Theory of Statistics - George Mason University

Exercises 153
1.85. Show that Doob’s martingale inequality (1.281) implies Robbins’s likelihood
ratio martingale inequality (1.279).
1.86. Let {Mn} be a martingale, and let {Cn} be adapted to {σ(Mt : t ≤ n)}.
Let M0 = 0, and for n ≥ 1, let
Mn =
n
Cj(Mj − Mj−1).
j=1
Show that Mn is a martingale.
The sequence Mn is called the martingale transform of {Mn} by {Cn}.
1.87. Let X1, X2, . . . be a sequence of independent random variables over a
common probability space such that for each E(X2 i ) < ∞. Show that the
sequence of partial sums
Yn =
n
(Xi − E(Xi))
i=1
is a martingale.
1.88. Let X be a random variable that is normally distributed with mean µ and
variance σ 2 . Show that the entropy of X is at least as great as the entropy
of any random variable with finite mean µ and finite variance σ 2 and having
a PDF that is dominated by Lebesgue measure. (See Exercise 1.32a.)
1.89. Show that the axioms for coherency given on page 139 define a linear
ordering, that is, a total ordering, on A. (See page 614.)
Theory of Statistics c○2000–2013 James E. Gentle