O T O Statistics and Probability 11. Let 1.2.... be a sequence of iid zero-mean random variables. Let (a,) be a sequence of positive constants such that E a, = 1. Define the sequence of random variables Xo, X1, X2, ... via the recursion Xo = (0, X, = (1 - a,)X-1 + a, En n= 1, 2,... (a) Show that the collection {4;} is uniformly integrable. (b) Use proposition 3.36 to show that {X,} is uniformly integrable. Hint: For a convex function f : R - R and a e (0, 1), f(ax + (1- a)y) < af(x) + (1 - a)f(y).
O T O Statistics and Probability 11. Let 1.2.... be a sequence of iid zero-mean random variables. Let (a,) be a sequence of positive constants such that E a, = 1. Define the sequence of random variables Xo, X1, X2, ... via the recursion Xo = (0, X, = (1 - a,)X-1 + a, En n= 1, 2,... (a) Show that the collection {4;} is uniformly integrable. (b) Use proposition 3.36 to show that {X,} is uniformly integrable. Hint: For a convex function f : R - R and a e (0, 1), f(ax + (1- a)y) < af(x) + (1 - a)f(y).
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![O T O
Statistics and Probability
11. Let j. 2... be a sequence of iid zero-mean random variables. Let (a,) be
a sequence of positive constants such that E, a, = 1. Define the sequence of
random variables Xp, X1, X2, ... via the recursion
Xo = (0, X, = (1 a,)X-1 + an En n = 1, 2, ...
(a) Show that the collection (E} is uniformly integrable.
(b) Use proposition 3.36 to show that {X,} is uniformly integrable. Hint: For a
convex function : R - R and ar e (0, 1), f(ax + (1 - a)y) s af(x) + (1 -
a)f(y).
Proposition 3.36: Conditions for Uniform Integrability
Let K be a collection of random variables.
1. If K is a finite collection of integrable random variables, then it is UI.
2. If |X| < Y for all X e K and some integrable Y, then K is UI.
3. K is UI if and only if there is an increasing convex function f such that
f(x)
lim
sup E f(|X]) < 0o.
XEK
= 00
and
4. K is UI if supxex B|X|+e < oo for some ɛ > 0.
5. K is UI if and only if
(a) supxer B|X| < o, and
(b) for every ɛ > 0 there is a 8 > 0 such that for every event H:
P(H) s 8 = sup E|X|1H SE.
XEK
(3.37)
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Statistics and Probability
11. Let j. 2... be a sequence of iid zero-mean random variables. Let (a,) be
a sequence of positive constants such that E, a, = 1. Define the sequence of
random variables Xp, X1, X2, ... via the recursion
Xo = (0, X, = (1 a,)X-1 + an En n = 1, 2, ...
(a) Show that the collection (E} is uniformly integrable.
(b) Use proposition 3.36 to show that {X,} is uniformly integrable. Hint: For a
convex function : R - R and ar e (0, 1), f(ax + (1 - a)y) s af(x) + (1 -
a)f(y).
Proposition 3.36: Conditions for Uniform Integrability
Let K be a collection of random variables.
1. If K is a finite collection of integrable random variables, then it is UI.
2. If |X| < Y for all X e K and some integrable Y, then K is UI.
3. K is UI if and only if there is an increasing convex function f such that
f(x)
lim
sup E f(|X]) < 0o.
XEK
= 00
and
4. K is UI if supxex B|X|+e < oo for some ɛ > 0.
5. K is UI if and only if
(a) supxer B|X| < o, and
(b) for every ɛ > 0 there is a 8 > 0 such that for every event H:
P(H) s 8 = sup E|X|1H SE.
XEK
(3.37)
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Add a caption.
> My group
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