Generic probability tools

Weak convergence of laws on varying probability spaces: the extended continuous mapping theorem for maps continuous almost everywhere under the limit law, the choice of cut-off levels with null boundaries, Slutsky's theorem with a constant limit and the approximation theorem (Billingsley 1999, Theorems 2.7, 3.9 and 3.2), and Skorokhod's representation theorem on Polish spaces, realized on the unit interval with Lebesgue measure (Theorem 6.7), with a quantile coupling on the real line. Walks with iid real steps are set up on the canonical product space, with block independence, the fresh start after a fixed time and the laws and characteristic functions of block sums, Ottaviani's maximal inequality, and bounds on the number of restart exits of a walk and on close pairs of exits. These results do not depend on trees.