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The convenience of this result is that we can bound the worst case deviation of a random walk at any point of time using its value at the end of time interval.
Proof
The following argument employs discrete martingales.
As argued in the discussion of Doob's martingale inequality, the sequence is a martingale.
Define as follows. Let , and
Billingsley, Patrick (1995). Probability and Measure. New York: John Wiley & Sons, Inc. ISBN0-471-00710-2. (Theorem 22.4)
Feller, William (1968) [1950]. An Introduction to Probability Theory and its Applications, Vol 1 (Third ed.). New York: John Wiley & Sons, Inc. xviii+509. ISBN0-471-25708-7.
Kahane, Jean-Pierre (1985) [1968]. Some random series of functions (Second ed.). Cambridge: Cambridge University Press. p. 29-30.