While by definition the increments of a Brownian motion are independent, it is unclear to me whether (that implies that) the random variables $W_t$ and $W_s$ are independent for $t \neq s$. While these random variables have different density functions, they are defined on the same state space, and I am not sure whether they are independent.
Independence of Brownian motion
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stochastic-processes
brownian-motion
random-variables
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0R$i$ght, poorly formulated. – 2012-11-17
2 Answers
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Suppose $W_t$ and $W_s$ are independent for $t\neq s$. Then for $0\leq s
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Take $t, then you can write $W(s)=W(t)+(W(s)-W(t))$, from which you immediately see that they are not independent.
It is the increments which are independent. Take e.g. $t, then $W(t)-W(0)$ and $W(s)-W(t)$ are independent.