97. Long-horizon marginal control is not full-record control
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The previous record bound grows with \(T\). A different question can have a uniform answer: how far apart are the distributions of the signal at a single future time? A lower density bound permits this distinction to be made exactly.
Let \(P,Q\) be Markov kernels on a common state space. Suppose \(P(s,\cdot)\geq\rho\mu(\cdot)\) for every \(s\), where \(\mu\) is one probability measure and \(0<\rho\leq1\). Suppose also \(\sup_s\|P(s,\cdot)-Q(s,\cdot)\|_{\mathrm{TV}}\leq\Delta\).
Proposition 86: Uniform marginal and rolling-window stability
For initial laws at distance \(e_0\), the one-time marginal laws satisfy
For \(k\) consecutive observations starting at time \(t\), their joint laws are at distance at most \(\min(1,e_t+(k-1)\Delta)\). A uniform small marginal bound does not in general bound the total variation of arbitrarily long complete records away from one.
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Proof
For \(\rho<1\), write \(P(s,\cdot)=\rho\mu+(1-\rho)R(s,\cdot)\), where \(R\) is a Markov kernel. Applying a Markov kernel cannot increase total variation, by integration of bounded test functions. Thus \(\|\nu P-\widetilde\nu P\|_{\mathrm{TV}}\leq(1-\rho)\|\nu-\widetilde\nu\|_{\mathrm{TV}}\). Adding and subtracting \(\widetilde\nu P\) yields \(e_{t+1}\leq(1-\rho)e_t+\Delta\). Iterate this inequality. The case \(\rho=1\) follows directly. Applying the finite-record telescoping argument from the initial time \(t\) gives the rolling-window statement.
For the final assertion, take two distinct independent Bernoulli laws with parameters strictly between zero and one. Each is a Markov kernel with a positive common minorization. At any time their marginal distance is the fixed parameter difference. For a full record, test whether its empirical mean lies above the midpoint between the parameters. Under each law, Chebyshev's inequality makes the error probability tend to zero as the record length grows. The total variation of the record laws therefore tends to one. \(\square\)
This is the standard minorization/contraction mechanism [R24], with its error accounting made explicit. It is not a stability theorem for arbitrary feedback policies that remember the full history. The reference model is autonomous and its current scalar observation is the Markov state. It also does not bound long products of unnormalized likelihood ratios.
With \(e_0=0\), \(\rho=31/125\) and \(\Delta=2437/90000\), the reference confidence rectangle gives
This is a nontrivial all-time bound on one-time marginal observation distributions. The complete-record bound is still \(\min(1,2437T/90000)\), which becomes trivial at integer horizon 37. The model does not exchange those two different claims. In live filtering after a shared observed value, its state is exactly known and the next-observation law has the separate uniform upper bound \(2437/90000\).