Suggestion: Set aside complex variables and do measure theory instead. For the Fourier and Laplace transforms, cover those via measure theory. For measure theory, H. Royden,
Real Analysis and the first half of W. Rudin,
Real and Complex Analysis.Then take a course in graduate probability which is based on measure theory. Good authors are Loeve, Neveu, Breiman, Chung, among others. So, learn about the cases of convergence including almost sure convergence, a good version of the central limit theorem, proved carefully, the weak and strong laws of large numbers, ergodic theory, and martingale theory.
Learn stochastic differential equations.
Learn some potential theory.
Then do some work in optimization and optimization under uncertainty.
Then take a course in graduate probability which is based on measure theory. Good authors are Loeve, Neveu, Breiman, Chung, among others. So, learn about the cases of convergence including almost sure convergence, a good version of the central limit theorem, proved carefully, the weak and strong laws of large numbers, ergodic theory, and martingale theory.
Learn stochastic differential equations.
Learn some potential theory.
Then do some work in optimization and optimization under uncertainty.