Unformatted text preview: Problem 4 (19 pts). Let
Denote denote a standard Brownian motion. a. (5 points) Let
is a constant
∑
∑
Hint: Prove that there
that doesn’t depend on n or
a
. so that:
 . For the rest of the problem let
denote the rational numbers in
and for each
let the order statistics of
be
denoted by:
[ Example: If then
.] b. (5 points)
(i). Prove: ∑
bab
as
∞.
(
)
(ii). How can you improve the result in (i) if there exist ℱ , a decreasing
sequence of σalgebras, so that {∑
(
) ℱ } is a
backwards martingale?
c...
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 Spring '12
 STAFF
 Probability, Probability theory

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