The convolution theorem refines the local minimax bound. A regular estimator's limit cannot be narrower than the efficient Gaussian law: it must equal that law convolved with an independent residual distribution. The theorem is a structural statement about all local alternatives, not a variance calculation at one parameter.
Regular estimator sequences
Definition. In a LAN model with rate \(r_n\), an estimator \(T_n\) is regular at \(\theta_0\) if a probability law \(L\) exists such that, for every fixed \(h\), \[ r_n\{T_n-(\theta_0+r_n^{-1}h)\} \rightsquigarrow L \] under \(P_{n,\theta_0+r_n^{-1}h}\), and the limit does not depend on \(h\).
Regularity says that local parameter shifts are correctly tracked by the estimator. Equivalently, \[ r_n(T_n-\theta_0) \rightsquigarrow L(\cdot-h) \] under the local law indexed by \(h\).
Hájek–Le Cam convolution theorem
Theorem. Suppose the model is LAN at \(\theta_0\) with nonsingular information \(I\), and \(T_n\) is regular with limit \(L\). Then a probability measure \(M\) exists such that \[ L=N_p(0,I^{-1})*M. \] Equivalently, one may represent a variable with law \(L\) as \[ I^{-1/2}Z+V, \] where \(Z\sim N_p(0,I_p)\) and \(V\sim M\) are independent.
Proof outline. Pass to the Gaussian shift limit experiment jointly with the estimator. Regularity implies that the limiting Markov rule \(\delta\) is translation equivariant: \[ \mathcal L_h\{\delta(Y)-h\}=L \quad\text{for every }h. \] Let \(Y=h+I^{-1/2}Z\). Equivariance implies that the conditional characteristic function of \(\delta(Y)-Y\), after averaging appropriately over translations, does not depend on \(h\). The joint characteristic function factors as \[ \varphi_L(t) = e^{-\frac12t^\top I^{-1}t}\varphi_M(t) \] for a positive-definite function \(\varphi_M\). Bochner's theorem identifies \(\varphi_M\) as the characteristic function of a probability measure \(M\). Thus \(L\) is the stated convolution. \(\square\)
The deep step is positivity of the quotient characteristic function. It arises from the experiment structure and equivariance, not from formal division alone.
Covariance and convex-loss consequences
Corollary. If \(L\) has finite second moments, then \[ \operatorname{Cov}(L) = I^{-1}+\operatorname{Cov}(M) \succeq I^{-1}. \] Equality holds exactly when \(M\) is degenerate, up to a possible fixed asymptotic bias.
Corollary. If \(L\) is centered and \(\ell\) is convex, then \[ E\ell(I^{-1/2}Z+V) \geq E\ell(I^{-1/2}Z) \] whenever \(EV=0\).
Proof. Conditional Jensen inequality gives \[ E\{\ell(I^{-1/2}Z+V)\mid Z\} \geq \ell(I^{-1/2}Z+EV). \] If \(EV=0\), take expectations. \(\square\)
Efficient asymptotic linear estimators
Proposition. If \[ r_n(T_n-\theta_0) = I^{-1}\Delta_n+o_{P_{\theta_0}}(1), \] then under every fixed local alternative, \[ r_n\{T_n-(\theta_0+r_n^{-1}h)\} \rightsquigarrow N_p(0,I^{-1}). \] Hence the estimator is regular with no residual convolution noise.
Proof. Under the local alternative, \(\Delta_n\rightsquigarrow N(Ih,I)\) by the third lemma, and contiguity transfers the \(o_p(1)\) remainder. Therefore \[ I^{-1}\Delta_n-h \rightsquigarrow N(0,I^{-1}). \] \(\square\)
Superefficiency and failure of regularity
Example (Hodges estimator). If \(\hat\theta_n\) is efficient for a scalar parameter, define \[ T_n= \begin{cases} 0,&|\hat\theta_n|\leq n^{-1/4},\\ \hat\theta_n,&|\hat\theta_n|>n^{-1/4}. \end{cases} \] At \(\theta_0=0\), \[ \sqrt n\,T_n\to_p0, \] apparently beating the information bound. But under \(\theta_n=h/\sqrt n\), the threshold \(n^{-1/4}\) is much wider than the local displacement, so \[ \sqrt n(T_n-\theta_n)\to_p-h. \] The limit depends on \(h\); \(T_n\) is not regular. Its risk is poor on intermediate neighborhoods around zero.
The convolution theorem does not prohibit pointwise superefficiency. It says that improvement cannot persist with stable behavior across all contiguous local alternatives.
Asymptotic admissibility
Definition. Informally, a sequence \(T_n\) is locally asymptotically admissible if no competing sequence has asymptotic local risk no larger at every \(h\) and strictly smaller at some \(h\), under the chosen mode of risk convergence.
No-residual-noise efficiency is necessary for optimality among regular estimators under many convex losses, but it is not sufficient for admissibility among all procedures. In dimension \(p\geq3\), the identity estimator in the Gaussian shift is minimax and has covariance \(I^{-1}\), yet James–Stein shrinkage dominates it under squared loss. Consequently, an efficient estimator sequence can fail an asymptotic admissibility criterion that transfers the full Gaussian-shift decision problem.
Three distinct optimality statements
The conclusions should be separated:
- Efficiency: the regular limit has no extra convolution factor.
- Local asymptotic minimaxity: risk attains the unrestricted Gaussian minimax value over expanding local sets.
- Asymptotic admissibility: no limiting Gaussian-shift rule dominates the induced rule.
They coincide in some one-dimensional regular problems but diverge under shrinkage, constraints, nonconvex loss, or irregular procedures.