## Weak convergence of the number of zero increments in the random walk with barrier.(English)Zbl 1320.60110

Summary: We continue the line of research of random walks with a barrier initiated by A. Iksanov and M. Möhle [Adv. Appl. Probab. 40, No. 1, 206–228 (2008; Zbl 1157.60041)]. Assuming that the tail of the step of the underlying random walk has a power-like behavior at infinity with the exponent $$-\alpha$$, $$\alpha\in(0,1)$$, we prove that $$V_n$$ the number of zero increments before absoprtion in the random walk with the barrier $$n$$, properly centered and normalized, converges weakly to the standard normal law. Our result complements the weak law of large numbers for $$V_n$$ proved in [A. Iksanov and P. Negadajlov, “On the number of zero increments of random walks with a barrier”, Discrete Math. Theor. Comput. Sci., Proc. 2008, 243–250 (2008), https://hal.inria.fr/hal-01194687].

### MSC:

 60G50 Sums of independent random variables; random walks 60C05 Combinatorial probability 60G09 Exchangeability for stochastic processes 60F05 Central limit and other weak theorems

Zbl 1157.60041
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