Theory of Statistics - George Mason University

1.6 Stochastic Processes 133
X0 has any given distribution with V(X0) > 0. The sequence {Xt : EXt =
EXt−1, VXt = t−1
k=0 VXk} is not a Markov chain.
A Markov chain that is not a martingale, for example, is {Xt : Xt
2Xt−1}, where X0 has any given distribution with E(X0) = 0.
A common application of martingales is as a model for stock prices. As
a concrete example, we can think of a random variable X1 as an initial sum
(say, of money), and a sequence of events in which X2, X3, . . . represents a
sequence of sums with the property that each event is a “fair game”; that
is, E(X2|X1) = X1 a.s., E(X3|X1, X2) = X2 a.s., . . .. We can generalize this
somewhat by letting Dn = σ(X1, . . ., Xn), and requiring that the sequence be
such that E(Xn|Dn−1) a.s.
= Xn−1.
Doob’s Martingale Inequality
A useful property of submartingales is Doob’s martingale inequality. This
inequality is a more general case of Kolmogorov’s inequality (B.11), page 841,
and the Hájek-Rènyi inequality (B.12), both of which involve partial sums
that are martingales.
Theorem 1.69 (Doob’s Martingale Inequality)
Let {Xt : t ∈ [0, T]} be a submartingale relative to {Dt : t ∈ [0, T]} taking
nonnegative real values; that is, 0 ≤ Xs ≤ E(Xt|Dt) for s, t. Then for any
constant ɛ > 0 and p ≥ 1,
Pr sup Xt ≥ ɛ
0≤t≤T
d
=
≤ 1
ɛ pE(|XT | p ). (1.281)
Proof. ***fix
Notice that Doob’s martingale inequality implies Robbins’s likelihood ratio
martingale inequality (1.279).
Azuma’s Inequality
extension of Hoeffding’s inequality (B.10), page 840
1.6.7 Empirical Processes
Given a random sample X1, . . ., Xn with the order statistics X(1), . . ., X(n),
we can form a conditional discrete distribution with CDF
⎧
⎨0
x < X(1)
Fn(x) = k/n X(k) ≤ x < X(k+1) for 1 ≤ k < n (1.282)
⎩
1 X(n) ≤ x.
Theory of Statistics c○2000–2013 James E. Gentle