A note on Spitzer identity for random walk

Let (Sn)n⩾0 be a random walk evolving on the real line and introduce the first hitting time of the half-line (a,+∞) for any real a: τa=min{n⩾1:Sn>a}. The classical Spitzer identity (1960) supplies an expression for the generating function of the couple (τ0,Sτ0). In 1998, Nakajima [Joint distribution of the first hitting time and first hitting place for a random walk. Kodai Math. J. 21 (1998) 192-200.] derived a relationship between the generating functions of the random couples (τ0,Sτ0) and (τa,Sτa) for any positive number a. In this note, we propose a new and shorter proof for this relationship and complement this analysis by considering the case of an increasing random walk. We especially investigate the Erlangian case and provide an explicit expression for the joint distribution of (τa,Sτa) in this situation.