The Łukasiewicz walk and the stable-domain hypothesis (H_α) #
Source lines 174–181, 699–713, 719–730, 842–853. The walk has i.i.d. steps X_i with
P(X_1 = i) = μ_{i+1} for i ≥ -1, and Ψ_m(i) = P(S_m = i).
The standing hypothesis (H_α) ("μ is critical and in the domain of attraction of an
α-stable law, α ∈ (1,2], with μ_1 ≠ 1") is expressed through normalized weak
convergence: (K_1+…+K_n−n)/a_n converges in distribution to the law ν
with Laplace exponent ∫ e^{-λ y} dν = e^{λ^α}. The separately cited truncated-moment
characterization is not included as a redundant assumption. The normalizing sequence a is part
of the
data because Assumption (S) mentions a_n.
The definitions stepLaw, walkLaw, SlowlyVarying, StableDomain and InStableDomain are
in PoissonDeclumping.Statements.Defs (from the upstream statement contract); this module adds
canonical probability-measure instances and walkProb.