Avoid common mistakes on your manuscript.
Correction to: Probab. Theory Relat. Fields (2007) 138:325–361 https://doi.org/10.1007/s00440-006-0025-2
A problem appears in the end of Lemma 8 page 355. The first part of the Lemma remains true but the second weak convergence result for the random predictor is not exact in general. In fact \(Z_{i,n}\) is not a martingale difference sequence with respect to the filtration generated by \(\left( X_{1},\varepsilon _{1},\ldots ,X_{n},\varepsilon _{n}\right) \) and the CLT mentioned in the end of page 355 cannot be invoked.
A new version of the second part of Lemma 8 is given below. An additional assumption denoted (1) below is also required for Theorem 2 to hold. It is satisfied in a wide range of examples and applications.
Let \(X_{i}=\sum _{l=1}^{+\infty }\sqrt{\lambda _{l}}\xi _{l,i}e_{l}\) be the Karhunen-Loeve expansion of \(X_{i}\) given at page 334. Assume that the sequence of the squared principal component satisfies the weak law of large numbers: when L tends to infinity,
then the second part of Lemma 8 holds namely:
Assumption (1) holds for Gaussian X and more generally when the principal components \(\xi _{l}\)’s are independent.
FormalPara ProofIn order to clarifiy we set below \(X_{0}=X_{n+1}\), \(s_{L,i}=\frac{1}{L}\sum _{l=1}^{L}\xi _{l,i}^{2}\) and denote \({\mathbb {E}}_{i}\) the expectation w.r.t. the couple \(\left( X_{i},\varepsilon _{i}\right) \). The derivation relies on proving classical pointwise convergence for the characteristic function of \(S_{n}=\frac{1}{\sqrt{ns_{n}}}\sum _{i=1}^{n}Z_{i,n}\) with \(Z_{i,n}=\left\langle \Gamma ^{\dag }X_{i},X_{0}\right\rangle \varepsilon _{i}\).
We prove the result above in the specific case of PCA-spectral cut then \(f_{n}\left( x\right) =1/x\) for \(x\ge \lambda _{k_{n}}\) (then \(s_{n}=k_{n}\)). The reader will check that it does not alter the generality of the statement.
Then with \(\varphi _{S_{n}}\left( t\right) ={\mathbb {E}}\left( \exp \left( itS_{n}\right) \right) \)
With the above notations on Karhunen-Loeve expansion for X
where \(\left( \xi _{l,1}\right) _{1\le l\le k_{n}}\) is independent from \(\left( \xi _{l,0}\right) _{1\le l\le k_{n}}\). Simple computations give, \({\mathbb {E}}_{1}\left[ Z_{1,n}\right] =0\), \({\mathbb {E}}_{1}\left[ Z_{1,n}^{2}\right] =\sigma _{\varepsilon }^{2}\sum _{l=1}^{k_{n}}\xi _{l,0}^{2}\), and Cauchy-Schwarz inequality yields
Taken from Jensen’s inequality, the bound
leads to \({\mathbb {E}}_{1}\left( \sum _{l=1}^{k_{n}}\xi _{l,1}^{2}\right) ^{3/2}\le k_{n}^{1/2} \sum _{l=1}^{k_{n}}{\mathbb {E}}\left| \xi _{l,1}\right| ^{3}\le M^{3/4}k_{n}^{3/2}\) where M appears in assumption \(\left( A.3\right) \) page 334 and finally to
Then a Taylor expansion for the characteristic function of \(\left\langle \Gamma ^{\dag }X_{1},X_{0}\right\rangle \varepsilon _{1}\) is
where \(H_{n}\left( t\right) ={\mathbb {E}}_{1}\left[ {\varepsilon }_{1}^{3}\left\langle \Gamma ^{\dag }X_{1},X_{0}\right\rangle ^{3}\exp \left( i\tau _{t}\left\langle \Gamma ^{\dag }X_{1},X_{0}\right\rangle \varepsilon _{1}\right) \right] \) for some \(\tau _{t}\in \left( 0,t/\sqrt{nk_{n}}\right) \) is a remainder term in the Taylor’s expansion of \(\varphi _{Z_{1,n}}\). Hence, we get from (2),
Remind that \(s_{k_{n},0}=\frac{1}{k_{n}}\sum _{l=1}^{k_{n}}\xi _{l,0}^{2}\). From the equations above we can write,
where this time: \(\sup _{\left( n,t\right) }\left| {\widetilde{H}}_{n}\left( t\right) \right| \le M^{3/4}{\mathbb {E}}\left| {\varepsilon }_{1}\right| ^{3}\). Then, with assumption (7) page 334 in Theorem 2, we have that \(k_{n}^{3}/n\) tends to zero when n tends to infinity so that,
when \(\frac{1}{k_{n}}\sum _{l=1}^{k_{n}}\xi _{l,0}^{2}\) is an \(O_{{\mathbb {P}}}\left( 1\right) \). In order to conclude we have to take expectation with respect to \(X_{0}\) and integrate to the limit.
First of all it is plain from (3), the assumption on \(\frac{1}{k_{n}}\sum _{l=1}^{k_{n}}\xi _{l,0}^{2}\) and the continuous mapping theorem that
Besides \(\left[ {\mathbb {E}}_{1}\exp \left( \frac{it}{\sqrt{nk_{n}}} Z_{1,n}\right) \right] ^{n}\le 1\mathbb {\ }\)almost surely and is uniformly integrable with respect to \({\mathbb {E}}_{0}\). We can conclude that
which concludes the proof of the Lemma. \(\square \)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Cardot, H., Mas, A. & Sarda, P. Correction: CLT in functional linear regression models. Probab. Theory Relat. Fields 187, 519–522 (2023). https://doi.org/10.1007/s00440-023-01215-7
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-023-01215-7