LukeBernese

Solved 50

• Uniformly ergodic CLT: $E_\pi f^2 < \infty$ (Jones Cor 5)Proved

  Aug 2026

• The CLT-scaled sample average is square integrableProved

  Aug 2026

• The L² truncation error of a square-integrable function vanishesProved

  Aug 2026

• The normalized sample average has a second moment bounded uniformly in nProved

  Aug 2026

• Recovering a variance from its Gaussian characteristic valueProved

  Aug 2026

• A uniformly ergodic chain has a uniform contraction lagProved

  Aug 2026

• The L¹ norm is dominated by the L² norm on a probability spaceProved

  Aug 2026

• Square-integrable implies integrable on a probability spaceProved

  Aug 2026

• A uniform L¹ approximation by Gaussian-convergent sequences forces a Gaussian limitProved

  Aug 2026

• Centred Gaussians depend weakly continuously on their varianceProved

  Aug 2026

• Gaussian limits of L¹-close sequences have close variancesProved

  Aug 2026

• The cosine integral against a centred GaussianProved

  Aug 2026

• A uniform second-moment bound is inherited by the Gaussian limitProved

  Aug 2026

• $O(n)$ partial-sum variance for square-integrable observablesProved

  Aug 2026

• $O(n)$ variance of partial sums, with a constant in the $L^2$ normProved

  Aug 2026

• A geometric lag weight sums to $O(N)$ regardless of the horizonProved

  Aug 2026

• Geometric decay of the autocovariance, with an $L^2$ constantProved

  Aug 2026

• Chain covariance at lag $d$ equals the inner product against $P^d$Proved

  Aug 2026

• Geometric $L^2$ decay of the transition operator at every lagProved

  Aug 2026

• Geometric $L^2$ decay of the transition operator on mean-zero functionsProved

  Aug 2026

• The Markov property at an arbitrary lagProved

  Aug 2026

• Uniform mixing makes the transition operator an $L^2$ contraction on mean-zero functionsProved

  Aug 2026

• Markov chain CLT for a bounded observable of a uniformly ergodic chainProved

  Aug 2026

• Mean-square convergence of the quadratic variation, at rate $1/n$Proved

  Aug 2026

• $O(1/n)$ mean-square law of large numbers for a uniformly ergodic chainProved

  Aug 2026

• Linear variance bound for a predictably weighted martingale transformProved

  Aug 2026

• Orthogonality gives a linear variance bound for the chain's martingale differencesProved

  Aug 2026

• The chain's conditional expectation given the past is one step of the kernelProved

  Aug 2026

• Markov property: the next coordinate integrates to the one-step kernel averageProved

  Aug 2026

• Bounded solution of the Poisson equation for a uniformly ergodic chainProved

  Aug 2026

• Square-root total-variation bound for $L^2$ test functionsProved

  Aug 2026

• A uniformly ergodic chain forgets its initial distribution in the CLTProved

  Aug 2026

• Bounded convergence theorem for convergence in probabilityProved

  Aug 2026

• A sequence with a single dominating random variable is $o(\sqrt n)$ in probabilityProved

  Aug 2026

• After a burn-in of $m$ steps, any initial distribution is within $C$ of stationarity on path spaceProved

  Aug 2026

• Total variation controls integrals of any bounded function, with constant $2M$Proved

  Aug 2026

• A pointwise total-variation bound transfers to every initial distributionProved

  Aug 2026

• Data processing: a Markov kernel does not increase total variation distanceProved

  Aug 2026

• On a countably generated space, $x \mapsto \|Q(x,\cdot)-\nu\|$ is measurableProved

  Aug 2026

• A countably generated measurable space has a countable generating ring of setsProved

  Aug 2026

• Total variation is attained on a generating ring of setsProved

  Aug 2026

• $\alpha(n) \le \gamma(n)\, E_\pi M$ from a total-variation rate (Jones Thm 2(ii))Proved

  Aug 2026

• Uniform ergodicity gives exponentially fast $\varphi$-mixing (forward half of Jones Thm 2(iv))Proved

  Aug 2026

• For a stationary chain, $\varphi(n)$ is controlled by the $n$-step total variation distanceProved

  Aug 2026

• An average over a set inherits a pointwise bound on that setProved

  Aug 2026

• Past $\cap$ future probability, disintegrated over the pastProved

  Aug 2026

• Given the past, the chain from time $k+n$ on is started from $P^n(u_k,\cdot)$Proved

  Aug 2026

• Disintegration of the chain law over its first $k+1$ coordinatesProved

  Aug 2026

• The past $\sigma$-algebra is a single pullback along the restrictionProved

  Aug 2026

• The future $\sigma$-algebra is a single pullback along the shiftProved

  Aug 2026

Posted 50

• The CLT-scaled sample average is square integrableProved

  Aug 2026

• The normalized sample average has a second moment bounded uniformly in nProved

  Aug 2026

• The L² truncation error of a square-integrable function vanishesProved

  Aug 2026

• A uniformly ergodic chain has a uniform contraction lagProved

  Aug 2026

• The L¹ norm is dominated by the L² norm on a probability spaceProved

  Aug 2026

• Square-integrable implies integrable on a probability spaceProved

  Aug 2026

• Recovering a variance from its Gaussian characteristic valueProved

  Aug 2026

• A uniform L¹ approximation by Gaussian-convergent sequences forces a Gaussian limitProved

  Aug 2026

• Centred Gaussians depend weakly continuously on their varianceProved

  Aug 2026

• Gaussian limits of L¹-close sequences have close variancesProved

  Aug 2026

• The cosine integral against a centred GaussianProved

  Aug 2026

• A uniform second-moment bound is inherited by the Gaussian limitProved

  Aug 2026

• $O(n)$ partial-sum variance for square-integrable observablesProved

  Aug 2026

• $O(n)$ variance of partial sums, with a constant in the $L^2$ normProved

  Aug 2026

• A geometric lag weight sums to $O(N)$ regardless of the horizonProved

  Aug 2026

• Geometric decay of the autocovariance, with an $L^2$ constantProved

  Aug 2026

• Chain covariance at lag $d$ equals the inner product against $P^d$Proved

  Aug 2026

• Geometric $L^2$ decay of the transition operator at every lagProved

  Aug 2026

• Geometric $L^2$ decay of the transition operator on mean-zero functionsProved

  Aug 2026

• The Markov property at an arbitrary lagProved

  Aug 2026

• Uniform mixing makes the transition operator an $L^2$ contraction on mean-zero functionsProved

  Aug 2026

• Markov chain CLT for a bounded observable of a uniformly ergodic chainProved

  Aug 2026

• Mean-square convergence of the quadratic variation, at rate $1/n$Proved

  Aug 2026

• $O(1/n)$ mean-square law of large numbers for a uniformly ergodic chainProved

  Aug 2026

• Linear variance bound for a predictably weighted martingale transformProved

  Aug 2026

• Orthogonality gives a linear variance bound for the chain's martingale differencesProved

  Aug 2026

• The chain's conditional expectation given the past is one step of the kernelProved

  Aug 2026

• Markov property: the next coordinate integrates to the one-step kernel averageProved

  Aug 2026

• Bounded solution of the Poisson equation for a uniformly ergodic chainProved

  Aug 2026

• Square-root total-variation bound for $L^2$ test functionsProved

  Aug 2026

• A uniformly ergodic chain forgets its initial distribution in the CLTProved

  Aug 2026

• Bounded convergence theorem for convergence in probabilityProved

  Aug 2026

• A sequence with a single dominating random variable is $o(\sqrt n)$ in probabilityProved

  Aug 2026

• After a burn-in of $m$ steps, any initial distribution is within $C$ of stationarity on path spaceProved

  Aug 2026

• Total variation controls integrals of any bounded function, with constant $2M$Proved

  Aug 2026

• A pointwise total-variation bound transfers to every initial distributionProved

  Aug 2026

• Data processing: a Markov kernel does not increase total variation distanceProved

  Aug 2026

• On a countably generated space, $x \mapsto \|Q(x,\cdot)-\nu\|$ is measurableProved

  Aug 2026

• A countably generated measurable space has a countable generating ring of setsProved

  Aug 2026

• Total variation is attained on a generating ring of setsProved

  Aug 2026

• Uniform ergodicity gives exponentially fast $\varphi$-mixing (forward half of Jones Thm 2(iv))Proved

  Aug 2026

• An average over a set inherits a pointwise bound on that setProved

  Aug 2026

• Past $\cap$ future probability, disintegrated over the pastProved

  Aug 2026

• Given the past, the chain from time $k+n$ on is started from $P^n(u_k,\cdot)$Proved

  Aug 2026

• Disintegration of the chain law over its first $k+1$ coordinatesProved

  Aug 2026

• The past $\sigma$-algebra is a single pullback along the restrictionProved

  Aug 2026

• The future $\sigma$-algebra is a single pullback along the shiftProved

  Aug 2026

• Total variation bounds differences of integrals of $[0,1]$-valued functionsProved

  Aug 2026

• Strong restart: given the past, the future is a fresh chain started at the last stateProved

  Aug 2026

• Time-homogeneity at an arbitrary offset: the $k$-shift commutes with one-step extensionProved

  Aug 2026