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\begin{latin}
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IN THE NAME OF ALLAH
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Ph.D . Thesis Proposal
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\textbf{Renewal Fields and its Application}
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\rl{ما برای بودن آمدیم}
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\large By
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\large
\textbf{Mansoor Karamzadeh}
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\large Supervisors
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\textbf{1 - Ahmad Reza Soltani}
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\textbf{Shiraz University}
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\textbf{2 - Heydar Ali Mardani-Fard}
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\textbf{Yasouj University}
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\textbf{Department of Statistics}
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\textbf{Shiraz University}
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\textbf{Shiraz , Iran.}
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\textbf{Autumn 2017}
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\subsection*{1 . Introduction}
Renewal and regenerative processes are models of stochastic phenomena in which an event (or combination of events) occurs repeatedly over time , and the times between occurrences are i.i.d . Models of such phenomena typically focus on determining limiting averages for costs or other system parameters , or establishing whether certain probabilities or expected values for a system converge over time , and evaluating their limits . More details are given in [9] and [18] .
\subsection*{References}
\addcontentsline{toc}{chapter}{References}
$[1]$ Athreya , Krishna B. , David McDonald , and Peter Ney . " Limit theorems for semi-Markov processes and renewal theory for Markov chains. " The Annals of Probability (1978) : 788-797 . \\
$[2]$ Ball , Frank . " Central limit theorems for multivariate semi-Markov sequences and processes , with applications. " Journal of applied probability 36 , no . 2 (1999) : 415-432.\\
$[3]$ Bickel , Peter J. , and Joseph A . Yahav . " Renewal theory in the plane. " The Annals of Mathematical Statistics (1965) : 946-955.\\
$[4]$ Blischke , Wallace . Product warranty handbook . CRC Press , 1995 . \\
$[5]$ Bracewell , Ronald . " Two-dimensional convolution. " In Fourier Analysis and Imaging , pp . 174-203 . Springer US , 2003 . \\
$[6]$ Brown , Mark , and Sheldon M . Ross . Renewal reward processes . No . TR-93 . C
FORNIA UNIV BERKELEY OPERATIONS RESEARCH CENTER , 1969 . \\
$[7]$ Cinlar , Erhan . " Markov renewal theory. " Advances in Applied Probability 1 , no . 2 (1969) : 123-187 . \\
$[8]$ Cinlar , Erhan . " Exceptional paper—Markov renewal theory : A survey. " Management Science 21 , no . 7 (1975) : 727-752 . \\
$[9]$ Cinlar , Erhan . Introduction to stochastic processes . Courier Corporation , 2013 . \\
$[10]$ Davies , Katherine F. , and W . John Braun . " A two‐dimensional Markov renewal process. " Environmetrics 22 , no . 1 (2011) : 98-108 . \\
$[11]$ Davies , Katherine F. , and B . Gail Ivanoff . " Nonparametric estimation for a two-dimensional renewal process. " Electronic Journal of Statistics 6 (2012) : 1449-1476 . \\
$[12]$ Demongeot , J. , and J . Fricot . " Random Fields And Spatial Renewal Potentials. " In Disordered Systems and Biological Organization , pp . 71-84 . Springer , Berlin , Heidelberg , 1986 . \\
$[13]$ Farrell , R . H . " Limit theorems for stopped random walks. " The Annals of Mathematical Statistics (1964) : 1332-1343.\\
$[14]$ Hunter , Jeffrey J . " Renewal theory in two dimensions : basic results. " Advances in Applied Probability 6 , no . 2 (1974) : 376-391.\\
$[15]$ Hunter , Jeffrey J . " Renewal theory in two dimensions : asymptotic results. " Advances in Applied Probability 6 , no . 3 (1974) : 546-562.\\
$[16]$ Hunter , Jeffrey J . " Renewal theory in two dimensions : Bounds on the renewal function. " Advances in Applied Probability 9 , no . 3 (1977) : 527-541.\\
$[17]$ Janssen , J. , and J . M . Reinhard . " Some duality results for a class of multivariate semi-Markov processes. " Journal of Applied Probability 19 , no . 1 (1982) : 90-98 . \\
$[18]$ Karlin , Samuel . A first course in stochastic processes . Academic press , 2014 . \\
$[19]$ Masuda , Yasushi , and Ushio Sumita . " A multivariate reward process defined on a semi-Markov process and its first-passage-time distributions. " Journal of Applied Probability 28 , no . 2 (1991) : 360-373 . \\
$[20]$ Pyke , Ronald . " Markov renewal processes : definitions and preliminary properties. " The Annals of Mathematical Statistics (1961) : 1231-1242 . \\
$[21]$ Pyke , Ronald . " Markov renewal processes with finitely many states. " The Annals of Mathematical Statistics (1961) : 1243-1259 . \\
$[22]$ Pyke , Ronald , and Ronald Schaufele . " Limit theorems for Markov renewal processes. " The Annals of Mathematical Statistics (1964) : 1746-1764 . \\
$[23]$ Smith , Walter L . " II.—Asymptotic Renewal Theorems. " Proceedings of the Royal Society of Edinburgh Section A : Mathematics 64 , no . 1 (1953) : 9-48 . \\
$[24]$ Smith , Walter L . " Regenerative stochastic processes. " In Proceedings of the Royal Society of London A : Mathematical , Physical and Engineering Sciences , vol . 232 , no . 1188 , pp . 6-31 . The Royal Society , 1955 . \\
$[25]$ Soltani , A . Reza . " Reward processes with nonlinear reward functions. " Journal of applied probability 33 , no . 4 (1996) : 1011-1017 . \\
$[26]$ Soltani , A . Reza , and K . Khorshidian . " Reward processes for semi-Markov processes : asymptotic behaviour. " Journal of applied probability 35 , no . 4 (1998) : 833-842 . \\
$[27]$ Tijms , Henk C . " Renewal‐Reward Processes. " A first course in stochastic models (2003) : 33-79.\\
$[28]$ Soltani , A . R. , and H . Ghasemi . " Semi-Markov and reward fields. " Statistics \& Probability Letters 95 (2014) : 71-76 . \\
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