Long excursions of a random walk (Q5952043)

Some properties of ``long'' excursions of a simple symmetric random walk on the plane and on higher dimensions are considered. For \(d=2\), denote by \(\rho_n\) the \(n\)th return time to zero, and \(T_n= (\rho_{n+1}-\rho_n)/ \rho_n\), \(T_n^*= (\rho_{n+1}- \rho_n)/ \rho_{n+1}\). The a.s. intervals for \(T_n\) and \(T_n^*\) are considered and for the sum \(\sum^n_{k=0} T_k^*\) convergence to the normal distribution and a.s. statements similar to the strong law of large numbers as well as the law of the iterated logarithm are proved. If \(d\geq 3\), some a.s. limit for the length of the longest completed excursion (away from any point) at time \(n\) is presented.