 convergence. It does not supply the changing actual score law, constrained posterior centers or finite-mesh criterion proved here.

The stronger quantitative nonlattice assumptions discussed earlier in this note remain absent. The result uses explicit count-line point probabilities instead of a universal microscopic score density. This bounded source comparison is not a global originality certificate. The theorem fixes amplitude and beta, makes no actual-moment or finite onset claim, and does not assert another topology, arbitrary phase attainment, adaptive parameters, growing windows or efficiency.

Dense central meshes and homogeneous posterior noise

Chapter 47 of the window-phase volume considers intensities between the reopening scale and the finite-mesh scale. The coefficient of log(Q) may vary arbitrarily inside a fixed compact subinterval of (0,3). The exact compensated window mesh tends to zero, while the number of selected count groups diverges. Every selected standardized count point simultaneously tends to the center of the two-Poisson profile. These are separate conditions: a dense window mesh alone would not imply a spatially constant variance clock.

The repo-derived conclusion is full-sequence homogeneous Brownian noise for both actual pair/path experiments at the exact-mesh scale, with the same endpoint and its independent Gaussian bridge projection. All centers are the exact fixed-cardinality posterior centers. The proof establishes a uniform relative actual occupancy estimate over the growing selected integer set. Its union bound costs only a fixed power of Q. The exact floor-count identity then makes the normalized variance clock linear, with no convergent rounding-phase assumption. One full posterior-vector comparison and a separate simultaneous weighted center estimate transfer the grouped auxiliary functional limit at this smaller scale. The old whole-line normalization gives the zero process on the same window. The equivalent simpler normalization retains exact q; dropping its bounded oscillating factor is not a proved equivalence.

Whitt’s Theorem 2.1(ii) supplies the classical functional-limit step. The proof checks whole-group fourth moments, the expected maximum squared group jump and predictable bracket jumps. Independent Brownian continuation after the window endpoint permits literal use of the half-line theorem, followed by restriction at a continuous limiting endpoint. No martingale property is imposed on the exact conditioned posterior and no independence is imposed on actual rows. The Bernoulli local estimate and conditional representation cited above remain the classical tools for the posterior comparison.

U. Einmahl and D. M. Mason, “Gaussian approximation of local empirical processes indexed by functions,” Probability Theory and Related Fields 107 (1997), 283–311, Theorem 1.1, provides a relevant general local-Gaussian mechanism. Its iid sample has a fixed law; the local mass and conditional local laws, envelope conditions and asymptotic equicontinuity satisfy its assumptions (A), (S) and (F). At vanishing local mass the limiting covariance is the local second-product integral. The general theorem does not require a density, so discrete observations alone do not exclude it. It does not establish the present changing actual-row comparison or the fixed-cardinality posterior center estimate. Those obligations remain necessary even when a general array theorem supplies the final Gaussian step.

L. Pasquazzi, “Functional Central Limit Theorems for Conditional Poisson sampling Designs,” arXiv:1905.01021v2, Theorem 6, is a related conditional bounded-Lipschitz limit theorem. Its assumption A2* requires every canonical inclusion probability to have a fixed positive lower bound with high probability. The complete array here has minimum probability at most q/M, tending to zero. Restriction to the central window does not make its posterior label total fixed; the complement conditioning ratio must still be handled. This comparison concerns that theorem and does not dismiss the paper’s other designs allowing smaller probabilities.

The fixed Liouville increment law still has super-polynomially close Fourier resonances and fails the quantitative nonlattice hypotheses discussed earlier. The local homogeneous posterior noise therefore does not establish a generic microscopic score density. The bounded primary-source comparison is not a global originality certificate. Amplitude, beta, exponent compact and remainder bounds stay fixed; there is no claim at the exponent endpoints, of actual moment convergence, arbitrary phase attainment, useful finite onset, growing windows, old-field jointness, adaptation or efficiency.