Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
The Control of the Controlled N-Agent Dynamics is Adapted to the Observation Filtration
lemmalem:n-agent-control-observation-adapted-2026aProbabilityAdopt the setting of the definition of a solution of the controlled -agent dynamics on : natural numbers , , , , a transition-rate family , an observation-rate family , a horizon , an…Conditional Expectation Minimizes Weighted Mean-Square Estimation Error
lemmalem:conditional-mean-square-optimality-2026aProbabilityLet be a probability space, let be a sub--algebra of , let be a natural number, and let be a tuple of square-integrable random variables on . For each…Existence of Asymptotically Optimal-Value Observation-Driven Policies
theoremthm:asymptotically-optimal-value-policies-2026aProbabilityAdopt the setting, hypotheses (H1)--(H4), and notation of the approximate Kalman filter and policy lemma, for the fluctuation LQG data of the stationary mean-field triple , whose stationary co-state has value at time : in particular the…The Limiting Cost Along the Approximate Kalman Policy Is the Optimal Value of the Fluctuation LQG Problem
corollarycor:kalman-policy-limit-is-lqg-value-2026aProbabilityAdopt the setting, hypotheses (H1)--(H4), and notation of the approximate Kalman filter and policy lemma for the fluctuation LQG data of the stationary mean-field triple : in particular the natural numbers , , , the horizon , the matrice…Cost Limit Along the Approximate Kalman Policy
propositionprop:kalman-policy-cost-limit-2026aProbabilityAdopt the setting, hypotheses (H1)--(H4), and notation of the approximate Kalman filter and policy lemma, for the fluctuation LQG data of the stationary mean-field triple : in particular the cost extension of the population cost data is part of the data, the fami…Mean-Square Error Covariance of the Approximate Kalman Filter
lemmalem:kalman-filter-error-covariance-2026aProbabilityAdopt the setting, hypotheses (H1)--(H4), and notation of the approximate Kalman filter and policy lemma: the fluctuation LQG data of the stationary mean-field triple with matrices , , , ,…Weighted Compensated Sums over the Observation Events of the Controlled N-Agent Dynamics
lemmalem:n-agent-weighted-observation-sums-2026aProbabilityAdopt the setting of the compensated counters of the controlled -agent dynamics: a transition-rate family with rate bound on states with control dimension , an observation-rate family with rate bound and channels, a ho…Mean-Square Linearization Residual of the Observation Fluctuation Process
lemmalem:observation-linearization-residual-2026aProbabilityAdopt the setting of the fluctuation processes of the controlled -agent dynamics: a transition-rate family on states with control dimension , an observation-rate family with observation channels and rate bound , a horizon…The Approximate Kalman Filter and Policy for the Controlled N-Agent Dynamics
lemmalem:approximate-kalman-policy-2026aProbabilityAdopt the setting and notation of the definition of the fluctuation LQG data of a stationary mean-field triple: natural numbers , , , the transition-rate family with rate bound and its extension , the cost extension — whos…Jump Representation and Positive Semidefiniteness of the Aggregate Fluctuation Covariance
lemmalem:fluctuation-covariance-psd-2026aProbabilityLet and be natural numbers with and , let be a transition-rate family on states with control dimension and rate bound , and let be its aggregate fluctuation covariance. Write for the probability simplex, write…Fluctuation LQG Data of a Stationary Mean-Field Triple
definitiondef:fluctuation-lqg-data-2026aProbabilityLet , , with rate bound , with derivative bound , , with second-derivative bound , , , and the stationary co-state be as in the definition of the fluctuation linear-quadratic cost functional,…Regularity and Derivative Bounds of the Extended Aggregate Observation Drift
lemmalem:extended-observation-drift-regularity-2026aProbabilityLet and be natural numbers with and , let be an observation-rate family on states with observation channels and rate bound , let be a…Extended Aggregate Observation Drift
definitiondef:extended-aggregate-observation-drift-2026aProbabilityLet and be natural numbers with and , let be an observation-rate family on states with observation channels, and let be a twice continuously differentiable extension of…Twice Continuously Differentiable Extension of an Observation-Rate Family
definitiondef:c2-observation-rate-extension-2026aProbabilityLet and be natural numbers with and , let be an observation-rate family on states with observation channels and rate bound , let be the probability simplex, and let…Multiplier Identities and Interval Estimates for the Compensated Counters of the Controlled N-Agent Dynamics
lemmalem:n-agent-multiplier-interval-estimates-2026aProbabilityAdopt the setting of the compensated counters of the controlled -agent dynamics: a transition-rate family with rate bound on states with control dimension , an observation-rate family with rate bound and channels, a ho…Factorial Moments and Moments of Every Order of the Poisson Distribution
lemmalem:poisson-factorial-moments-2026aProbabilityLet be a probability space, let be a real number, let be a random variable on with the Poisson distribution with parameter , and let be a natural number. Write for the expectation and use the…A Priori Fourth-Moment Bound for the State Fluctuation Process
lemmalem:fluctuation-fourth-moment-bound-2026aProbabilityAdopt the setting of the fluctuation processes of the controlled -agent dynamics: a transition-rate family with rate bound on states with control dimension , an observation-rate family , a horizon , an -agent driving system…Moment Bounds for the Aggregate Compensated Counters of the Controlled N-Agent Dynamics
lemmalem:n-agent-counter-fourth-moment-2026aProbabilityAdopt the setting of the compensated counters of the controlled -agent dynamics with agents, states, observation channels, and control dimension : a transition-rate family with rate bound , an observation-rate family with rate…Mean Deviation Bound for the Aggregate Fluctuation Covariance along a Mean-Field Trajectory Pair
lemmalem:fluctuation-covariance-deviation-2026aProbabilityAdopt the setting of the fluctuation processes of the controlled -agent dynamics: a transition-rate family with rate bound on states with control dimension , an observation-rate family , a horizon , an -agent driving system…Mean-Square Continuous Dependence for Linear Stochastic Differential Equations Driven by the Same Brownian Motion
lemmalem:linear-sde-continuous-dependence-2026aProbabilityLet be a probability space, let be real, let be natural numbers, and let and be linear stochastic differential equations with additive Wiener noise on that share the coefficient…