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Let ''X''<sub>1</sub>, ''X''<sub>2</sub>, ... be a sequence of [[independent and identically distributed random variables]], each with expected value 0 and finite variance, and let
:<math>S_{n} = X_{1} + \
Then there is a non-[[decreasing]] (a.k.a. weakly increasing) sequence ''τ''<sub>1</sub>, ''τ''<sub>2</sub>, ... of stopping times such that the <math>W_{\tau_{n}}</math> have the same joint distributions as the partial sums ''S''<sub>''n''</sub> and ''τ''<sub>1</sub>, ''τ''<sub>2</sub> − ''τ''<sub>1</sub>, ''τ''<sub>3</sub> − ''τ''<sub>2</sub>, ... are independent and identically distributed random variables satisfying
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