The Method Of Dense Cycle Conditioning, Its Application, Computation And A Result On Concentration

This dissertation contains works on three different directions. In the first direction, three different problems have been solved. The fundamental theme of these problems are to consider the log-likelihood ratio of certain processes under local perturbations. It is shown that in these cases below certain threshold the log-likelihood ratio can be approximated by log-likelihood ratio restricted to a certain class of statistics called the signed cycles". These statistics were considered by the author in order to study contiguity for planted partition model in dense case. Details are given in Chapter 2. The sparse case is known in the literature by a paper of Mossel et al. These statistics found further applications in statistics and statistical physics where two other problems were solved. One might look at Chapters 3 and 1 for details. The second direction of this thesis is to show computability of these cycle statistics. It is proved that the signed cycles" statistics can be approximated by certain linear spectral statistics of high dimensional random matrices. The proof techniques are highly motivated by a paper of Anderson and Zeitouni. One can have a look at Chapter 4 for details. In the third direction a problem of concentration inequality is considered. A Bernstein type concentration inequality is proved for statistics which are generalizations of a statistics introduced by Hoeffding. It is proven using the method exchangeable pairs introduced by Chatterjee. One might look at Chapter 6 for details.