If you’ve typed into Google, I know exactly where you are. You’re likely a graduate student, a self-studying quant, or a researcher staring down the barrel of "Measure theory? Martingales? Doob’s optional stopping?"
Let (Y_1, Y_2, \dots) be i.i.d. with mean 0, and define (X_n = \sum_k=1^n Y_k^2). This is a submartingale (since (Y_k^2 \ge 0)). Then: stochastic process doob pdf download install
For those interested in delving deeper into stochastic processes and Doob's contributions, several online resources and academic papers are available. Here are some steps to find relevant PDF materials: If you’ve typed into Google, I know exactly where you are
: Doob is most famous for his work on martingales , a class of stochastic processes that model fair games and are now essential in mathematical finance and potential theory. If you’ve typed into Google