Exercise given at the end of Leure.
Letµbe a critical and aperiodic offspring diribution onZ+, with finite and positive variance σ. Recall that:
∗ Wn=X+· · ·+Xnis the random walk such thatP(X=i) =µ(i+) fori ≥ −.
∗ T=andTk = inf{i > Tk−;Wi≥WTk−}fork≥are the weak record times ofW.
∗ Hn=|{≤k≤n−;Wk= mink≤j≤nWj}|forn≥is the associated height process.
(i) CalculateE hWTi
.
(ii) SetIn= min≤i≤nWi. Show that
Hn Wn−In
(P)
−→
n→∞
σ, where (P) means that the convergence holds in probability.
