Conditional moments of the actual repetition statistic
Abstract
The native ancestral-pair multiplicities determine both conditional repetition moments.
Let V and O be arbitrary types, let G be a connected locally finite undirected simple graph with a neighbour at each vertex, and let P be a probability measure on O. The original RepetitionVarianceLinear.Model G P supplies measurable, mutually independent initial opinions, fresh opinions, refresh controls and actual-neighbour choices with the prescribed fair, noisy and uniform marginal laws and with noise strictly between zero and one. Its nvm is the original synchronous recursion: a copy at time n uses the preceding time layer. No stationary or deterministic initial law is substituted.
For a root u, a natural lag d at least one and any natural t, including zero, write W for RepetitionVarianceLinear.W M.coordinate u d t and N=t+d. The source and coin functions are those of AncestralSources on the disjoint native label type Label V. Put a_i=source M.coordinate i u and b_i=source M.coordinate (i+d) u. The real-valued count C M u d t is the sum, over Icc 1 t, of the indicators of a_i=b_i. For a finite label set e, multiplicity M u d t e is the natural cardinality of the indices in Icc 1 t for which a_i differs from b_i and the unordered two-element set containing a_i and b_i equals e. Thus orientation is forgotten, repeated occurrences are counted and coalesced pairs are excluded.
The field H M is the comap of the complete vector of native coordinates c satisfying positiveControl c. These are precisely refresh(n+1,v) and choice(n+1,v), for every natural n and every v in V. The vector has its dependent product sigma-field. Consequently H is exactly the sigma-field generated by all strictly positive-time refresh and neighbour-choice coordinates, including times and vertices outside the observed rectangle. It contains no auxiliary time-zero coordinate by definition. Source labels and the displayed count and multiplicities are H-measurable.
Set E_N=pairs G u N=(RectangularSupport.L G u N).powersetCard 2. The original rectangular support theorem puts all a_i and b_i in this deterministic finite label set. Every nonzero global pair multiplicity therefore belongs to E_N, even on an infinite graph. The character of e is the product, over its labels, of sgn(coin M.coordinate l); for a two-element set it is the product of its two fair signs.
Theorem 1.1 (Both native conditional moment identities).
Proof. Machine-checked in Lean as D5/S3/StatisticalMechanics/NoisyVoter/ConditionalRepetitionMoments.conditional_repetition_moments (✓ std3). ∎
Source. Repository-derived.
Acknowledgement. I. Benjamini, H. Helman Tov and M. Zhukovskii (2023). Global information from local observations of the noisy voter model on a graph. DOI: 10.1214/23-AOP1637. URL: https://arxiv.org/abs/2207.01224.
Commentary.
In the displayed statement, AE(P,f,g) means equality of the two real functions P-almost everywhere, and condExp(P,f,H) denotes the conditional expectation P[f|H]. The symbols W, C and m denote exactly the native functions defined above, with m(M,u,d,t,e,w) the natural multiplicity cast to a real number. The second expectation centers by the original conditional expectation of W itself; it is not a variance under a chosen pointwise conditional probability kernel.
The full positive-control vector is independent of the entire native opinion-coin family by independence of disjoint coordinate subfamilies. Fixed pair-character means and cross means are supplied by fixed_pair_sign_mean and fixed_four_sign_mean at their original owner. Conditional expectation independence turns these into conditional orthogonality on H. This includes identical pairs, disjoint pairs and pairs sharing one label; it assumes neither independence of observed opinions nor joint independence of all pair characters.
The exact pointwise regrouping of W is (t+C)/2 plus one half of the sum of multiplicity times character. Bounded H-measurable multiplicities pull out of conditional expectations. The first moment removes the character sum, and the square of that sum leaves precisely its diagonal terms. Finite intersections of probability-one events give simultaneous conditional identities without choosing values on zero-probability control histories. The native W is in L2 on the original probability space: W_measurable, W_bound and W_sample_eq apply directly to its coordinate family. Each multiplicity is at most t, so every finite product integrated in the proof is integrable.
When t=0, W, C and every multiplicity vanish, and both identities hold as empty-sum statements. Arbitrary finite and infinite connected locally finite graphs remain within the same telescope. The unconditional total-variance formula, its exact nonnegative remainder, its equality criterion and optimality of the universal lower constant require further arguments.