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arxiv: 2604.03868 · v1 · submitted 2026-04-04 · 📡 eess.SY · cs.RO· cs.SY

Risk-Constrained Belief-Space Optimization for Safe Control under Latent Uncertainty

Clinton Enwerem , John S. Baras , Calin Belta This is my paper

Pith reviewed 2026-05-13 16:55 UTC · model grok-4.3

classification 📡 eess.SY cs.ROcs.SY
keywords risk-constrained controlbelief-space planningCVaRMPPIlatent uncertaintysafe controlprobabilistic safety
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The pith

Enforcing a CVaR constraint on safety margins within belief-space MPPI control delivers a probabilistic safety guarantee under latent uncertainty.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a risk-sensitive belief-space Model Predictive Path Integral controller for partially observed systems whose dynamics and safety depend on an unobserved latent parameter. It augments standard MPPI planning with a Conditional Value-at-Risk constraint applied to the trajectory safety margin, computed over the belief distribution maintained by the controller. The resulting method optimizes a risk-regularized performance objective while explicitly limiting tail risk of safety violations. Three formal properties follow: the CVaR constraint yields a probabilistic safety guarantee, the controller reverts to the risk-neutral optimum as the risk weight approaches zero, and a union-bound argument lifts the per-horizon guarantee to repeated receding-horizon executions. In a vision-guided dexterous stowing simulation with pose uncertainty exceeding clearance limits, the approach reaches 82 percent success with zero contact violations at high risk aversion, outperforming risk-neutral and chance-constrained baselines.

Core claim

The central claim is that a risk-constrained belief-space MPPI controller, which plans trajectories under a belief distribution over a latent parameter while enforcing a CVaR constraint on the induced safety-margin distribution, simultaneously achieves a probabilistic safety guarantee and recovers the risk-neutral optimum as risk sensitivity vanishes.

What carries the argument

The CVaR constraint applied to the distribution of trajectory safety margins induced by the belief over the latent parameter, inside the receding-horizon MPPI optimization loop.

If this is right

  • The CVaR constraint directly implies a probabilistic safety guarantee for each planning horizon.
  • The controller recovers the risk-neutral optimum as the risk weight in the objective tends to zero.
  • A union-bound argument extends the per-horizon safety guarantee to cumulative safety across repeated solves.
  • In the dexterous stowing task, high risk aversion produces 82 percent success with zero exterior contact forces, versus 55 percent and 50 percent for the risk-neutral and chance-constrained baselines.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Tuning the risk weight offers a continuous dial between performance and tail-risk aversion that may generalize to other belief-space planners.
  • The same CVaR mechanism could be inserted into alternative sampling-based or gradient-based belief-space methods beyond MPPI.
  • Hardware experiments with real-time vision-driven belief updates would test whether the simulated safety improvement persists under sensor noise and model mismatch.

Load-bearing premise

The maintained belief distribution over the latent parameter accurately represents the true uncertainty, so that the computed CVaR on safety margins correctly bounds the actual tail risk.

What would settle it

Repeated closed-loop trials in which the empirical frequency of safety-margin violations exceeds the probabilistic bound implied by the chosen CVaR level and confidence would falsify the guarantee.

Figures

Figures reproduced from arXiv: 2604.03868 by Calin Belta, Clinton Enwerem, John S. Baras.

Figure 1
Figure 1. Figure 1: Running Example. (a). A robotic manipulator inserts a grasped object into an occupied receptacle with uncertain pose. (b). With low risk aversion (βs = 0.50), the controller allows trajectories that produce catastrophic contact forces. (c). With high risk aversion (βs = 0.95), the CVaR safety constraint maintains safer insertion margins and avoids contact. A. Related Work 1) Risk Measures & Risk-Sensitive … view at source ↗
Figure 2
Figure 2. Figure 2: Object Stowing Task. FR, FH, and FC denote the robot base, hand, and camera frames. Bobj is the object to be transported, Bslot denotes the receptacle, and Benv represents previously stowed objects inside the receptacle. The goal region Xgoal specifies admissible placement poses within the receptacle, and all frames are expressed with respect to FR. and taking the complement yields (24). Remark 5. The boun… view at source ↗
Figure 3
Figure 3. Figure 3: Representative Trajectory. Colored bands indicate task phases ( approach and insert). εp is the stowing proximity threshold. Both CVaR-MPPI (βs ≥ 0.90) configurations reach the goal while maintaining positive safety margins, whereas βs=0.50 and CCMPPI (δH=0.05) fail to complete insertion. risk thresholds (βc = βs ∈ {0.5, 0.9, 0.95}) and fixed cost￾risk weight λr=0.5, contrasting them with the CCMPPI base￾l… view at source ↗
Figure 4
Figure 4. Figure 4: shows the CVaR safety constraint evolution for each configuration. Because CVaRβs (−MH) evaluates the worst (1−βs) tail, a higher βs focuses on a narrower extreme, making the constraint harder to satisfy and moving the trace closer to the boundary. All three configurations maintain feasibility (≤ 0) throughout approach and insertion. The critical difference is that the loose βs=0.50 constraint per￾mits tra… view at source ↗
read the original abstract

Many safety-critical control systems must operate under latent uncertainty that sensors cannot directly resolve at decision time. Such uncertainty, arising from unknown physical properties, exogenous disturbances, or unobserved environment geometry, influences dynamics, task feasibility, and safety margins. Standard methods optimize expected performance and offer limited protection against rare but severe outcomes, while robust formulations treat uncertainty conservatively without exploiting its probabilistic structure. We consider partially observed dynamical systems whose dynamics, costs, and safety constraints depend on a latent parameter maintained as a belief distribution, and propose a risk-sensitive belief-space Model Predictive Path Integral (MPPI) control framework that plans under this belief while enforcing a Conditional Value-at-Risk (CVaR) constraint on a trajectory safety margin over the receding horizon. The resulting controller optimizes a risk-regularized performance objective while explicitly constraining the tail risk of safety violations induced by latent parameter variability. We establish three properties of the resulting risk-constrained controller: (1) the CVaR constraint implies a probabilistic safety guarantee, (2) the controller recovers the risk-neutral optimum as the risk weight in the objective tends to zero, and (3) a union-bound argument extends the per-horizon guarantee to cumulative safety over repeated solves. In physics-based simulations of a vision-guided dexterous stowing task in which a grasped object must be inserted into an occupied slot with pose uncertainty exceeding prescribed lateral clearance requirements, our method achieves 82% success with zero contact violations at high risk aversion, compared to 55% and 50% for a risk-neutral configuration and a chance-constrained baseline, both of which incur nonzero exterior contact forces.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript proposes a risk-sensitive belief-space Model Predictive Path Integral (MPPI) controller for partially observed systems whose dynamics and safety margins depend on a latent parameter represented by a belief distribution. The framework optimizes a risk-regularized objective subject to a Conditional Value-at-Risk (CVaR) constraint on trajectory safety margins over the receding horizon. It claims three properties: (1) the CVaR constraint implies a probabilistic safety guarantee, (2) the controller recovers the risk-neutral optimum as the risk weight tends to zero, and (3) a union-bound argument extends the per-horizon guarantee to cumulative safety. In physics-based simulations of a vision-guided dexterous stowing task with pose uncertainty, the method reports 82% success with zero contact violations at high risk aversion, compared to 55% and 50% for a risk-neutral MPPI and a chance-constrained baseline.

Significance. If the three properties are rigorously established and the simulation outcomes prove robust, the work provides a concrete method for incorporating tail-risk constraints into belief-space planning, offering a middle ground between risk-neutral and fully robust controllers. The explicit use of CVaR on safety margins and the recovery property are potentially useful for robotics applications where latent uncertainty (e.g., object pose or contact parameters) must be handled probabilistically without excessive conservatism.

major comments (3)
  1. [Abstract] Abstract: The three claimed properties are stated without derivation details or error analysis. In particular, property (1) asserts that the CVaR constraint on the trajectory safety margin implies a strict probabilistic safety guarantee, yet the MPPI implementation relies on finite Monte-Carlo sampling from the belief; the skeptic note correctly identifies that empirical CVaR estimates converge slowly for rare tail events and can be biased low, breaking the direct implication from CVaR ≤ 0 to the claimed probability bound. The manuscript must supply the exact statement of the guarantee together with a quantitative bound on the sampling error.
  2. [Abstract] Simulation results (final paragraph): The 82% success rate with zero contact violations is reported for 'high risk aversion,' but the manuscript does not specify the MPPI sample count, the number of particles used for CVaR estimation, the exact form of the belief distribution, or the risk-aversion parameter values. Without these, the empirical comparison to the 55% and 50% baselines cannot be assessed for statistical significance or sensitivity to sampling budget, undermining the claim that the method achieves the stated safety-performance trade-off.
  3. [Theoretical Properties] Property (3) and receding-horizon implementation: The union-bound argument is invoked to extend the per-horizon CVaR guarantee to cumulative safety across repeated solves. Because each solve re-samples from the updated belief and the horizon overlaps, the safety-margin random variables are statistically dependent; the manuscript must show that the union bound remains valid or provide a tighter concentration result that accounts for this dependence and the finite-sample CVaR approximation.
minor comments (2)
  1. [Method] Notation: The precise definition of the 'trajectory safety margin' (scalar or vector) and how it is evaluated under sampled latent parameters should be stated explicitly before the CVaR constraint is introduced.
  2. [Abstract] The abstract refers to 'vision-guided dexterous stowing' but provides no figure or description of the observation model that generates the belief; a brief statement of the sensor model would improve readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments on our manuscript. We address each major comment point by point below, providing clarifications where possible and indicating the specific revisions we will make to strengthen the presentation of the theoretical properties and empirical results.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The three claimed properties are stated without derivation details or error analysis. In particular, property (1) asserts that the CVaR constraint on the trajectory safety margin implies a strict probabilistic safety guarantee, yet the MPPI implementation relies on finite Monte-Carlo sampling from the belief; the skeptic note correctly identifies that empirical CVaR estimates converge slowly for rare tail events and can be biased low, breaking the direct implication from CVaR ≤ 0 to the claimed probability bound. The manuscript must supply the exact statement of the guarantee together with a quantitative bound on the sampling error.

    Authors: We agree that the abstract statements of the three properties would benefit from greater precision and supporting analysis. In the revised manuscript, we will expand the relevant section to include the exact mathematical statements of all three properties together with brief derivations. For property (1), we will explicitly distinguish the population-level guarantee (CVaR constraint with respect to the true belief distribution) from the finite-sample implementation, and we will add a quantitative error bound on the empirical CVaR estimator using standard concentration results for CVaR (e.g., via empirical-process theory or Hoeffding-type inequalities adapted to tail expectations). This will make the sampling-error caveat transparent. revision: yes

  2. Referee: [Abstract] Simulation results (final paragraph): The 82% success rate with zero contact violations is reported for 'high risk aversion,' but the manuscript does not specify the MPPI sample count, the number of particles used for CVaR estimation, the exact form of the belief distribution, or the risk-aversion parameter values. Without these, the empirical comparison to the 55% and 50% baselines cannot be assessed for statistical significance or sensitivity to sampling budget, undermining the claim that the method achieves the stated safety-performance trade-off.

    Authors: We accept that the abstract omitted critical implementation parameters needed for reproducibility and statistical assessment. In the revised version, we will augment the simulation paragraph with the following details: MPPI employs 2000 trajectory samples per optimization step, CVaR is estimated from 1000 particles drawn from the belief, the belief is a multivariate Gaussian over object pose whose covariance is obtained from the vision pipeline, and the risk-aversion level is set to α = 0.9. We will also add standard-error bars from 50 independent trials and a brief sensitivity study with respect to sample budget to support the reported performance gap. revision: yes

  3. Referee: [Theoretical Properties] Property (3) and receding-horizon implementation: The union-bound argument is invoked to extend the per-horizon CVaR guarantee to cumulative safety across repeated solves. Because each solve re-samples from the updated belief and the horizon overlaps, the safety-margin random variables are statistically dependent; the manuscript must show that the union bound remains valid or provide a tighter concentration result that accounts for this dependence and the finite-sample CVaR approximation.

    Authors: The referee correctly identifies that overlapping horizons and belief updates induce statistical dependence among the per-step safety-margin random variables. The classical union bound remains valid without requiring independence, and we will state this explicitly in the revision. To address the dependence more carefully, we will add a short paragraph noting that the bound is conservative under positive dependence and outlining how a martingale concentration inequality could yield a tighter result; we will also discuss the additional conservatism introduced by the finite-sample CVaR approximation and suggest a practical adjustment to the risk level α. These clarifications will be incorporated without altering the core claim. revision: partial

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The three properties are derived directly from the definition of the CVaR-constrained MPPI optimization problem using standard tail-risk implications and a union bound; these are external mathematical facts applied to the controller rather than reductions to fitted parameters or self-referential definitions. The belief distribution and safety margin are inputs to the formulation, not outputs renamed as predictions. No load-bearing self-citations or ansatzes are invoked to close the chain, and the simulation results are presented as empirical validation separate from the theoretical claims.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on standard belief-space assumptions for partially observed systems and the mathematical properties of CVaR; no free parameters or invented entities are explicitly introduced beyond the risk weight and belief distribution.

axioms (2)
  • domain assumption Dynamics, costs, and safety constraints depend on a latent parameter maintained as a belief distribution
    Stated in the problem setup for partially observed dynamical systems.
  • domain assumption CVaR constraint on trajectory safety margin provides a probabilistic safety guarantee
    Listed as one of the three established properties of the controller.

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discussion (0)

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Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages · 1 internal anchor

  1. [1]

    Vulcan Pick: A Robotic System for Picking Targeted Objects from Fabric Pods,

    K. Park, J. Kulick, A. Melkozerov, R. A. Vilagrasa, T. S. Lembono, V . Neubauer, A. Minichev, K. Turner, O. Agrigoroaiei, P. Klink,et al., “Vulcan Pick: A Robotic System for Picking Targeted Objects from Fabric Pods,” 2025

    work page 2025

  2. [2]

    Au- tonomous Vehicle Overtaking in a Bidirectional Mixed-Traffic Set- ting,

    F. M. Tariq, N. Suriyarachchi, C. Mavridis, and J. S. Baras, “Au- tonomous Vehicle Overtaking in a Bidirectional Mixed-Traffic Set- ting,” in2022 American Control Conference (ACC), pp. 3132–3139, IEEE, 2022

    work page 2022

  3. [3]

    LQG-MP: Optimized Path Planning for Robots with Motion Uncertainty and Imperfect State Information,

    J. Van den Berg, P. Abbeel, and K. Goldberg, “LQG-MP: Optimized Path Planning for Robots with Motion Uncertainty and Imperfect State Information,”International Journal of Robotics Research, vol. 30, no. 7, pp. 895–913, 2011

    work page 2011

  4. [4]

    Belief Control Barrier Functions for Risk-Aware Control,

    M. Vahs, C. Pek, and J. Tumova, “Belief Control Barrier Functions for Risk-Aware Control,”IEEE Robotics and Automation Letters, vol. 8, no. 12, pp. 8565–8572, 2023

    work page 2023

  5. [5]

    Coherent Measures of Risk,

    P. Artzner, F. Delbaen, J.-M. Eber, and D. Heath, “Coherent Measures of Risk,”Mathematical Finance, vol. 9, no. 3, pp. 203–228, 1999

    work page 1999

  6. [6]

    Optimization of Conditional Value- at-Risk,

    R. T. Rockafellar and S. Uryasev, “Optimization of Conditional Value- at-Risk,”Journal of Risk, vol. 2, pp. 21–42, 2000

    work page 2000

  7. [7]

    Shapiro, D

    A. Shapiro, D. Dentcheva, and A. Ruszczy ´nski,Lectures on Stochastic Programming: Modeling and Theory. SIAM, 2nd ed., 2014

    work page 2014

  8. [8]

    Risk-Sensitive and Robust Decision-Making: A CVaR Optimization Approach,

    Y . Chow, A. Tamar, S. Mannor, and M. Pavone, “Risk-Sensitive and Robust Decision-Making: A CVaR Optimization Approach,” in Advances in Neural Information Processing Systems (NeurIPS), 2015

    work page 2015

  9. [9]

    Risk-Sensitive Rein- forcement Learning with Exponential Criteria,

    E. Noorani, C. N. Mavridis, and J. S. Baras, “Risk-Sensitive Rein- forcement Learning with Exponential Criteria,”IEEE Transactions on Cybernetics, 2025

    work page 2025

  10. [10]

    Robust Stochastic Shortest-Path Planning via Risk-Sensitive Incremental Sam- pling,

    C. Enwerem, E. Noorani, J. S. Baras, and B. M. Sadler, “Robust Stochastic Shortest-Path Planning via Risk-Sensitive Incremental Sam- pling,” in2024 IEEE 63rd Conference on Decision and Control (CDC), pp. 1087–1094, IEEE, 2024

    work page 2024

  11. [11]

    Risk-Averse Stochastic Shortest Path Planning,

    M. Ahmadi, A. Dixit, J. W. Burdick, and A. D. Ames, “Risk-Averse Stochastic Shortest Path Planning,” in2021 60th IEEE Conference on Decision and Control (CDC), (Austin, TX, USA), pp. 5199–5204, IEEE, Dec. 2021

    work page 2021

  12. [12]

    Risk-Aware Motion Planning and Control Using CVaR-Constrained Optimization,

    A. Hakobyan, G. C. Kim, and I. Yang, “Risk-Aware Motion Planning and Control Using CVaR-Constrained Optimization,”IEEE Robotics and Automation Letters, vol. 4, pp. 3924–3931, Oct. 2019

    work page 2019

  13. [13]

    How Should a Robot Assess Risk? Towards an Axiomatic Theory of Risk in Robotics,

    A. Majumdar and M. Pavone, “How Should a Robot Assess Risk? Towards an Axiomatic Theory of Risk in Robotics,” inRobotics Research (Springer Proceedings in Advanced Robotics), 2020

    work page 2020

  14. [14]

    Risk-Aware Robotics: Tail Risk Measures in Planning, Control, and Verification

    P. Akella, A. Dixit, M. Ahmadi, L. Lindemann, M. P. Chapman, G. J. Pappas, A. D. Ames, and J. W. Burdick, “Risk-Aware Robotics: Tail Risk Measures in Planning, Control, and Verification.”

    work page

  15. [15]

    Chance-Constrained Information-Theoretic Stochastic MPC with Safety Shielding,

    H. Yin, P. Tsiotras, and K. Berntorp, “Chance-Constrained Information-Theoretic Stochastic MPC with Safety Shielding,” in IEEE Conference on Decision and Control (CDC), 2024

    work page 2024

  16. [16]

    Convex Approximations of Chance Constrained Programs,

    A. Nemirovski and A. Shapiro, “Convex Approximations of Chance Constrained Programs,”SIAM Journal on Optimization, vol. 17, no. 4, pp. 969–996, 2006

    work page 2006

  17. [17]

    Stochastic Model Predictive Control: An Overview and Perspectives for Future Research,

    A. Mesbah, “Stochastic Model Predictive Control: An Overview and Perspectives for Future Research,”IEEE Control Systems Magazine, vol. 36, no. 6, pp. 30–44, 2016

    work page 2016

  18. [18]

    Chance-Constrained Optimal Path Planning With Obstacles in Stochastic Environments,

    L. Blackmore, M. Ono, and B. C. Williams, “Chance-Constrained Optimal Path Planning With Obstacles in Stochastic Environments,” IEEE Transactions on Robotics, vol. 27, no. 6, pp. 1080–1094, 2011

    work page 2011

  19. [19]

    Information Theoretic MPC for Model-Based Reinforcement Learning,

    G. Williams, A. Aldrich, and E. A. Theodorou, “Information Theoretic MPC for Model-Based Reinforcement Learning,” inIEEE Interna- tional Conference on Robotics and Automation (ICRA), 2017

    work page 2017

  20. [20]

    Belief Space Planning Assuming Maximum Likelihood Observations,

    R. Platt, R. Tedrake, L. P. Kaelbling, and T. Lozano-P ´erez, “Belief Space Planning Assuming Maximum Likelihood Observations,” in Robotics: Science and Systems (RSS), 2010

    work page 2010

  21. [21]

    Planning and Acting in Partially Observable Stochastic Domains,

    L. P. Kaelbling, M. L. Littman, and A. R. Cassandra, “Planning and Acting in Partially Observable Stochastic Domains,”Artificial Intelligence, vol. 101, no. 1–2, pp. 99–134, 1998

    work page 1998

  22. [22]

    Thrun, W

    S. Thrun, W. Burgard, and D. Fox,Probabilistic Robotics. MIT Press, 2005

    work page 2005

  23. [23]

    FIRM: Sampling-Based Feedback Motion-Planning under Motion Uncertainty and Imperfect Measurements,

    A. Agha-mohammadi, S. Chakravorty, and N. Amato, “FIRM: Sampling-Based Feedback Motion-Planning under Motion Uncertainty and Imperfect Measurements,”The International Journal of Robotics Research, vol. 33, no. 2, pp. 268–304, 2014

    work page 2014

  24. [24]

    Control Barrier Function Based Quadratic Programs for Safety Critical Systems,

    A. D. Ames, X. Xu, J. W. Grizzle, and P. Tabuada, “Control Barrier Function Based Quadratic Programs for Safety Critical Systems,” IEEE Transactions on Automatic Control, vol. 62, no. 8, pp. 3861– 3876, 2017

    work page 2017

  25. [25]

    Safe Control for Discrete- time Stochastic Systems with Flexible Safe Bounds using Quadratic Control Barrier Functions,

    S. Fushimi, K. Hoshino, and Y . Nishimura, “Safe Control for Discrete- time Stochastic Systems with Flexible Safe Bounds using Quadratic Control Barrier Functions,”IFAC-PapersOnLine, vol. 59, no. 19, pp. 85–90, 2025

    work page 2025

  26. [26]

    High-Order Control Barrier Functions,

    W. Xiao and C. Belta, “High-Order Control Barrier Functions,”IEEE Transactions on Automatic Control, vol. 67, no. 7, pp. 3655–3662, 2022

    work page 2022

  27. [27]

    Safe Collective Control under Noisy Inputs and Competing Constraints via Non-Smooth Barrier Functions,

    C. Enwerem and J. S. Baras, “Safe Collective Control under Noisy Inputs and Competing Constraints via Non-Smooth Barrier Functions,” in2024 European Control Conference (ECC), pp. 3762–3768, IEEE, 2024

    work page 2024

  28. [28]

    Nonsmooth Barrier Func- tions With Applications to Multi-Robot Systems,

    P. Glotfelter, J. Cortes, and M. Egerstedt, “Nonsmooth Barrier Func- tions With Applications to Multi-Robot Systems,”IEEE Control Systems Letters, vol. 1, no. 2, pp. 310–315, 2017

    work page 2017

  29. [29]

    Learning for Feasible and Safe Control with Control Barrier Functions: A Tutorial,

    W. Xiao, C. Belta, and C. G. Cassandras, “Learning for Feasible and Safe Control with Control Barrier Functions: A Tutorial,”Cybernetics and Intelligence, 2026

    work page 2026

  30. [30]

    Cohen and C

    M. Cohen and C. Belta,Adaptive and Learning-Based Control of Safety-Critical Systems. Springer, 2023

    work page 2023

  31. [31]

    Safety-critical Control Under Partial Observability: Reach-Avoid POMDP meets Belief Space Control

    M. Vahs, J. Verhagen, and J. Tumova, “Safety-critical Control Un- der Partial Observability: Reach-Avoid POMDP meets Belief Space Control,”arXiv preprint arXiv:2603.10572, 2026

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  32. [32]

    Durrett,Probability: Theory and Examples

    R. Durrett,Probability: Theory and Examples. Cambridge University Press, 2019

    work page 2019

  33. [33]

    An Upper Bound for the Probability of a Union,

    D. Hunter, “An Upper Bound for the Probability of a Union,”Journal of Applied Probability, vol. 13, no. 3, pp. 597–603, 1976

    work page 1976

  34. [34]

    Smoothing and Differentiation of Data by Simplified Least Squares Procedures,

    A. Savitzky and M. J. Golay, “Smoothing and Differentiation of Data by Simplified Least Squares Procedures,”Analytical Chemistry, vol. 36, no. 8, pp. 1627–1639, 1964

    work page 1964