This work covers a range of topics in advanced probability and statistics, beginning with quantum stochastic calculus and extending to convergence theorems for empirical processes and U-statistics. It explores symmetry properties and nonparametric estimates of spectral densities for stationary random processes, as well as large deviations concerning first-passage times. The quasi-average approach is discussed in relation to the n-vector Curie-Weiss ferromagnet's limit states, alongside Markov approximation inequalities for lumped processes. The text addresses limit theorems for random variable sequences with random indices, continuity of local time in Markov processes, and change detection in ARMA models with nonstationary coefficients. It also covers nonparametric estimation from incomplete data and diffusion processes related to elliptic equations. The work delves into convergence rates in the Central Limit Theorem within Banach spaces, weak limits of probability measures, and optimal consumption in stochastic models. Further topics include limit theorems for hierarchical models, upper and lower estimates in empirical measures, and noncommutative integration on von Neumann algebras. The book discusses optimal controls in locally absolutely continuous measure changes, stable stochastic integrals, and smooth measures on infinite-dimensional manifolds. It concludes with insights on the asymptotic behavior of stochastic heat
