Pinning more general processes on a defect line

2 Pinning more general processes on a defect line 

The random walk S defines the pinning model of Section 1.2.1, but the arguments really used only the distribution K -) of the return times to zero. In particular we could generalize the model to any homogeneous Markov chain S to avoid trivialities we should assume for example that P Sn = 0) 0 for some n, but in general there is no reason to assume that 0 is recurrent for the chain. 

But one could go much beyond by defining the model just in terms of the sub-probability distribution if ( ) on N, extended to NU oo as we did before. Given an IID sequence g = 7n n6N with distribution if(-), we consider g as inter-arrival times and we define t as partial sums process associated to g i.e. tq = 0 and Tj = Hn) - t is a discrete renewal 

All the models we will consider essentially boil down to return time 

4 A more general lattice walk the jumps are not limited to taking values in —1,0,+1. The walk can cross the x-axis, or defect line, without touching it (in gray the contacts with the defect line). The renewal points to,ti,7, t3,T4,. .. are 0,4,5, 6,9.  

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