Estimation in Spectral Analysis - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 72

()

process. Thus, N →∞ F , θ becomes negligible compared to the second term and

we can write:

4

⎛ ⎞

2 σ 0

1

( )

ˆ

⎜ ⎟

C θ ≥

N

− 1

N ⎜ ⎟

0 R

⎝ ⎠

p

Before concluding this first part, here is an interesting result by Whittle which
expresses the normalized asymptotic Fisher information matrix in certain cases. Let
us consider a stationary Gaussian random process of zero mean and power spectral
density () .

x Sf

Then [DZH 86, POR 94]:

[ ]

− 1

()

F lim N ⎡ F N θ ⎤

⎣ ⎦

0

k , k ,

N →∞

[3.7]

() ()

∂ Sf ∂ Sf

1

1/ 2

1

x x

= ∫ df

2

2

()

∂θ ∂θ

− 1/ 2

Sf

k

x

This formula helps us, in particular, obtain extremely simple expressions in the
case of ARMA process (see [FRI 84a, FRI 84b] for example).

3.1.3. Sequence of estimators

The theoretical elements which have just been given are related to fixed
dimensional data vector N . In the context of random processes, we often study the
asymptotic behavior of estimators, that is to say when the dimension N of the data
vector increases 3 . This gives rise to an estimated sequence

ˆ θ and we study the

asymptotic behavior of this sequence, that is to say when N →∞ Before this we .

define the types of convergences considered.

Let ξ N be a sequence of random variables and a N a series of strictly positive real
numbers. We say that ξ N converges in probability to 0 if, whatever δ > 0 may be:

{ }

≥= lim P ξδ 0

N

N →∞

3 Another more pragmatic reason is that we rarely know how to carry out a statistical analysis
with finite Ν and most of the results require the hypothesis of a large number of samples
[STO 98].