LONG GAPS BETWEEN PRIMES 7
We write P for probability, and E for expectation. If X takes at most countably many values, we define
the essential range of X to be the set of all X such that P(X = X) is non-zero, thus X almost surely takes
values in its essential range. We also employ the following conditional expectation notation. If E is an event
of non-zero probability, we write
P(F |E) :=
P(F ∧ E)
P(E)
for any event F , and
E(X|E) :=
E(X1
E
)
P(E)
for any (absolutely integrable) real-valued random variable X. If Y is another random variable taking at
most countably many values, we define the conditional probability P(F |Y) to be the random variable that
equals P(F |Y = Y ) on the event Y = Y for each Y in the essential range of Y, and similarly define the
conditional expectation E(X|Y) to be the random variable that equals E(X|Y = Y ) on the event Y = Y .
We observe the idempotency property
(2.1) E(E(X|Y)) = EX
whenever X is absolutely integrable and Y takes at most countably many values.
We will rely frequently on the following simple concentration of measure result.
Lemma 2.1 (Chebyshev inequality). Let X, Y be independent random variables taking at most countably
many values. Let Y
′
be a conditionally independent copy of Y over X; in other words, for every X in
the essential range of X, the random variables Y, Y
′
are independent and identically distributed after
conditioning to the event X = X. Let F(X, Y) be a (absolutely integrable) random variable depending on
X and Y. Suppose that one has the bounds
(2.2) EF (X, Y) = α + O(εα)
and
(2.3) EF (X, Y)F (X, Y
′
) = α
2
+ O(εα
2
)
for some α, ε > 0 with ε = O(1). Then for any θ > 0, one has
(2.4) E(F (X, Y)|X) = α + O
6
(θ)
with probability 1 − O(
εα
2
θ
2
).
In practice, we will often establish (2.2) and (2.3) by first computing the conditional expectations
E(F (X, Y)|Y)
and
E(F (X, Y)F(X, Y
′
)|Y, Y
′
)
and then using (2.1). Thus we see that we can control the X-conditional expectation of F (X, Y) via the
Y-conditional expectation, provided that we can similarly control the Y, Y
′
-conditional expectation of
F (X, Y)F(X, Y
′
).
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