Kendall’s tau and conditional Kendall’s tau matrices are multivariate (conditional) dependence measures between the components of a random vector. For large dimensions, available estimators are computationally expensive and can be improved by averaging. Under structural assumptions on the underlying Kendall’s tau and conditional Kendall’s tau matrices, we introduce new estimators that have a significantly reduced computational cost while keeping a similar error level. In the unconditional setting we assume that, up to reordering, the underlying Kendall’s tau matrix is block-structured with constant values in each of the off-diagonal blocks. The estimators take advantage of this block structure by averaging over (part of) the pairwise estimates in each of the off-diagonal blocks. Derived explicit variance expressions show their improved efficiency. In the conditional setting, the conditional Kendall’s tau matrix is assumed to have a constant block structure, independently of the conditioning variable. Conditional Kendall’s tau matrix estimators are constructed similarly as in the unconditional case by averaging over (part of) the pairwise conditional Kendall’s tau estimators. We establish their joint asymptotic normality, and show that the asymptotic variance is reduced compared to the naive estimators. Then, we perform a simulation study which displays the improved performance of both the unconditional and conditional estimators. Finally, the estimators are used for estimating the value at risk of a large stock portfolio; backtesting illustrates the obtained improvements compared to the previous estimators.

In this thesis, the relation between the generator of the OrnsteinUhlenbeck process and the Hamiltonian of the quantum harmonic oscillator is used to derive a new understanding of the evolution of certain quantum states. More precisely, we transform the Hamiltonian with respect to the ground state and corresponding eigenvalue to find that it is equal to minus the generator of the Ornstein-Uhlenbeck process. Next, we use the knowledge of the evolution of distributions in the OrnsteinUhlenbeck process to obtain the time evolution of corresponding quantum states. Specifically, we derived that the evolution of normal distributions in the Ornstein-Uhlenbeck process remain normally distributed with varying mean and variance. Furthermore, the ground state of the harmonic oscillator is equal to the square root of the reversible distribution of the Ornstein-Uhlenbeck process. Combining these results gives us the evolution of quantum states with an almost Gaussian wave function. If we confine one degree of freedom in the end result, we obtain the coherent states of the quantum harmonic oscillator. These are Gaussian wave packets, which means that the probability density is Gaussian with constant variance and oscillating mean. Coherent states most closely resemble classical particles in the harmonic oscillator and minimise Heisenberg’s uncertainty principle.