A Course on Large Deviations with an Introduction to Gibbs Measures.

3.2. Cramér’s theorem 21
3.2. Cramér’s theorem
While proving the large deviation principle in several simple situations (e.g.
Example 1.1 and Exercises 2.5, 2.6, and 3.4) one notices similarities in the
arguments. Indeed, these results fall under the umbrella of a general large
deviation principle for the sample mean Sn/n of i.i.d. random variables.
(Recall that Sn = X1 + . . . + Xn.) This result, called Cramér’s theorem,
is one of the central results of large deviation theory and certainly the one
most frequently applied.
Cramér’s theorem is valid not only for real-valued random variables,
but also for R d -valued random vectors, and even for some infinite dimensional
random vectors. The infinite dimensional case will not be discussed
in the book. In this section we first develop the upper bound for the
one-dimensional case through a series of exercises. Then we state the onedimensional
Cramér’s theorem in full generality, and explore it further in
exercises. The notion of convexity makes a somewhat premature but small
appearance in this section, although it is not taken up seriously until Chapter
5.
Next we state the multidimensional theorem in full, and prove parts of
it. The final discussion of Cramér’s theorem takes place later in Section 5.3.
There, armed with some ideas from convex analysis, we prove the multidimensional
result under the natural assumption that the moment generating
function is finite in a neighborhood of the origin.
Let {Xn} be a sequence of i.i.d. real-valued random variables, and write
X for another random variable with the same distribution. Recall the moment
generating function M(θ) = E[e θX ] for θ ∈ R. Observe that M(θ) > 0
always and it can happen that M(θ) = ∞. Using Chebyshev’s inequality
(page 15 of [15]) write,
(3.3)
(3.4)
P {Sn ≥ nb} ≤ e −nθb E[e θSn ] = e −nθb M(θ) n , for θ ≥ 0,
P {Sn ≤ na} ≤ e −nθa E[e θSn ] = e −nθa M(θ) n , for θ ≤ 0.
From above we get immediately the upper bounds
1
lim
n→∞ n log P {Sn ≥ nb} ≤ − sup{θb
− log M(θ)},
θ≥0
1
lim
n→∞ n log P {Sn ≤ na} ≤ − sup{θa
− log M(θ)}.
θ≤0