content_image/305153.jpg
Φ(Λ , m, V ) for some non-negative-valued function Λ; a generalized function b ∈ Φ(Λ , m, V ) if there exists a function ¯ b ( ζ ) ∈ Φ( m, V ) such that also ¯ b ( ζ ) − 1 ∈ Φ( m, V ) and b = ¯ b ( ζ ) exp ( − Λ( ζ )) . Note that a linear combination of functions in Φ(Λ , m, V ) belongs to the same class. Define convergence: a sequence of b$_{n}$ ∈ Φ(Λ , m, V ) converges to zero if the corresponding sequence ¯ b$_{n}$ converges to zero in S $^{∗}$. 

Convergence in probability for a sequence of random functions, ε$_{n}$, in S ∗ is defined as follows: ( ε$_{n}$ − ε ) →$_{p}$ 0 in S ∗ if for any set ψ$_{1}$, ..., ψ$_{v}$ ∈ S the random vector of the values of the functionals converges: (( ε$_{n}$ − ε, ψ$_{1}$ ) , ..., ( ε$_{n}$ − ε, ψ$_{v}$ )) →$_{p}$ 0 . 

Lemma 2. If in model 1 φ$_{u}$ = b ∈ Φ(Λ , m, V ) , where Λ is a polynomial function of order no more than k, and ε$_{n}$ is a sequence of estimators of ε that are consistent in S ∗ : r$_{n}$ ( ε$_{n}$ − ε ) →$_{p}$ 0 in S ∗ at some rate r$_{n}$ → ∞ , then for any sequence of constants ¯ B$_{n}$ : 0 < ¯ B$_{n}$ < (ln r$_{n}$ ) 1 k and the corresponding set B$_{n}$ = { ζ : ‖ ζ ‖ < ¯ B$_{n}$ } the sequence of regularized estimators φ − 1 u ( ε$_{n}$ − ε ) I ( B$_{n}$ ) converges to zero in probability in S $^{∗}$. 

Proof. For n the value of the random functional 

<img src='content_image/305153.jpg'>

Multiplication by ¯ b − 1 ∈ Φ( m, V ) , that corresponds to φ$_{u}$ = b does not affect convergence thus ¯ b − $^{1}$( ζ ) r$_{n}$ ( ε$_{n}$ − ε ) converges to zero in probability in S $^{∗}$. To show that ( φ − 1 u ( ε$_{n}$ − ε ) I ( B$_{n}$ ) , ψ ) converges to zero it is sufficient to show that 

