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Finite-History Outage Certification for Partially Observed Fluid Antenna Receivers

Yuanjun Shen

arXiv:2610.21252v1eess.SPcs.ITmath.ITstat.AP

TL;DR

Finite histories leave uncertainty about the covariance underlying a partially observed receiver’s service-position decision. The paper separates history and integration uncertainty to certify a frozen rule; in its 32-position, five-probe example, all 282 risk groups are below target, with a separate comparison favoring the rule over two observed-port controls.

  • Problem

    Partial observations can select service ports, but finite historical data leave covariance uncertain, so a rule meeting its outage target under one fitted covariance may fail under another plausible covariance.

    Source: §I · p. 1

  • Method

    The procedure separates finite-history parameter uncertainty from integration uncertainty and combines Gaussian domination, shared conditional-probability anchors, and selector enclosures for continuous-region risk bounds.

    Source: §I · p. 2

  • Results

    All 282 risk groups lie below the outage target of 0.001; the current-rule risk interval is [7.6820, 9.1266] × 10^-4, with lower outage than two fixed observed-port controls.

    Source: A. Fixed Benchmark and Complete… · p. 7, §VIII · p. 9

  • Takeaways & Limitations

    One new independent history reused the certificate without new risk samples while retaining the rule and operating threshold.

    Source: §VIII · p. 9

    The numerical experiment does not qualify the same rule at a stricter outage probability than its fixed target of 0.001.

    Source: A. Fixed Benchmark and Complete… · p. 7

Abstract

from arXiv · show

Fluid antenna receivers can select a service position from partial port observations, but finite historical data leave uncertainty in the covariance used for that decision. We develop deployment-before-use reliability certification for a frozen partial-observation rule. A proper predictive density defines a history-based parameter region. Gaussian covariance envelopes, shared conditional-probability bounds and selector enclosures provide simultaneous risk bounds over continuous parameters and discontinuous port decisions. Separate history-coverage and integration budgets limit erroneous positive certification to 0.05 for a single qualification procedure under the stated stationary model and numerical assumptions. Confidence-set inclusion allows a new independent history to reuse the risk computation with the rule unchanged. In a structured Gaussian example with 32 candidate positions, five noisy probes and target outage 0.001, all 282 risk groups are qualified; the largest upper bound is 0.000999748. A separate coupled comparison gives a current-rule outage interval of $[7.6820,9.1266]\times10^{-4}$ and resolves lower outage than two observed-port controls using the same historical fit. One independently recalibrated history reuses the certificate without additional risk sampling. Unresolved and adverse-member controls delimit this reuse.

I. INTRODUCTION

The introduction motivates certification because partial observations leave service selection dependent on uncertain channel statistics. It presents a frozen-rule qualification procedure and a structured receiver setting for evaluating it.

I. INTRODUCTION

  • Fluid antenna systems provide alternative receiving positions with one radio-frequency chain, while partial-observation methods use spatial dependence to select even unprobed service ports.The introduction frames these methods as a response to the probing burden of observing every candidate position.Source: §I · p. 1
  • A fitted covariance may meet an outage target even when another statistically plausible covariance gives the frozen rule a larger outage probability.The paper asks whether the rule can be qualified before use while controlling erroneous positive certification.Source: §I · p. 1
  • The deployment-before-use procedure separates finite-history parameter uncertainty from integration uncertainty and controls erroneous positive certification for a frozen receiver rule.Its risk is averaged over future fading and probe noise, not a single block’s conditional probability.Source: §I · p. 2
  • Joint Gaussian domination, shared conditional-probability anchors, and selector enclosures bound risk across continuous parameter cells and discontinuous port decisions.The construction also uses bounded-variable integration and explicit numerical payments.Source: §I · p. 2
  • Fixed-policy recalibration checks whether a new history’s confidence set lies inside the already certified region, allowing the same rule to reuse its risk computation.The demonstrated example also includes unresolved and adverse-member controls.Source: §I · p. 2

A. Receiver and Observations

  • The receiver model uses a fixed transmit antenna, a single-RF-chain terminal, spatially correlated fading, and a channel that remains constant during probing and service selection.The unknown covariance parameter belongs to a specified angular-spectrum family.Source: A. Receiver and Observations · p. 2
  • Historical data fit a service rule that is frozen after fitting; its canonical selector minimizes disk probability over all candidate positions, including unobserved ones.The evaluated covariance family uses continuous mean-angle and spread parameters over a specified finite-angle domain.Source: A. Receiver and Observations · pp. 2–3

B. Two Probability Objects

The section distinguishes future policy outage from block-conditional outage and from certification error. It defines qualification over a history-supported parameter region and bounds the numerical evidence to a specified operating point and configuration.

B. Two Probability Objects

  • Policy risk under parameter ϕ averages outage over new block fading and probe noise, rather than conditioning on one realized observation or measuring certification error.The outage threshold is x = θ/γ, with θ defined from the spectral-efficiency target and average SNR.Source: B. Two Probability Objects · p. 3
  • Qualification verifies an outage upper bound throughout a history-supported parameter region, while reuse tests whether a new independent history supports that same region and unchanged policy.The section assigns qualification and reuse to separate stages of the method.Source: B. Two Probability Objects · p. 3
  • The numerical evidence fixes target probability pt = 0.001 and operating threshold x = 0.108; it does not qualify this same rule at a stricter probability.The target is a study input, not a communication-standard requirement.Source: B. Two Probability Objects · p. 3
  • The arguments apply to finite candidate counts and fixed probe sets, but each different configuration needs new covariance and selector enclosures; no port-count convergence or minimum-probe theorem is provided.The evidence is a complete worked configuration, not a general optimization of port or probe counts.Source: B. Two Probability Objects · p. 3

A. Finite-History Parameter Coverage

A proper predictive density defines a finite-history region of plausible parameters, while coverage bounds control the chance that the true parameter lies outside it. Certification combines this history coverage with simultaneous computational risk bounds and issues only when every eligible risk bound is below the target.

A. Finite-History Parameter Coverage

  • A proper density q, fixed before observing the history, defines a density-ratio region whose coverage follows from Markov’s inequality.The predictor can be a finite mixture even when the true parameter is not among its components.Source: A. Finite-History Parameter Coverage · p. 3
  • The likelihood-ratio coverage argument permits fitting the receiver policy on the same history because risk evaluation conditions on its realized, frozen rule.Freely fitting the numerator on those calibration observations does not preserve the integral identity.Source: A. Finite-History Parameter Coverage · p. 3
  • The retained parameter region is covered by continuous regions, and excluded boxes receive verified lower bounds to establish eligibility.Geometric coverage must include boundaries; this controls an intersection event, not conditional coverage given eligibility.Source: A. Finite-History Parameter Coverage · pp. 3–4, B. Issuance and Computational… · p. 4

B. Issuance and Computational Uncertainty

  • Theorem 1 bounds erroneous positive certification by the history-coverage and computational-failure budgets under its stated model and computation assumptions.Issuance requires the coverage event and every risk upper bound to fall below the target.Source: B. Issuance and Computational… · p. 4

C. A Prespecified Independent Contingency

A prespecified independent contingency can finish unresolved integration groups without pooling their earlier samples. The risk bounds rely on Gaussian covariance envelopes that apply across continuous parameter cells.

C. A Prespecified Independent Contingency

  • The contingency uses fresh independent inputs for unresolved groups, retains accepted primary bounds, and never pools unresolved primary moments with new samples.The two-family failure probability is bounded by δ1 + δ2, conditional on the history and primary results.Source: C. A Prespecified Independent… · p. 4
  • The implemented allocations are (α, δ1, δ2) = (0.024, 0.025, 0.001), giving a 0.05 single-issuance failure budget across 282 groups and three looks.Acceptance is tested only at prespecified looks; exhausting the time cap returns unresolved.Source: C. A Prespecified Independent… · p. 4

A. Joint Gaussian Domination

  • A Taylor enclosure and remainder bound covariance variation in Loewner order across each continuous parameter cell.Source and assembly errors must also be included.Source: A. Joint Gaussian Domination · p. 4
  • A positive-semidefinite observation-block correction gives all ports a common observation covariance, enabling an integrated joint-density bound via conditional disk probabilities.The bound factors the upper Gaussian into its observation marginal and conditional density.Source: A. Joint Gaussian Domination · p. 4

B. One Observation Reference per Group

One observation reference density per group supports a risk envelope over all covered continuous parameter cells. Shared conditional-probability bounds and selector enclosures reduce computation while retaining possible minimizing ports.

B. One Observation Reference per Group

  • The whole-group envelope bounds risk over every parameter in a covered group, with the maximum taken inside the integral rather than only at cell centers.The shared reference density uses a covariance that dominates each cell’s observation covariance.Source: B. One Observation Reference per Group · pp. 4–5

C. CDF Sharing and Selection Enclosure

  • A positive logarithmic series with explicit tails evaluates very small disk probabilities without cancellation.The disk probability is for a complex Gaussian characterized by its mean and variance.Source: C. CDF Sharing and Selection Enclosure · p. 5
  • A shared conditional anchor reduces CDF calls without equating the probabilities of different cells.Its density ratio is bounded on the disk.Source: C. CDF Sharing and Selection Enclosure · p. 5
  • The selector enclosure retains ports whose lower log-outage bound is no greater than the smallest upper bound, including possible ties.Maximizing over this set bounds the canonical rule without assuming continuity of its selected index.Source: C. CDF Sharing and Selection Enclosure · p. 5

D. Numerical Payments and Simultaneous Integration

The integration procedure combines cushioned sampled integrands with fixed-look empirical Bernstein bounds and simultaneous error allocations. Its guarantees depend on the stated model, independent-input, and arithmetic assumptions.

D. Numerical Payments and Simultaneous Integration

  • The sampled variable includes CDF and integrand cushions, input-transform coupling, and all possibly selected ports; uncertain inputs return the proven global cap.The sampled integrand includes a 10^-6 cushion on regular compact inputs, under the stated verification conditions.Source: D. Numerical Payments and Simultaneous… · p. 5
  • The scaled empirical Bernstein bound uses the sample mean and unbiased variance at a prespecified look, with a union bound over potential groups and looks.The fixed-look procedure permits acceptance only at allocated looks and requires fresh independent inputs for its contingency guarantee.Source: D. Numerical Payments and Simultaneous… · p. 5
  • Exact integer accumulation and outward evaluation support the arithmetic guarantee, which assumes the structured model, ideal independent inputs, and stated arithmetic semantics.The assumptions are not a general proof of a computer or pseudorandom generator.Source: D. Numerical Payments and Simultaneous… · p. 5
  • A final unresolved primary look can trigger a second family if time remains; unresolved contingency results end qualification, and exhausting the shared time cap returns unresolved.The mathematical budget and time/sample exit rules are fixed before confirmation.Source: D. Numerical Payments and Simultaneous… · p. 5

V. FIXED-POLICY RECALIBRATION

Fixed-policy recalibration reuses an existing risk computation when a new history’s confidence set is verified inside the original certified region and the unchanged rule and threshold still meet the target. The reuse keeps the original computational failure budget, while repeated announcements require separately allocated coverage budgets.

V. FIXED-POLICY RECALIBRATION

  • C(D1) is formed from a proper predictive density evaluated on independent calibration data, giving the true parameter a coverage guarantee of at least 1 − α.The calibration data are kept separate from the observations used to form the predictive density.Source: §V · pp. 5–6
  • A new certificate requires C(D1) ⊆ O and max_g U_g < p_t; under the stated conditions, erroneous issuance is bounded by α + δ.The policy π0, operating threshold, and certified region O remain fixed.Source: §V · p. 6
  • Inclusion is checked over every continuous leaf outside O, with deterministic refinement until all complement boxes are excluded.Center, gradient, remainder, source, inverse, and empirical-covariance errors are accounted for before exclusion.Source: §V · p. 6
  • Reusing the computation is not a new risk integration, but repeated recalibrations still require a total coverage budget across histories.Combining the original and new issuance statements uses separate coverage allocations; for L prespecified histories, the bound includes the sum of their allocations plus δ.Source: §V · p. 6

VI. NUMERICAL RESULTS

The fixed benchmark qualifies all 282 risk groups below the 0.001 target, with the maximum upper bound narrowly below it. A separate coupled comparison reports lower outage than two observed-port controls at one true parameter under the same historical fit.

A. Fixed Benchmark and Complete Certificate

  • The benchmark uses 32 candidate positions and five noisy probes, leaving 27 unobserved positions eligible for service.The five-probe layout was retained for the directional-family study; the study does not treat 32 as a demonstrated continuous-aperture limit or optimize probe count.Source: A. Fixed Benchmark and Complete… · p. 6
  • 0.0009997474561274761 was the maximum upper bound across all 282 continuous risk groups, below the 0.001 target.Figure 3 shows final simultaneous upper bounds normalized by the target; blue circles are primary bounds and orange triangles are contingency bounds.Source: A. Fixed Benchmark and Complete… · pp. 6–7, Figure 3 · p. 7
  • The target p_t = 10^-3 is a moderate rare-outage benchmark, and this experiment does not qualify the same rule at a stricter probability.The target was fixed before final confirmation; x = 0.108 was selected on separate design data.Source: A. Fixed Benchmark and Complete… · p. 7

B. Independent Performance Controls

  • At (30°, 15°), the current-rule risk interval was [0.0007682054, 0.0009126514], with negative paired risk-difference upper bounds against both observed-port controls.The upper bounds were −0.0001299014 against the strong five-observed-port rule and −0.0000492295 against the fixed six-observed-port rule; this comparison has a separate 0.05 failure budget.Source: B. Independent Performance Controls · p. 7
  • The observed-port comparison isolates the benefit of permitting unobserved service positions under the stated model, not a ranking of all probe schedules.The controls use the same historical fit and restrict service to observed positions; the comparison does not establish that a larger nested observation set worsens the optimal receiver.Source: B. Independent Performance Controls · p. 7

C. Fixed-Policy Recalibration Controls

The section distinguishes fixed-policy reuse from fresh risk qualification, then accounts separately for online actions and direct-label validation. The certificate bounds risk across the history-supported region, while its deployment costs and validation alternatives remain distinct.

C. Fixed-Policy Recalibration Controls

  • One new history’s confidence set falls inside the certified region, enabling reuse without new risk draws; an outside-member control leaves transfer unavailable, and an adverse member does not prove the true parameter unsafe.The in-region history records 169.967 s of preparation and verification; the off-grid member’s oracle-risk lower bound is 0.0011689753, above the 0.001 target.Source: C. Fixed-Policy Recalibration Controls · p. 7

D. Online Observation and Selection Costs

  • 87.810% of recorded blocks select an unobserved service position, and 95.760% change state after the last probe, so five initial probes do not capture every service action.The record covers 20,000 blocks and does not establish a switching-time or energy advantage.Source: D. Online Observation and Selection… · p. 8

E. Direct-Label Validation

  • At 25,000 independent service-event labels, the exact-binomial test’s pass probability is approximately 8.77%–27.64% across the comparison risk interval, versus 80.71%–100% at 800,000 labels.These are projections, not new label experiments or universal sample-size requirements; model-based qualification instead uses 25,000 historical port observations, and those budgets are not directly comparable as an acquisition-saving factor.Source: E. Direct-Label Validation · p. 8

A. What the Certificate Adds

  • The certificate checks outage throughout history-supported covariance uncertainty, while its continuous-parameter envelopes and selector enclosures handle discontinuous port decisions.The 282 simultaneous bounds support qualification across the retained region; the coupled comparison serves a separate purpose and has a separate failure budget.Source: A. What the Certificate Adds · p. 8

B. Operational Tradeoffs

The operational comparison favors allowing service at unobserved positions, while emphasizing that selection, activation, and training have costs. Reuse retains the frozen rule, but the guarantees depend on stated model and numerical assumptions.

VIII. CONCLUSION

  • Confidence-set inclusion reuses the risk computation without changing the rule or threshold, but each new announcement consumes history-coverage budget and valid coverage need not yield a positive decision.The original and new announcements together have a stated union-bound budget of 0.074; measured acquisition costs and channel-model validation are needed to establish physical throughput or energy benefit.Source: C. Applicability and Reuse · p. 9, §VIII · p. 9

C. Applicability and Reuse

  • The 0.05 erroneous-positive-certification budget applies under the stated stationary Gaussian model, independent-data conditions, and numerical contracts; changing fitted coefficients, threshold, or selection logic requires new qualification.Physical reliability claims also require separate validation for channel aging, model mismatch, and acquisition costs.Source: C. Applicability and Reuse · p. 9

VIII. CONCLUSION

  • The all-position rule has lower outage than both fixed observed-port controls in the coupled comparison, with risk interval [7.6820, 9.1266] × 10^-4.The observed-six control is not an optimized six-probe receiver; more eligible actions increase selection work, and unobserved service may require additional activation or training.Source: B. Operational Tradeoffs · p. 8, §VIII · p. 9

APPENDIX B CONDITIONAL ANCHOR BOUND

The appendix bounds one conditional disk probability by another using a density-ratio bound. It also controls inverse-approximation and source-perturbation errors before continuous-box exclusion.

APPENDIX B CONDITIONAL ANCHOR BOUND

  • The conditional anchor bound upper-bounds disk probability Fc by e^Λc Fa, then intersects that bound with Fc ≤ 1 without requiring equal means.The density-ratio bound combines quadratic and linear terms over the disk.Source: Appendix B · p. 9

APPENDIX C INVERSE RESIDUAL AND SOURCE PERTURBATION

  • The inverse-residual bound separates inverse residual, covariance-source, and short matrix-product errors before continuous-box exclusion.It applies when the stored inverse approximation has residual r < 1 and the source perturbation satisfies εβ/(1 − r) < 1.Source: Appendix C · p. 9

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