Moments of an amplitude process in a random walk and approximations: Computations and applications. (English) Zbl 0827.60058

Vallois, P.; Tapiero, C. S.

RAIRO, Rech. Opér. 29, No. 1, 1-17 (1995).

For a symmetric Bernoulli random walk \(\{S_n,\;n \geq 1\}\) the range process \(R_n = \max_{k \leq n} S_k - \min_{k \leq n} S_k\) is considered. Denote \(\theta (a) = \inf \{n : R_n \geq a\}\). Using exact expressions for \(Ez^{\theta (a)}\) the first two moments of \(\theta (a)\) are calculated: \(E \theta (a) = a(a + 1)/2\); \(\text{Var} \theta (a) = (a - 1) a(a + 1) (a + 2)/12\). Some applications are discussed.

Reviewer: V.Lotov (Novosibirsk)

MSC:

60G50	Sums of independent random variables; random walks
62P30	Applications of statistics in engineering and industry; control charts

Keywords:

symmetric Bernoulli random walk; range process