Value and position of large weak left-to-right maxima for samples of geometrically distributed variables

We consider words of length n, where the characters are independently generated with a geometric distribution. An element is a weak left-to-right maximum if it is larger than or equal to all the elements to its left. The positions of these maxima are counted from the right-hand side of the word hence the term large. We investigate two parameters, the value and the position of the rth left-to-right maxima for fixed r   and n→∞n→∞.