What motivated Joe to develop deep results in applied probability? In the interview, Joe reveals it was his desire to convert common probabilistic intuition into mathematics (statements and proofs) that drove him to discover his theorems.
The wikipedia entry on martingales is also interesting. Doob's soiree into martingale theory coincided with the Great Depression of 1929.
Other notable statistical activity at that time included Kolmogorov's axiomatisation of probability in 1933. Read Doob's bio here.
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