@article{multiplicativeCoalescentAldous,
	Annote = {Copio la siguiente frase de la introduccion pues creo que el proyecto de investigacion va en ese sentido.

" Conversely, for readers with a background in modern process theory, we should emphasize that our results open up challenging problems in rederiving, via process techniques, existing random graph asymptotics formulas."

Un proyecto que me interesa es revisar su quinta seccion, pues son puros argumentos de calculo estocastico.

Me preocupa un poco el siguiente parrafo de la subseccion 6.5 sobre problemas abiertos.

"(3) Is there an explicit construction of the entire standard multiplicative coalescent in terms of familiar stochastic processes, avoiding any weak con- vergence argument? In other words, our results imply that there exists a two- parameter process 
$\paren{B^t_s}_{s,t\geq 0}$ which for fixed $t$ is distributed as $B^t$ and whose excursion lengths evolve as the multiplicative coalescent. But we do not know how to directly define such a process. Janson [15] describes the two-parameter point process giving the times t and component  sizes where multicyclic components arise; we would like Jansons process to be included in a two-parameter process description. Using (1) and (2) with the same $W$ for each $t$ definitely does not work; we need the excursions to merge in a much more complicated way. It seems intuitive that for a fixed pair $t_1 < t_2$ 
one cannot define a bivariate Markov process 
$\paren{B^{t_1}_s,B^{t_2}_s}_{0\leq s<\infty}$ such that the joint distribution of the excursion lengths is the distribution of $\paren{X_{t_1},X_{t_2}}$ . Specifically, if the construction of $Z^t_n$ is made simultaneously for $t= t_1$ and $t_2$ , then Theorem 3 has a bivariate version which leads to a joint distribution 
$\paren{B^{t_1}_s,B^{t_2}_s}$, but the joint process is not Markov because coalescence of clusters between $t_1$ and $t_2$ leads to excursions of $B_{t_1}$ becoming embedded within earlier (in terms of $s$) excursions of $B^{t_2}$ , which is incompatible with the Markov property. },
	Author = {Aldous, David},
	Coden = {APBYAE},
	Date-Added = {2011-06-07 16:43:16 -0500},
	Date-Modified = {2017-10-11 13:14:40 +0000},
	Fjournal = {The Annals of Probability},
	Impreso = {1},
	Issn = {0091-1798},
	Journal = {Ann. Probab.},
	Keywords = {Curso Gr{\'a}ficas Aleatorias},
	Local-Url = {file://localhost/Users/Gero/Documents/Arti%CC%81culos/multiplicative%20coalescent.pdf},
	Mrclass = {60C05 (60J50)},
	Mrnumber = {1434128},
	Mrreviewer = {Endre Csaki},
	Number = {2},
	Pages = {812--854},
	Title = {Brownian excursions, critical random graphs and the multiplicative coalescent},
	Volume = {25},
	Year = {1997},
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In other words, our results imply that there exists a two- parameter process \u0000$\\paren{B^t_s}_{s,t≥0}$\u0000 which for fixed $t$ is distributed as $B^t$ and whose excursion lengths evolve as the multiplicative coalescent. But we do not know how to directly define such a process. Janson [15] describes the two-parameter point process giving the times t and component sizes where multicyclic components arise; we would like Jansons process to be included in a two-parameter process description. Using (1) and (2) with the same $W$ for each $t$ definitely does not work; we need the excursions to merge in a much more complicated way. It seems intuitive that for a fixed pair $t_1 < t_2$ one cannot define a bivariate Markov process \u0000\u0000$\\paren{B^{t_1}_s,B^{t_2}_s}_{0≤s<\\infty}$ such that the joint distribution of the excursion lengths is the distribution of \u0000$\\paren{X_{t_1},X_{t_2}}$ \u0000\u0000. Specifically, if the construction of $Z^t_n$ is made simultaneously for $t= t_1$ and $t_2$ , then Theorem 3 has a bivariate version which leads to a joint distribution $\u0000\u0000\\paren{B^{t_1}_s,B^{t_2}_s}$, but the joint process is not Markov because coalescence of clusters between $t_1$ and $t_2$ leads to excursions of $B_{t_1}$ becoming embedded within earlier (in terms of $s$) excursions of $B^{t_2}$ , which is incompatible with the Markov property. ","author":[{"propositions":[],"lastnames":["Aldous"],"firstnames":["David"],"suffixes":[]}],"coden":"APBYAE","date-added":"2011-06-07 16:43:16 -0500","date-modified":"2017-10-11 13:14:40 +0000","fjournal":"The Annals of Probability","impreso":"1","issn":"0091-1798","journal":"Ann. 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In other words, our results imply that there exists a two- parameter process \n\u0000$\\paren{B^t_s}_{s,t\\geq 0}$\u0000 which for fixed $t$ is distributed as $B^t$ and whose excursion lengths evolve as the multiplicative coalescent. But we do not know how to directly define such a process. Janson [15] describes the two-parameter point process giving the times t and component sizes where multicyclic components arise; we would like Jansons process to be included in a two-parameter process description. Using (1) and (2) with the same $W$ for each $t$ definitely does not work; we need the excursions to merge in a much more complicated way. It seems intuitive that for a fixed pair $t_1 < t_2$ \none cannot define a bivariate Markov process \n\u0000\u0000$\\paren{B^{t_1}_s,B^{t_2}_s}_{0\\leq s<\\infty}$ such that the joint distribution of the excursion lengths is the distribution of \u0000$\\paren{X_{t_1},X_{t_2}}$ \u0000\u0000. Specifically, if the construction of $Z^t_n$ is made simultaneously for $t= t_1$ and $t_2$ , then Theorem 3 has a bivariate version which leads to a joint distribution \n$\u0000\u0000\\paren{B^{t_1}_s,B^{t_2}_s}$, but the joint process is not Markov because coalescence of clusters between $t_1$ and $t_2$ leads to excursions of $B_{t_1}$ becoming embedded within earlier (in terms of $s$) excursions of $B^{t_2}$ , which is incompatible with the Markov property. },\n\tAuthor = {Aldous, David},\n\tCoden = {APBYAE},\n\tDate-Added = {2011-06-07 16:43:16 -0500},\n\tDate-Modified = {2017-10-11 13:14:40 +0000},\n\tFjournal = {The Annals of Probability},\n\tImpreso = {1},\n\tIssn = {0091-1798},\n\tJournal = {Ann. 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