The concept of Local Asymptotic Normality (LAN) – introduced by Lucien LeCam – is one of the most important and fundamental ideas of the general asymptotic statistical theory. The LAN property is of particular importance in the asymptotic theory of testing, estimation and discriminant analysis. Many statistical models have got likelihood ratios which are locally asymptotic normal – that is the likelihood ratio processes of those models are asymptotically similar to those for the normal location parameter.

Let and be two sequences of probability measures on . Suppose there is a sequence , of sub -algebras of s.th. and . Let be the restriction of to and let be the Radon-Nikodym density taken on of the part of that is dominated by . Put

where and .

The logarithm of likelihood ratio

taken on is then

since .

(LeCam 1986). Suppose that under the following conditions are satisfied

- L1:
- L2: ,
- L3: , and
- L4: for some . then

*Proof:* Note that

where if . Note that

Hence

For some define

hence using L1

and

Since

Note also that

Hence using L3, L4

Thus can be expressed as

where

Since we can use the *dominated convergence theorem* to have

which implies that

Note that is a martingale difference array and that

and

Hence

Note that is i) uniformly bounded in norm, ii) and iii) .

Hence by applying McLeish’s central limit theorem we finally prove that

References:

L. Le Cam, G.L. Yang (2000). Asymptotics in Statistics. Springer-Verlag, New York

A.W. van der Vaart (2000). Asymptotic Statistics. Cambridge University Press

D. L. McLeish. Dependent Central Limit Theorems and Invariance Principles.

Ann. Probab. Volume 2, Number 4 (1974), 620-628.

M. Taniguchi, Y. Kakizawa (1998). Asymptotic Theory of Statistical Inference for Time Series. Springer-Verlag, New York

(notes based mostly on the latter text)

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