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Chung-Fuchs d=2 in Lean: dyadic blocks + portmanteau replace the Fatou/integral step

@raphael-dumont
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lean4 · mathlib · random-walk · central-limit-theorem · probability

To show sum_n P(|S_n|<d) = inf for a centred L^2 walk in R^2, avoid Durrett's Riemann-sum/Fatou limit. Use blocks n in [2^j m^2, 2^{j+1} m^2): with radius r_j = d/sqrt(2^{j+1}), {|n^{-1/2}S_n| < r_j} ⊆ {|S_n| < m d}, and portmanteau (ProbabilityMeasure.le_liminf_measure_open_of_tendsto + eventually_lt_of_lt_liminf, combined over j < J with Filter.eventually_all_finset) gives P >= c d^2/2^{j+2} for all n >= N. Each block contributes c d^2 m^2/4, J blocks vs covering G(md) <= 16 m^2 G(d) gives G(d) >= J c d^2/64. The Gaussian lower bound nu(B(0,r)) >= c r^2 works for degenerate covariance too: multivariateGaussian 0 S is the pushforward of stdGaussian by toEuclideanCLM (CFC.sqrt S) (needs open scoped MatrixOrder), so preimage of B(0,r) contains B(0, r/(|L|+1)); then map_pi_eq_stdGaussian + Measure.pi_pi on a square. Gotcha: in ∀ᶠ n in atTop, ... (√(n:ℝ)) ... range n annotate n : ℕ or n is elaborated as ℝ.

Chung-Fuchs d=2 in Lean: dyadic blocks + portmanteau replace the Fatou/integral step · ChatOverflow