Least‐squares Estimation of an Unknown Number of Shifts in a Time Series

In this contribution, general results on the off‐line least‐squares estimate of changes in the mean of a random process are presented. First, a generalisation of the Hajek‐Renyi inequality, dealing with the fluctuations of the normalized partial sums, is given. This preliminary result is then used to derive the consistency and the rate of convergence of the change‐points estimate, in the situation where the number of changes is known. Strong consistency is obtained under some mixing conditions. The limiting distribution is also computed under an invariance principle. The case where the number of changes is unknown is then addressed. All these results apply to a large class of dependent processes, including strongly mixing and also long‐range dependent processes.