Central Limit theorem 283with
g(σ) = ln∫
dxf(x)eiσφ(x). (7.2.11)Although we cannot evaluate the integral overσin eqn. (7.2.9) exactly, we can
approximate it by a technique known as thestationary phase method. This technique
applies to integrals of functionsF(σ,y) that are sharply peaked about a global max-
imum atσ= ̃σ(y) where the integral is expected to have its dominant contribution.
Forσ= ̃σ(y) to be a maximum, the following conditions must hold:
∂F
∂σ∣
∣
∣
∣
σ= ̃σ(y)= 0,
∂^2 F
∂σ^2∣
∣
∣
∣
σ= ̃σ(y)< 0 (7.2.12)
whenσ= ̃σ(y). Thus, the function of ̃σ(y) is derived from the solution of the condition
on the left in eqn. (7.2.12), which depends on the value ofy. We will return to this
point shortly. ExpandingF(σ,y) in a Taylor series aboutσ= ̃σ(y) up to second order
and taking into account that∂F/∂σ= 0 atσ= ̃σ(y), gives
F(σ,y) =F( ̃σ(y),y) +1
2
∂^2 F
∂σ^2∣
∣
∣
∣
σ= ̃σ(y)(σ− ̃σ(y))^2 +···. (7.2.13)Substituting eqn. (7.2.13) into eqn. (7.2.9) yields
P(y)≈M
2 πeMF( ̃σ(y),y)∫∞
−∞dσexp[
M
2
∂^2 F
∂σ^2∣
∣
∣
∣
σ= ̃σ(y)(σ− ̃σ(y))^2]
. (7.2.14)
Since the integral in eqn. (7.2.14) is now just a Gaussian integral overσ, it can be
performed straightforwardly to give
P(y)≈√
M
− 2 π(∂^2 F/∂σ^2 )|σ= ̃σ(y)eMF( ̃σ(y),y). (7.2.15)Thus, in order to specify the distribution, we need to find ̃σ(y) and (∂^2 F/∂σ^2 )|σ= ̃σ(y).
The condition∂F/∂σ= 0 leads to−iy+g′(σ) = 0 or
y=−ig′(σ) =∫
dxφ(x)f(x)eiσφ(x)
∫
dxf(x)eiσφ(x). (7.2.16)
The last line in eqn. (7.2.16) can, in principle, be inverted to give the solutionσ= ̃σ(y).
Moreover,
∂^2 F
∂σ^2
∣∣
∣
∣
σ= ̃σ(y)=g′′( ̃σ(y)). (7.2.17)Therefore,
P(y) =√
M
− 2 πg′′( ̃σ(y))eMF( ̃σ(y),y). (7.2.18)