The maximum of a symmetric next neighbor walk on the nonnegative integers

• Autores: Ora E. Percus, Jerome K. Percus
• Localización: Journal of Applied Probability, ISSN-e 0021-9002, Vol. 51, Nº. 1, 2014, págs. 162-173
• Idioma: inglés
• DOI: 10.1017/s0021900200010159
• Texto completo no disponible (Saber más ...)
• Resumen

  • We consider a one-dimensional discrete symmetric random walk with a reflecting boundary at the origin. Generating functions are found for the two-dimensional probability distribution P{Sn = x, max1=j =nSn = a} of being at position x after n steps, while the maximal location that the walker has achieved during these n steps is a. We also obtain the familiar (marginal) one-dimensional distribution for Sn = x, but more importantly that for max1=j=n Sj = a asymptotically at fixed a ²/n. We are able to compute and compare the expectations and variances of the two one-dimensional distributions, finding that they have qualitatively similar forms, but differ quantitatively in the anticipated fashion