Correction to “Modeling Time-Varying Random Objects and Dynamic Networks”

This article refers to:

Modeling Time-Varying Random Objects and Dynamic Networks

Paromita Dubey^a and Hans-Georg Müller^b

^aDepartment of Data Sciences and Operations, University of Southern California, Los Angeles, CA; ^bDepartment of Statistics, University of California, Davis, Davis, CA

CONTACT Paromita Dubey [email protected] Department of Data Sciences and Operations, University of Southern California, Los Angeles, CA.

The last inequality in the proof of Lemma 2 in the supplementary materials of Dubey and Müller (2021) is incorrect, as we discovered when addressing an inquiry by Jiazhen Xu, Australian National University. The corrected Lemma 2 in the updated supplement gives the rate$$\begin{array}{c}\underset{s,t\in [0,1]}{\sup }\left|\widehat{C}(s,t)-\tilde{C}(s,t)\right|\\ \hspace{1em}=O_P\left(\max \left\{{\left(\frac{\sqrt{\hspace{0.17em}\log \hspace{0.17em}n}}{n}\right)}^{\frac{1}{2(\beta -1)}},\frac{1}{\sqrt{n}}\right\}\right)\end{array}$$where β is as defined in assumption (A2). This impacts the rates of convergence in Theorems 1 and 2 in Dubey and Müller (2021). The corrected results are stated hereafter.

Theorem 1.

Under assumptions (A1)–(A4),$$\begin{array}{c}\underset{s,t\in [0,1]}{\sup }\left|\widehat{C}(s,t)-C(s,t)\right|\\ \hspace{1em}=O_P\left(\max \left\{{\left(\frac{\sqrt{\hspace{0.17em}\log \hspace{0.17em}n}}{n}\right)}^{\frac{1}{2(\beta -1)}},\frac{1}{\sqrt{n}}\right\}\right).\end{array}$$

Theorem 2.

Under assumptions (A1)–(A6), we have$$\begin{array}{c}|{\widehat{\lambda }}_j-{\lambda }_j|\\ \hspace{1em}=O_P\left(\max \left\{{\left(\frac{\sqrt{\hspace{0.17em}\log \hspace{0.17em}n}}{n}\right)}^{\frac{1}{2(\beta -1)}},\frac{1}{\sqrt{n}}\right\}\right),\\ \underset{s\in [0,1]}{\sup }\left|{\widehat{\varphi }}_j(s)-{\varphi }_j(s)\right|\\ \hspace{1em}=O_P\left(\frac{1}{{\delta }_j}\max \left\{{\left(\frac{\sqrt{\hspace{0.17em}\log \hspace{0.17em}n}}{n}\right)}^{\frac{1}{2(\beta -1)}},\frac{1}{\sqrt{n}}\right\}\right)\mathrm{},\\ |{\widehat{B}}_{ij}-B_{ij}|\\ \hspace{1em}=O_P\left(\max \left\{\frac{1}{{\delta }_j},1\right\}\max \left\{{\left(\frac{\sqrt{\hspace{0.17em}\log \hspace{0.17em}n}}{n}\right)}^{\frac{1}{2(\beta -1)}},\frac{1}{\sqrt{n}}\right\}\right)\mathrm{}.\end{array}$$

Everything else remains intact.