\def\filename{aptpub.cls}
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\def\filedate{8 January 2004}
%% This is aptpub.cls, public version
%% Copyright (C) 2001 Applied Probability Trust.
%% Written by Stephen Webb.
%% Updated by Maria Kelly and by David Cruickshank
%% All rights reserved.
%% Copying of this file is authorized only if
%% you make absolutely no changes to your copy.
%%
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%%
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%%
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%%
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%%
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%%
% The Acknowledgements section
%%
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%%
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%%
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%%
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%%
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%%
% These are the various available environments
%%
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%%
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%%
% For Theorem 1.1, Lemma 1.1, Corollary 1.1, Propostion 2.1,...
%%
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\newtheorem{prop}{\noindent Proposition}[section]
\newtheorem{conj}{\noindent Conjecture}[section]
\newtheorem{assert}{\noindent Assertion}[section]
\newtheorem{assumpt}{\noindent Assumption}[section]
\newtheorem{cond}{\noindent Condition}[section]
%%
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%%
% For Example 1, Definition 1, ...
%%
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{\theorembodyfont{\rmfamily} \newtheorem{problem}{\noindent Problem}}
{\theorembodyfont{\rmfamily} \newtheorem{observation}{\noindent Observation}}
{\theorembodyfont{\rmfamily} \newtheorem{result}{\noindent Result}}
{\theorembodyfont{\rmfamily} \newtheorem{remark}{\noindent Remark}}
{\theorembodyfont{\rmfamily} \newtheorem{remarks}[remark]{\noindent Remarks}}
{\theorembodyfont{\rmfamily} \newtheorem{note}{\noindent Note}}
{\theorembodyfont{\rmfamily} \newtheorem{notes}[note]{\noindent Notes}}
%%
% For Example 1.1, Definition 1.1, ...
%%
{\theorembodyfont{\rmfamily} \newtheorem{ex}{\noindent Example}[section]}
{\theorembodyfont{\rmfamily} \newtheorem{defn}{\noindent Definition}[section]}
{\theorembodyfont{\rmfamily} \newtheorem{prob}{\noindent Problem}[section]}
{\theorembodyfont{\rmfamily} \newtheorem{obs}{\noindent Observation}[section]}
{\theorembodyfont{\rmfamily} \newtheorem{res}{\noindent Result}[section]}
{\theorembodyfont{\rmfamily} \newtheorem{rem}{\noindent Remark}[section]}
{\theorembodyfont{\rmfamily} \newtheorem{rems}[rem]{\noindent Remarks}}
{\theorembodyfont{\rmfamily} \newtheorem{nte}{\noindent Note}[section]}
{\theorembodyfont{\rmfamily} \newtheorem{ntes}[nte]{\noindent Notes}}
%%
% For unnumbered Example, Definition, Remark,...
%%
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%%
% The proof environment
%%
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               {\par \normalfont \topsep6\p@\@plus2\p@\relax
                \trivlist
                \item[\hspace{17pt}\itshape #1{.}]\ignorespaces
               }{\endtrivlist}
%%
% Various useful definitions
%%
\def\var{\mathop{\rm var}\nolimits}    %variance
\def\cov{\mathop{\rm cov}\nolimits}    %covariance
\def\corr{\mathop{\rm corr}\nolimits}  %correlation
%
\newcommand{\rd}{\mathrm{d}} %differential d
\DeclareMathOperator\sd{d\!} %differential d preceeded by a thin space, when needed
\newcommand{\ri}{\mathrm{i}} %imaginary i
\newcommand{\re}{\mathrm{e}} %exponential e
\DeclareMathOperator\sP{P}   %probability, treated as an operator
\newcommand{\rP}{\mathrm{P}} %probability, when spacing isn't a worry
\DeclareMathOperator\sE{E}   %expectation, treated as an operator
\newcommand{\rE}{\mathrm{E}} %expectation, when spacing isn't a worry
%
\newcommand{\trans}{^\top}   %matrix transpose
%
\newcommand{\cvgpr}{\xrightarrow{\text{\upshape\tiny P}}}   %convergence in probability
\newcommand{\cvgwk}{\xrightarrow{\text{\upshape\tiny W}}}   %weak convergence
\newcommand{\cvgdist}{\xrightarrow{\text{\upshape\tiny D}}} %convergence in distribution
\newcommand{\eqdist}{\stackrel{\text{\upshape\tiny D}}{=}}  %equality in distribution
%
\hyphenpenalty=1000
%
\chardef\ii="10
%%
% The name--date reference list style
%%
\newenvironment{harvreferences}
               {\section*{References}
                \begin{footnotesize}
                \begin{list}{}{\setlength{\itemsep}{0pt}
                               \setlength{\parsep}{0pt}
                               \setlength{\labelwidth}{0pt}
                               \setlength{\leftmargin}{12pt}
                               \setlength{\labelsep}{0pt}}
                \setlength{\itemindent}{-12pt}
               }{\end{list}\end{footnotesize}}
%%
% bold italic font for use in vectors etc
%%
\DeclareMathAlphabet{\bi}{OT1}{ptm}{b}{it}
%
\newcommand{\balpha}{\boldsymbol{\alpha}}
\newcommand{\bbeta}{\boldsymbol{\beta}}
\newcommand{\bgamma}{\boldsymbol{\gamma}}
\newcommand{\bdelta}{\boldsymbol{\delta}}
\newcommand{\bepsilon}{\boldsymbol{\epsilon}}
\newcommand{\bzeta}{\boldsymbol{\zeta}}
\newcommand{\bfeta}{\boldsymbol{\feta}}
\newcommand{\btheta}{\boldsymbol{\theta}}
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\newcommand{\bkappa}{\boldsymbol{\kappa}}
\newcommand{\blambda}{\boldsymbol{\lambda}}
\newcommand{\bmu}{\boldsymbol{\mu}}
\newcommand{\bnu}{\boldsymbol{\nu}}
\newcommand{\bxi}{\boldsymbol{\xi}}
\newcommand{\bpi}{\boldsymbol{\pi}}
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\newcommand{\bvartheta}{\boldsymbol{\vartheta}}
\newcommand{\bvaromega}{\boldsymbol{\varomega}}
\newcommand{\bvarrho}{\boldsymbol{\varrho}}
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\newcommand{\bell}{\boldsymbol{\ell}}
\newcommand{\bimath}{\boldsymbol{\imath}}
\newcommand{\bjmath}{\boldsymbol{\jmath}}
%
\newcommand{\binfty}{\boldsymbol{\infty}}
\newcommand{\bnabla}{\boldsymbol{\nabla}}
\newcommand{\bdot}{\boldsymbol{\cdot}}
%%
% Short cuts for some commonly referenced journals
%%
\newcommand{\AAP}{\emph{Adv. Appl. Prob.}}
\newcommand{\ADM}{\emph{Ann. Discrete Math.}}
\newcommand{\AM}{\emph{Acta Math.}}
\newcommand{\AMM}{\emph{Amer. Math. Monthly}}
\newcommand{\AMS}{\emph{Ann. Math. Statist.}}
\newcommand{\AP}{\emph{Ann. Prob.}}
\newcommand{\AS}{\emph{Amer. Statistician}}
\newcommand{\ApS}{\emph{Appl. Statist.}}
\newcommand{\AnS}{\emph{Ann. Statist.}}
\newcommand{\ASM}{\emph{Acta Sci. Math.}}
%
\newcommand{\B}{\emph{Biometrika}}
\newcommand{\BAMS}{\emph{Bull. Amer. Math. Soc.}}
\newcommand{\BIMA}{\emph{Bull. Inst. Math. Appl.}}
%
\newcommand{\CB}{\emph{Comput. Bull.}}
\newcommand{\CJ}{\emph{Comput. J.}}
\newcommand{\CCERO}{\emph{Cahiers Centre Etudes Rech. Op\'erat.}}
\newcommand{\CJS}{\emph{Canad. J. Statist.}}
\newcommand{\CMA}{\emph{Comput. Math. Appl.}}
\newcommand{\CMP}{\emph{Commun. Math. Phys.}}
\newcommand{\CPAM}{\emph{Commun. Pure Appl. Math.}}
\newcommand{\CS}{\emph{Commun. Statist.}}
\newcommand{\CRA}{\emph{C. R. Acad. Sci. Paris}}
%
\newcommand{\DE}{\emph{Differential Equns}}
\newcommand{\DMJ}{\emph{Duke Math. J.}}
%
\newcommand{\E}{\emph{Econometrica}}
%
\newcommand{\IETAC}{\emph{IEEE Trans. Automatic Control}}
\newcommand{\IETC}{\emph{IEEE Trans. Commun.}}
\newcommand{\IETIT}{\emph{IEEE Trans. Inf. Theory}}
\newcommand{\IETR}{\emph{IEEE Trans. Rel.}}
\newcommand{\IJC}{\emph{Int. J. Control}}
\newcommand{\IJGT}{\emph{Int. J. Game Theory}}
\newcommand{\IJM}{\emph{Israel J. Math.}}
%
\newcommand{\JASA}{\emph{J. Amer. Statist. Soc.}}
\newcommand{\JAM}{\emph{J. Appl. Mech.}}
\newcommand{\JAMS}{\emph{J. Austral. Math. Soc.}}
\newcommand{\JAP}{\emph{J. Appl. Prob.}}
\newcommand{\JDE}{\emph{J. Differential Equns}}
\newcommand{\JET}{\emph{J. Econom. Theory}}
\newcommand{\JIMA}{\emph{J. Inst. Math. Appl.}}
\newcommand{\JISA}{\emph{J. Indian Statist. Assoc.}}
\newcommand{\JLMS}{\emph{J. London Math. Soc.}}
\newcommand{\JMA}{\emph{J. Multivar. Anal.}}
\newcommand{\JMAA}{\emph{J. Math. Anal. Appl.}}
\newcommand{\JMB}{\emph{J. Math. Biol.}}
\newcommand{\JMP}{\emph{J. Math. Phys.}}
\newcommand{\JOTA}{\emph{J. Optimization Theory Appl.}}
\newcommand{\JPE}{\emph{J. Political Econ.}}
\newcommand{\JRSS}{\emph{J. R. Statist. Soc.}}
\newcommand{\JS}{\emph{J. SIAM}}
\newcommand{\JSCS}{\emph{J. Statist. Comput. Simulation}}
\newcommand{\JSR}{\emph{J. Statist. Res.}}
\newcommand{\JTB}{\emph{J. Theoret. Biol.}}
\newcommand{\JTSA}{\emph{J. Time Series Anal.}}
%
\newcommand{\MAMS}{\emph{Mem. Amer. Math. Soc.}}
\newcommand{\MB}{\emph{Math. Biosci.}}
\newcommand{\MC}{\emph{Math. Comput.}}
\newcommand{\MF}{\emph{Math. Finance}}
\newcommand{\MN}{\emph{Math. Nachr.}}
\newcommand{\MOR}{\emph{Math. Operat. Res.}}
\newcommand{\MS}{\emph{Management Sci.}}
%
\newcommand{\NMJ}{\emph{Nagoya Math. J.}}
\newcommand{\N}{\emph{Nature}}
\newcommand{\NRLQ}{\emph{Naval Res. Logist. Quart.}}
%
\newcommand{\OR}{\emph{Operat. Res.}}
\newcommand{\ORQ}{\emph{Operat. Res. Quart.}}
%
\newcommand{\PTRF}{\emph{Prob. Theory Rel. Fields}}
\newcommand{\PTRS}{\emph{Phil. Trans. R. Soc.}}
\newcommand{\PAMS}{\emph{Proc. Amer. Math. Soc.}}
\newcommand{\PCPS}{\emph{Proc. Camb. Phil. Soc.}}
\newcommand{\PLMS}{\emph{Proc. London Math. Soc.}}
\newcommand{\PNAS}{\emph{Proc. Nat. Acad. Sci. USA}}
\newcommand{\PRS}{\emph{Proc. R. Soc.}}
%
\newcommand{\QAM}{\emph{Quart. Appl. Math.}}
\newcommand{\QS}{\emph{Queueing Systems}}
%
\newcommand{\RES}{\emph{Rev. Econ. Stud.}}
\newcommand{\RISI}{\emph{Rev. Int. Statist. Inst.}}
\newcommand{\RSA}{\emph{Rev. Statist. Appl.}}
%
\newcommand{\Sa}{\emph{Sankhy\=a}}
\newcommand{\SJAM}{\emph{SIAM J. Appl. Math.}}
\newcommand{\SJMA}{\emph{SIAM J. Math. Anal.}}
\newcommand{\SN}{\emph{Statist. Neerlandica}}
\newcommand{\SPA}{\emph{Stoch. Proc. Appl.}}
\newcommand{\SR}{\emph{SIAM Rev.}}
%
\newcommand{\TAMS}{\emph{Trans. Amer. Math. Soc.}}
\newcommand{\TPA}{\emph{Theory Prob. Appl.}}
\newcommand{\TPB}{\emph{Theoret. Popn Biol.}}
%
\newcommand{\ZW}{\emph{Z. Wahrscheinlichkeitsth.}}
%%
%%