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Is there a name for a process defined this way: for all $t \in [0,\infty)$, $X_t$ is a Bernoulli RV with p=1/2, and the $X_t$ are mutually independent. Basically an IID sequence of coin flips in continuous time.

And a follow-up question: suppose the coin is weighted by a fixed weight $p$ but $p$ is unknown initially. Is it true that the law of large numbers implies that at any time $t>0$ one can identify the weight of the coin with probability 1 after observing the process up to time $t$?

Edit: To be more precise about the follow-up question, assume that the coin weighting is drawn from some continuous distribution on $[0,1]$ with full support.

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Are you sure you want this, or something on the random walk created from the Bernoulli variables? What you describe would just be a Bernoulli variable in continuous time, and I'm not quite sure how that would be of interest over the discrete case. –  gnometorule Dec 2 '12 at 17:50

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