This paper utilizes the quantum master equation in particle number representation and calculates the characteristic polynomial of the Liouville superoperator to study the properties of various order cumulants of full counting statistics in a single quantum dot coupled with a topological superconducting nanowire structure. The results show that the cumulants exhibit a step-like distribution with voltage, the cumulative amount of each order increases when the voltage is twice the energy difference between different parity levels of the system; the sum of the first-order cumulants of each sub-channel equals the total current of all channels, but the interference between energy levels leads to non-trivial addition rules for higher-order cumulants between sub-channels and the total channel. When quantum dots and nanowires are weakly coupled at the remote end, Majorana zero modes are more likely to appear, and Majorana zero modes can participate in transport through lower energy levels; The Coulomb interaction alters the degeneracy state of energy levels, thereby changing the number of steps in the voltage-dependent cumulants.

The shot noise of electron transport processes in a hybrid system of double quantum dots coupled to topological superconducting nanowire is studied. There may be various coherent tunneling mechanisms in the system, and we investigate the shot noise in four cases of mixed systems, when the Majorana bound state and the Poorman's Majorana bound state do not exist, and when they exist respectively and simultaneously. It is found that two coherent tunneling mechanisms corresponding to the two bound states result in significant enhancement of the low and zero frequency shot noise, but affect the depth and dynamics behavior of the characteristic valley of shot noise significantly differently. It can provide evidence to distinguish Majorana bound states and Poorman's Majorana bound state. In addition, the shot noise spectral is decomposed. Studying the components, one finds that system-environment interactions, as well as single electron transport, competes with electron coherent tunneling. This paper contributes to a deeper understanding of the nature of Majorana bound states.

The waiting time distribution of electron transport in a topological superconductor coupled double quantum dots is studied by using the master equation. The existence of Majorana fermions in the system is characterized by the correlation between the number of peaks in the joint waiting time distribution and the waiting time of continuous electron jumps. It is found that with the synchronous increase of superconducting proximity effect and elastic cotunneling, the oscillation frequency of the joint waiting time distribution increases higher, and the waiting time correlation changes more frequent in unit time. It indicates that when the superconducting proximity effect and elastic cotunneling intensity increase compared to the tunneling rate, electrons move more frequently in the superconductor and are harder to tunnel out.