Nuclear physics is essential in extracting important information, such as investigating the fundamental properties of neutrinos (absolute neutrino mass scale), neutrino mixing and CP properties of neutrinos. It is a formidable task to accurately evaluate the relevant nuclear transitions of electron capture, single beta decay and double beta decay. Among the sophisticated nuclear structure approaches, which are commonly used to calculate matrix elements of weak nuclear processes, Quasiparticle Random Phase Approximation (QRPA) plays an important role. This master thesis aims at providing an understanding of the limitations and shortfalls of realistic nuclear transition calculations within the standard QRPA, as well as proposing a possible improvement for this many-body approach. For this, a exactly solvable model that possesses the main qualitative features of a realistic Hamiltonian, is exploited. It is shown that the QRPA method allows only a single from many excited states, corresponding to a Hamiltonian of the proton-neutron Lipkin model, to be described and that beta strengths to higher excited states play an important role. The consequencies of the quasiboson approximation, which violates the Pauli exclusion principle, are discussed with the aid of fermion-boson mapping Marumori procedure. For the first time (to our knowledge) an extension of the standard QRPA approach is proposed that includes nonlinear terms in the definition of the phonon operator. This allows us to describe, simultaneously, the first and third excited states and to calculate the beta transitions to these states from ground state. The reasonable results obtained in this master thesis encourage further studies with the help of the QRPA with nonlinear operator within schematic models and to further apply this method for realistic calculations. Keywords: standard QRPA, nonlinear phonon operator, nuclear models, neutrinoless double beta decay, schematic models, beta decay, beta decay strengths, Fermi and Gamow-Teller transitions