Correction Limit theorems for coupled continuous time random walks
aop635.tex
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% Institute of Mathematical Statistics (IMI)
% Journal "The Annals of Probabability"

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\documentclass[aop,citesort,MSNbibl,dvips]{arximspdf}

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% article settings
\doi{10.1214/10-AOP635}
\volume{40}
\issue{2}
\pubyear{2012}
\firstpage{890}
\lastpage{891}


\begin{document}
%
\begin{frontmatter}

\title{Correction
Limit theorems for coupled continuous time random walks}\vspace*{12pt}
\runtitle{Correction}
\pdftitle{Correction
Limit theorems for coupled continuous time random walks}

\textit{Ann. Probab.} \textbf{32} (2004) 730--756

\begin{aug}
\author[A]{\fnms{Peter} \snm{Kern}\ead[label=e1]{kern@math.uni-duesseldorf.de}},
\author[B]{\fnms{Mark M.} \snm{Meerschaert}\corref{}\ead[label=e2]{mcubed@stt.msu.edu}}\\ and
\author[C]{\fnms{Hans-Peter} \snm{Scheffler}\ead[label=e3]{scheffler@mathematik.uni-siegen.de}}
\runauthor{P. Kern, M. M. Meerschaert and H.-P. Scheffler}
\affiliation{Heinrich Heine University, Michigan State University
and University~of~Siegen}
\address[A]{P. Kern\\
Mathematisches Institut\\
Heinrich Heine University\\
40225 D\"{u}sseldorf\\
Germany\\
\printead{e1}} %adresu isvedimo komanda gale!
\address[B]{M. M. Meerschaert\\
Department of Statistics and Probability\\
Michigan State University\\
East Lansing, Michigan 48824\\
USA\\
\printead{e2}}
\address[C]{H.-P. Scheffler\\
Fachbereich Mathematik\\
University of Siegen\\
57068 Siegen\\
Germany\\
\printead{e3}}
\end{aug}

% HISTORY:
\received{\smonth{8} \syear{2010}}
\revised{\smonth{11} \syear{2010}}

% ABSTRACT

% KEYWORDS

\end{frontmatter}

The converse portion of Theorem 2.2 requires an additional condition,
that the probability measure $\omega$ is such that (2.10) assigns
finite measure to sets bounded away from the origin. The argument on
page 735 must consider~$B_1$ and $B_2$ such that {at least one} is bounded
away from zero, not just the case where both are bounded away from zero.
%This was the flaw in the original proof.
The condition on~$\omega$ ensures that the integral on page 735 l.--2 is
finite, which is obviously necessary.

%Lemma 3.3 is true as stated. The proof of Lemma 3.3 argues that (3.17)
%holds as $c\to\infty$ uniformly in $0\leq s\leq s_1$ using Laplace
%transforms. This argument is suspect since the functions involved are
%not monotone, so that the continuity
%Theorem for Laplace-transforms, Theorem 2a on page 433 in
%
%First note that $\tilde b(c)(c^{-1}\rho)\to\bar\phi$ (vague
%convergence) where $\phi(r,\infty)=r^{-\beta}/\Gamma(1-\beta)$ is the L
%convergence criterion for nonnormal limits (see, e.g., Theorem 3.3.8
%in \cite{thebook}), which states that $c\cdot(b(c)\rho)\to\bar\phi$.
%Then $\tilde b(c)\cdot(b(\tilde b(c))\rho)\to\bar\phi$ as well, and
%then $\tilde b(c)(c^{-1}\rho)\to\bar\phi$ follows from (3.3).
%
%Next we will use the argument on p.744 at the end of the proof of
%Lemma 3.3, which states that if functions converge uniformly on a
%compact interval and the measures converge vaguely, then the integrals
%also converge. Write
%(H_s(t)-H_s(t-\tau))\frac{\beta\tau^{-\beta-1}}{\Gamma(1-
%=\int_0^\infty(H_s(t)-H_s(t-\tau))d\bar\phi(\tau)\]
%using a well-known formula for the fractional derivative (see, e.g.,
%b(c)\cdot(\varepsilon_0-(c^{-1}\rho))](t)
%=\int_0^\infty(F_{\tilde b(c)s}^{(c)}(t)-F_{\tilde
%b(c)s}^{(c)}(t-\tau))\tilde b(c)(c^{-1}\rho)(\tau)\]
%and now we would like to show that for any $s(c)\to s$ in the interval
%$0\leq s\leq s_1$ we have
%as $c\to\infty$ uniformly in $J(t)=\{\tau:0\leq\tau\leq t\}$ for any
%$t>0$. To see this, first note that $H_s(\tau)=P(A(s)\in M,D(s)\leq
%Then $H_s(\tau)$ is also uniformly continuous on $0\leq\tau\leq t$.
%Given $\varepsilon>0$, choose $\delta>0$ such that
%|H_s(\tau)-H_s(\tau')|<\varepsilon/2 \text{whenever $|\tau-\tau'|<
% for $\tau,\tau'\in J(t)$. Next choose a finite set $\{\tau_n\}$ such
%that $\tau_{n+1}=\tau_n+\delta/2$.
% %Note that this implies
%%\begin{equation}\label{eq1}
%%|H_s(\tau_{n+1})-H_s(\tau_n)|<\varepsilon/2 \text{ for all $n$.}
%%\end{equation}
%Since $\{\tau_n\}$ is a finite set, then using the fact that (3.12)
%holds uniformly on compact subsets of $s\geq0$, for some $c_0>0$ we
%also have for all $c>c_0$ that
%|F_{\tilde b(c)s(c)}^{(c)}(\tau_n)-H_s(\tau_n)|<\varepsilon/2
%Given $\tau\in J(t)$, choose $n$ such that $\tau_n\leq\tau<
%using the monotonicity of $F_{\tilde b(c)s(c)}^{(c)}(\tau)$, and then
%using the monotonicity of $F_{\tilde b(c)s(c)}^{(c)}(\tau)$, and then
%follows that
%uniformly in $\tau\in J(t)$ for all $c>0$ sufficiently large. In other
%words, we have shown that $F_{\tilde b(c)s}^{(c)}(\tau)\to H_s(\tau)$
%uniformly on $0\leq\tau\leq t$ and $0\leq s\leq s_1$ as $c\to\infty$.
%
%Now in order to show that (3.17) holds uniformly in $0\leq s\leq s_1$
%as $c\to\infty$, first note that
%b(c)s(c)}^{(c)}(t-\tau))\tilde b(c)(c^{-1}\rho)(\tau)
%using the vague convergence, since both integrands are constant for $
%) \to(H_s(t)-H_s(t-\tau))\]
%uniformly in $0\leq\tau\leq t$, and this along with the vague
%convergence implies that
%b(c)s(c)}^{(c)}(t-\tau))\tilde b(c)(c^{-1}\rho)(\tau)
%using the same argument as on p.744. This shows that (3.17) holds as $c
%words, the convergence is uniform on compact subsets of $s\geq0$.

The limit process in Theorem 3.4 should read $A(E(t)-)$. If $A(t)$
and~$D(t)$ are dependent, this is a different process than $A(E(t))$. To
clarify the argument, note that
%
%e1 ###
\begin{equation}\label{mistake}\qquad
\lim_{h\downarrow0}\frac1hP\{A(s)\in M, s<E(t)\leq s+h\}= P\{A(s-)\in
M | E(t)=s\} p_t(s),
\end{equation}
%
where $p_t$ is the density of $E(t)$, since $s<E(t)$ in the
conditioning event. For an alternative proof, see Theorem 3.6 in Straka
and Henry \cite{StrakaHenry}. Theorem~4.1 in \cite{coupleCTRW} gives
the density of $A(E(t)-)$.
Examples 5.2--5.6 in \cite{coupleCTRW} provide governing equations for
the CTRW limit process $M(t)=A(E(t)-)$ in some special cases with
simultaneous jumps. Especially, Example 5.5 considers the case where
$Y_i=J_i$ so that $A(t)$ is a stable subordinator and $E(t)=\inf\{
x>0\dvtx A(x)>t\}$ is its inverse or first passage time process. The beta
density for $A(E(t)-)$ given in that example agrees with the result in
Bertoin \cite{bertoin}, page~82. Note that here we have $A(E(t)-)<t$ and
$A(E(t))>t$ almost surely for any $t>0$, by \cite{bertoin}, Chapter III, Theorem~4.

% imsref loaded by lrinkeviciute, 2011-01-26 11:16:55
\begin{thebibliography}{4}
% BibTex style file: ims.bst, 2010-01-14
% Default style options (sort=0,type=number).
% Used options (sort=1,type=number).

%b1 ###
\bibitem{coupleCTRW}
\begin{barticle}[mr]
\bauthor{\bsnm{Becker-Kern},~\bfnm{Peter}\binits{P.}},
  \bauthor{\bsnm{Meerschaert},~\bfnm{Mark~M.}\binits{M.~M.}} \AND
  \bauthor{\bsnm{Scheffler},~\bfnm{Hans-Peter}\binits{H.-P.}}
(\byear{2004}).
\btitle{Limit theorems for coupled continuous time random walks}.
\bjournal{Ann. Probab.}
\bvolume{32}
\bpages{730--756}.
\bid{doi={10.1214/aop/1079021462}, mr={2039941}}
\end{barticle}
\endbibitem

%b2 ###
\bibitem{bertoin}
\begin{bbook}[mr]
\bauthor{\bsnm{Bertoin},~\bfnm{Jean}\binits{J.}}
(\byear{1996}).
\btitle{L\'evy Processes}.
\bseries{Cambridge Tracts in Mathematics}
\bvolume{121}.
\bpublisher{Cambridge Univ. Press}, \baddress{Cambridge}.
\bid{mr={1406564}}
\end{bbook}
\endbibitem

%b3 ###
\bibitem{StrakaHenry}
\begin{barticle}[auto:STB|2010-11-18|09:18:59]
\bauthor{\bsnm{Straka},~\bfnm{P.}\binits{P.}} \AND
  \bauthor{\bsnm{Henry},~\bfnm{B.~I.}\binits{B.~I.}}
(\byear{2011}).
\btitle{Lagging and leading coupled continuous time random
walks, renewal times and their joint limits}.
\bjournal{Stochastic Process. Appl.}
\bvolume{121}
\bpages{324--336}.
%DOI:
\end{barticle}
\endbibitem

\end{thebibliography}

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\printaddresses

\end{document}
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1208.3372.extracted_bib
\begin{thebibliography}{4}
% BibTex style file: ims.bst, 2010-01-14
% Default style options (sort=0,type=number).
% Used options (sort=1,type=number).

%b1 ###
\bibitem{coupleCTRW}
\begin{barticle}[mr]
\bauthor{\bsnm{Becker-Kern},~\bfnm{Peter}\binits{P.}},
  \bauthor{\bsnm{Meerschaert},~\bfnm{Mark~M.}\binits{M.~M.}} \AND
  \bauthor{\bsnm{Scheffler},~\bfnm{Hans-Peter}\binits{H.-P.}}
(\byear{2004}).
\btitle{Limit theorems for coupled continuous time random walks}.
\bjournal{Ann. Probab.}
\bvolume{32}
\bpages{730--756}.
\bid{doi={10.1214/aop/1079021462}, mr={2039941}}
\end{barticle}
\endbibitem

%b2 ###
\bibitem{bertoin}
\begin{bbook}[mr]
\bauthor{\bsnm{Bertoin},~\bfnm{Jean}\binits{J.}}
(\byear{1996}).
\btitle{L\'evy Processes}.
\bseries{Cambridge Tracts in Mathematics}
\bvolume{121}.
\bpublisher{Cambridge Univ. Press}, \baddress{Cambridge}.
\bid{mr={1406564}}
\end{bbook}
\endbibitem

%b3 ###
\bibitem{StrakaHenry}
\begin{barticle}[auto:STB|2010-11-18|09:18:59]
\bauthor{\bsnm{Straka},~\bfnm{P.}\binits{P.}} \AND
  \bauthor{\bsnm{Henry},~\bfnm{B.~I.}\binits{B.~I.}}
(\byear{2011}).
\btitle{Lagging and leading coupled continuous time random
walks, renewal times and their joint limits}.
\bjournal{Stochastic Process. Appl.}
\bvolume{121}
\bpages{324--336}.
%DOI:
\end{barticle}
\endbibitem

\end{thebibliography}