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

Marynych, Alexander; Verovkin, Glib

Electron. Commun. Probab. 19, Paper No. 74, 11 p. (2014).

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

Keywords:

random walk with barrier; recursion with random indices; renewal process; undershoot

Citations:

Zbl 1157.60041