Lognormal law for a renormalization chain arising in search theory and the modelling of descent algorithms (Q1091929)

This paper is motivated by random search models. It is desired to find an object at an unknown location which is successively approached by some specified process. How far will we be from the object at the n-th probe? The process assumed is that if \(X_ n=x\) is the distance from the target of the n-th probe, then the distance from the target of the \((n+1)st\) probe is \[ P(X_{n+1}\in dy| X_ n=x)=\frac{f(y)}{F(x)},\quad 0<y<x, \] where f: [0,1]\(\to R\) is a nonegative integrable function and F(x) is the integral of f from 0 to x. It is shown that it is more tractable to consider log \(X_ n\) rather that \(X_ n\), and a limit theorem is found in relation to the process. This leads to a discussion of random descent models which exhibit linear convergence.