Chapter 7
Asymptotic Statistics
7.1. Generalities
Asymptotic statistics is the study of decision rules when the number of observations tends to infinity.
Theoretically, the asymptotic model may be described as follows: one considers a statistical model written as 
, a sequence 
of sub-σ-algebras of 
, and a sequence (dn, n ≥ 1) of 
-adapted decision rules (i.e. dn is 
-measurable for all n ≥ 1).
The decision space being provided with a distance δ, we say that (dn) is convergent in probability if:
where aθ denotes the “correct” decision when the value of the parameter is θ.
In the usual case, where (Xn,n ≥ 1) is a sample from Pθ, θ ∈ Θ, we have 
, 
, and , and the convergence in probability may be rewritten ...
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