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^α/l^α with α≥0. For α=0, we derive the probability distribution of T_loc exactly, which is expressed as a series of δ functions in which the coefficients can be interpreted as the probability of multiple revisits of the RTP to the origin starting from the origin. For general α, we show that the typical fluctuations of T_loc scale with time as T_loc∼t^1+α/2+α for large t and their probability distribution possesses a scaling behavior described by a scaling function which we have computed analytically. Second, we study the statistics of T_loc until the RTP makes a first passage to x=M(>0). In this case, we also show that the probability distribution can be expressed as a series sum of δ functions for all values of α(≥0) with coefficients originating from appropriate exit problems.
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