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

Preliminary examples
and generalities
2.1. Informal large deviations
Chapter 2
One common use of large deviations is to find an estimate good enough
for the purpose at hand; e.g. proving a limit theorem. The following is an
illustration of such a situation.
Example 2.1. Let {Xn} be an i.i.d. sequence with E[e θX ] < ∞ for each
θ close to 0 (i.e. |θ| < δ for some δ > 0). Assume E[X] = 0. We would
like to show that Sn/n p → 0 P -almost surely, for any p > 1/2. When
p ≥ 1 this follows from the strong law of large numbers. Let us thus assume
p ∈ (1/2, 1). Next, for t ≥ 0 Chebyshev’s inequality (see, for example, page
15 of Durrett’s textbook [15]) implies
P {Sn ≥ εn p } ≤ E[e tSn−εtnp
] = exp{−εtn p + n log E[e tX ]}.
The exponential moment assumption on X implies that E[|X| k ]t k /k! is
summable, for t ∈ [0, δ). Recalling that E[X] = 0, we see that there exists
a δ0 > 0 and a constant c such that
E[e tX ] = E[e tX − tX] ≤ 1 +
when t ∈ [0, δ0]. Then, taking t = εnp
2nc
∞
k=2
t k
k! E[|X|k ] ≤ 1 + ct 2 ,
and n large enough,
P {Sn ≥ εn p } ≤ exp{−εtn p + n log(1 + ct 2 )}
≤ exp{−εtn p + nct 2 } = exp
− ε2
4c n2p−1
.
11