Prove Lemma 35.8. Prove that the finite-dimensional distributions of Z n in Theorem 35.10 converge to those of a Brownian bridge. Let Nt(A) be a Poisson point process with respect to the measure space...

Prove Lemma 35.8.

Prove that the finite-dimensional distributions of Z_n
in Theorem 35.10 converge to those of a Brownian bridge.

Let Nt(A) be a Poisson point process with respect to the measure space (S, m) and let As, s > 0, be an increasing sequence of subsets of S with m(As) → ∞ as s → ∞. Does

converge weakly with respect to
  What is the limit? This can be applied to get central limit theorems for the number of down crossings of a Brownian motion, for example.