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Unformatted text preview: rocess, but rather than investigat
ing the interval (0, t] for a ﬁxed t, we want to inves
tigate (0, t] where t is selected by the sample path
up until t.
It is somewhat tricky to formalize this, since t b e
comes a rv which is a function of {X (t); τ ≤ t}. This
approach seems circular, so we have to b e careful.
We consider only discretetime processes {Xi; i ≥ 1}. 10 Let J b e a p ositive integer rv that describes when
a sequence X1, X2, . . . , is to b e stopped.
At trial 1, X1(ω ) is observed and a decision is made,
based on X1(ω ), whether or not to stop. If we stop,
J (ω ) = 1
At trial 2 (if J (ω ) = 1), X2(ω ) is observed and a
decision is made, based on X1(ω ), X2(ω ), whether
or not to stop. If we stop, J (ω ) = 2.
At trial 3 (if J (ω ) = 1, 2), X3(ω ) is observed and
a decision is made, based on X1(ω ), X2(ω ), X3(ω ),
whether or not to stop. If we stop, J (ω ) = 3, etc.
At each trial n (if stopping has not yet o ccurred),
Xn is observed and a decision (based on X1 . . . , Xn)
is made; if we stop, then J (ω ) = n. 11 Def: A stopping trial (or stopping time) J for {Xn; n ≥
1}, is a p ositive integervalued rv such that for each
n ≥ 1, the indicator rv I{J =n} is a function of
{X1, X2, . . . , Xn}.
A p ossibly defective stopping trial is the same ex
cept that J might b e defective.
We visualize ‘conducting’ successive trials X1, X2, . . . ,
until some n at which the event {J = n} o ccurs; fur
ther trials then cease. It is simpler conceptually to
visualize stopping the observation of trials after the
stopping trial, but continuing to conduct trials.
Since J is a (possibly defective) rv, the events {J =
1}, {J = 2}, . . . are disjoint. 12 Example 1: A gambler goes to a casino and gambles
until broke.
Example 2: Flip a coin until 10 successive heads
appear.
Example 3: Test an hypothesis with repeated tri
als until one or the other hypothesis is suﬃciently
probable a p osteriori.
Example 4: Observe successive renewals in a re
newal process until Sn ≥ 100. 13 Suppose...
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This note was uploaded on 01/13/2012 for the course ELECTRICAL 6.262 taught by Professor Staff during the Fall '11 term at MIT.
 Fall '11
 staff

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