{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T12:58:28Z","timestamp":1740142708213,"version":"3.37.3"},"reference-count":7,"publisher":"Informa UK Limited","license":[{"start":{"date-parts":[[2007,11,22]],"date-time":"2007-11-22T00:00:00Z","timestamp":1195689600000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/501100008306","name":"University of Wroc\u0142aw","doi-asserted-by":"crossref","award":["MTKD-CT-2004-13389"],"award-info":[{"award-number":["MTKD-CT-2004-13389"]}],"id":[{"id":"10.13039\/501100008306","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Applied Mathematics and Decision Sciences"],"published-print":{"date-parts":[[2007,11,22]]},"abstract":"<jats:p>A Cox process <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E1\"><mml:mrow><mml:msub><mml:mi>N<\/mml:mi><mml:mrow><mml:mtext>Cox<\/mml:mtext><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math> directed by a stationary random measure <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E2\"><mml:mi>\u03be<\/mml:mi><\/mml:math> has second\nmoment <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E3\"><mml:mrow><mml:mtext>var<\/mml:mtext><mml:mtext>\u00a0<\/mml:mtext><mml:msub><mml:mi>N<\/mml:mi><mml:mrow><mml:mtext>Cox<\/mml:mtext><\/mml:mrow><\/mml:msub><mml:mrow><mml:mo>(<\/mml:mo><mml:mrow><mml:mn>0<\/mml:mn><mml:mo>,<\/mml:mo><mml:mrow><mml:mi>t<\/mml:mi><mml:mo>]<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:mrow><mml:mo>=<\/mml:mo><mml:mtext>E<\/mml:mtext><mml:mrow><mml:mo>(<\/mml:mo><mml:mrow><mml:mi>\u03be<\/mml:mi><mml:mrow><mml:mo>(<\/mml:mo><mml:mrow><mml:mn>0<\/mml:mn><mml:mo>,<\/mml:mo><mml:mrow><mml:mi>t<\/mml:mi><mml:mo>]<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:mrow><\/mml:mrow><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mo>+<\/mml:mo><mml:mtext>var<\/mml:mtext><mml:mtext>\u00a0<\/mml:mtext><mml:mi>\u03be<\/mml:mi><mml:mrow><mml:mo>(<\/mml:mo><mml:mrow><mml:mn>0<\/mml:mn><mml:mo>,<\/mml:mo><mml:mrow><mml:mi>t<\/mml:mi><mml:mo>]<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:mrow><\/mml:mrow><\/mml:math>, where by\nstationarity <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E4\"><mml:mtext>E<\/mml:mtext><mml:mo>(<\/mml:mo><mml:mi>\u03be<\/mml:mi><mml:mo>(<\/mml:mo><mml:mn>0<\/mml:mn><mml:mo>,<\/mml:mo><mml:mi>t<\/mml:mi><mml:mo>]<\/mml:mo><mml:mo>)<\/mml:mo><mml:mo>=<\/mml:mo><mml:mo>(<\/mml:mo><mml:mtext>const<\/mml:mtext><mml:mo>.<\/mml:mo><mml:mo>)<\/mml:mo><mml:mi>t<\/mml:mi><mml:mo>=<\/mml:mo><\/mml:math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E5\"><mml:mtext>E<\/mml:mtext><mml:mo>(<\/mml:mo><mml:msub><mml:mi>N<\/mml:mi><mml:mtext>Cox<\/mml:mtext><\/mml:msub><mml:mo>(<\/mml:mo><mml:mn>0<\/mml:mn><mml:mo>,<\/mml:mo><mml:mi>t<\/mml:mi><mml:mo>]<\/mml:mo><mml:mo>)<\/mml:mo><\/mml:math>, so long-range dependence (LRD) properties of\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E6\"><mml:msub><mml:mi>N<\/mml:mi><mml:mrow><mml:mtext>Cox<\/mml:mtext><\/mml:mrow><\/mml:msub><\/mml:math> coincide with LRD properties of the random measure <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E7\"><mml:mi>\u03be<\/mml:mi><\/mml:math>.\nWhen <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E8\"><mml:mi>\u03be<\/mml:mi><mml:mrow><mml:mo>(<\/mml:mo><mml:mi>A<\/mml:mi><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mo>=<\/mml:mo><mml:mstyle displaystyle=\"true\"><mml:mrow><mml:msub><mml:mo>\u222b<\/mml:mo><mml:mi>A<\/mml:mi><\/mml:msub><mml:mrow><mml:msub><mml:mi>\u03bd<\/mml:mi><mml:mrow><mml:mi>J<\/mml:mi><mml:mrow><mml:mo>(<\/mml:mo><mml:mi>u<\/mml:mi><mml:mo>)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:mrow><\/mml:mstyle><mml:mtext>d<\/mml:mtext><mml:mi>u<\/mml:mi><\/mml:math> is determined by a density that depends\non rate parameters <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E9\"><mml:msub><mml:mi>\u03bd<\/mml:mi><mml:mi>i<\/mml:mi><\/mml:msub><mml:mrow><mml:mo>(<\/mml:mo><mml:mrow><mml:mi>i<\/mml:mi><mml:mo>\u2208<\/mml:mo><mml:mi>\ud835\udd4f<\/mml:mi><\/mml:mrow><mml:mo>)<\/mml:mo><\/mml:mrow><\/mml:math> and the current state <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E10\"><mml:mi>J<\/mml:mi><mml:mrow><mml:mo>(<\/mml:mo><mml:mo>\u22c5<\/mml:mo><mml:mo>)<\/mml:mo><\/mml:mrow><\/mml:math>\nof an <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E11\"><mml:mi>\ud835\udd4f<\/mml:mi><\/mml:math>-valued stationary irreducible Markov renewal process (MRP) for\nsome countable state space <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E12\"><mml:mi>\ud835\udd4f<\/mml:mi><\/mml:math> (so <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E13\"><mml:mi>J<\/mml:mi><mml:mrow><mml:mo>(<\/mml:mo><mml:mi>t<\/mml:mi><mml:mo>)<\/mml:mo><\/mml:mrow><\/mml:math> is a stationary semi-Markov\nprocess on <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E14\"><mml:mi>\ud835\udd4f<\/mml:mi><\/mml:math>), the random measure is LRD if and only if each (and then\nby irreducibility, every) generic return time <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E15\"><mml:msub><mml:mi>Y<\/mml:mi><mml:mrow><mml:mi>j<\/mml:mi><mml:mi>j<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mrow><mml:mo>(<\/mml:mo><mml:mrow><mml:mi>j<\/mml:mi><mml:mo>\u2208<\/mml:mo><mml:mi mathvariant=\"double-struck\">X<\/mml:mi><\/mml:mrow><mml:mo>)<\/mml:mo><\/mml:mrow><\/mml:math> of the\nprocess for entries to state <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E16\"><mml:mi>j<\/mml:mi><\/mml:math> has infinite second moment, for which a\nnecessary and sufficient condition when <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E17\"><mml:mi>\ud835\udd4f<\/mml:mi><\/mml:math> is finite is that at least\none generic holding time <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E18\"><mml:msub><mml:mi>X<\/mml:mi><mml:mi>j<\/mml:mi><\/mml:msub><\/mml:math> in state <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E19\"><mml:mi>j<\/mml:mi><\/mml:math>, with distribution function (DF)\\\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E20\"><mml:msub><mml:mi>H<\/mml:mi><mml:mi>j<\/mml:mi><\/mml:msub><\/mml:math>, say, has infinite second moment (a simple example shows that this\ncondition is not necessary when <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E21\"><mml:mi>\ud835\udd4f<\/mml:mi><\/mml:math> is countably infinite).\nThen, <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E22\"><mml:msub><mml:mi>N<\/mml:mi><mml:mrow><mml:mtext>Cox<\/mml:mtext><\/mml:mrow><\/mml:msub><\/mml:math> has the same Hurst index as the MRP <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E23\"><mml:msub><mml:mi>N<\/mml:mi><mml:mrow><mml:mtext>MRP<\/mml:mtext><\/mml:mrow><\/mml:msub><\/mml:math> that counts the jumps\nof <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E24\"><mml:mi>J<\/mml:mi><mml:mrow><mml:mo>(<\/mml:mo><mml:mo>\u22c5<\/mml:mo><mml:mo>)<\/mml:mo><\/mml:mrow><\/mml:math>, while as <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E25\"><mml:mi>t<\/mml:mi><mml:mo>\u2192<\/mml:mo><mml:mi>\u221e<\/mml:mi><\/mml:math>,  for finite <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E26\"><mml:mi>\ud835\udd4f<\/mml:mi><\/mml:math>,\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E27\"><mml:mtext>var<\/mml:mtext><mml:mtext>\u00a0<\/mml:mtext><mml:msub><mml:mi>N<\/mml:mi><mml:mrow><mml:mtext>MRP<\/mml:mtext><\/mml:mrow><\/mml:msub><mml:mrow><mml:mo>(<\/mml:mo><mml:mrow><mml:mn>0<\/mml:mn><mml:mo>,<\/mml:mo><mml:mrow><mml:mi>t<\/mml:mi><mml:mo>]<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:mrow><mml:mo>\u223c<\/mml:mo><mml:mn>2<\/mml:mn><mml:msup><mml:mi>\u03bb<\/mml:mi><mml:mn>2<\/mml:mn><\/mml:msup><mml:mstyle displaystyle=\"true\"><mml:mrow><mml:msubsup><mml:mo>\u222b<\/mml:mo><mml:mn>0<\/mml:mn><mml:mi>t<\/mml:mi><\/mml:msubsup><mml:mrow><mml:mi>\ud835\udca2<\/mml:mi><mml:mrow><mml:mo>(<\/mml:mo><mml:mi>u<\/mml:mi><mml:mo>)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:mrow><\/mml:mstyle><mml:mtext>d<\/mml:mtext><mml:mi>u<\/mml:mi><\/mml:math>,\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E28\"><mml:mtext>var<\/mml:mtext><mml:mtext>\u00a0<\/mml:mtext><mml:msub><mml:mi>N<\/mml:mi><mml:mrow><mml:mtext>Cox<\/mml:mtext><\/mml:mrow><\/mml:msub><mml:mrow><mml:mo>(<\/mml:mo><mml:mrow><mml:mn>0<\/mml:mn><mml:mo>,<\/mml:mo><mml:mrow><mml:mi>t<\/mml:mi><mml:mo>]<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:mrow><mml:mo>\u223c<\/mml:mo><mml:mn>2<\/mml:mn><mml:msup><mml:mstyle displaystyle=\"true\"><mml:mrow><mml:msubsup><mml:mo>\u222b<\/mml:mo><mml:mn>0<\/mml:mn><mml:mi>t<\/mml:mi><\/mml:msubsup><mml:mrow><mml:mstyle displaystyle=\"true\"><mml:munder><mml:mo>\u2211<\/mml:mo><mml:mrow><mml:mi>i<\/mml:mi><mml:mo>\u2208<\/mml:mo><mml:mi>\ud835\udd4f<\/mml:mi><\/mml:mrow><\/mml:munder><mml:mrow><mml:mrow><mml:mo>(<\/mml:mo><mml:mrow><mml:msub><mml:mi>\u03bd<\/mml:mi><mml:mi>i<\/mml:mi><\/mml:msub><mml:mo>\u2212<\/mml:mo><mml:mover accent=\"true\"><mml:mi>\u03bd<\/mml:mi><mml:mo stretchy=\"true\">\u00af<\/mml:mo><\/mml:mover><\/mml:mrow><mml:mo>)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:mstyle><\/mml:mrow><\/mml:mrow><\/mml:mstyle><mml:mn>2<\/mml:mn><\/mml:msup><mml:msub><mml:mi>\u03d6<\/mml:mi><mml:mi>i<\/mml:mi><\/mml:msub><mml:msub><mml:mi>\u210b<\/mml:mi><mml:mi>i<\/mml:mi><\/mml:msub><mml:mrow><mml:mo>(<\/mml:mo><mml:mi>t<\/mml:mi><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mtext>du<\/mml:mtext><\/mml:math>,\nwhere\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E29\"><mml:mover accent=\"true\"><mml:mi>\u03bd<\/mml:mi><mml:mo stretchy=\"true\">\u00af<\/mml:mo><\/mml:mover><mml:mo>=<\/mml:mo><mml:mstyle displaystyle=\"true\"><mml:msub><mml:mo>\u2211<\/mml:mo><mml:mi>i<\/mml:mi><\/mml:msub><mml:mrow><mml:msub><mml:mi>\u03d6<\/mml:mi><mml:mi>i<\/mml:mi><\/mml:msub><mml:msub><mml:mi>\u03bd<\/mml:mi><mml:mi>i<\/mml:mi><\/mml:msub><mml:mo>=<\/mml:mo><mml:mtext>E<\/mml:mtext><\/mml:mrow><\/mml:mstyle><mml:mrow><mml:mo>[<\/mml:mo><mml:mrow><mml:mi>\u03be<\/mml:mi><mml:mrow><mml:mo>(<\/mml:mo><mml:mrow><mml:mn>0<\/mml:mn><mml:mo>,<\/mml:mo><mml:mrow><mml:mn>1<\/mml:mn><mml:mo>]<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:mrow><\/mml:mrow><mml:mo>]<\/mml:mo><\/mml:mrow><\/mml:math>,\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E30\"><mml:msub><mml:mi>\u03d6<\/mml:mi><mml:mi>j<\/mml:mi><\/mml:msub><mml:mo>=<\/mml:mo><mml:mtext>Pr<\/mml:mtext><mml:mrow><mml:mo>{<\/mml:mo><mml:mrow><mml:mi>J<\/mml:mi><mml:mrow><mml:mo>(<\/mml:mo><mml:mi>t<\/mml:mi><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mo>=<\/mml:mo><mml:mi>j<\/mml:mi><\/mml:mrow><mml:mo>}<\/mml:mo><\/mml:mrow><mml:mo>,<\/mml:mo><mml:mrow><mml:mn>1<\/mml:mn><mml:mo>\/<\/mml:mo><mml:mi>\u03bb<\/mml:mi><\/mml:mrow><mml:mo>=<\/mml:mo><mml:mstyle displaystyle=\"true\"><mml:msub><mml:mo>\u2211<\/mml:mo><mml:mi>j<\/mml:mi><\/mml:msub><mml:mrow><mml:msub><mml:mover accent=\"true\"><mml:mi>p<\/mml:mi><mml:mo stretchy=\"true\">\u02c7<\/mml:mo><\/mml:mover><mml:mi>j<\/mml:mi><\/mml:msub><mml:msub><mml:mi>\u03bc<\/mml:mi><mml:mi>j<\/mml:mi><\/mml:msub><\/mml:mrow><\/mml:mstyle><\/mml:math>,\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E31\"><mml:msub><mml:mi>\u03bc<\/mml:mi><mml:mi>j<\/mml:mi><\/mml:msub><mml:mo>=<\/mml:mo><mml:mtext>E<\/mml:mtext><mml:mrow><mml:mo>(<\/mml:mo><mml:mrow><mml:msub><mml:mi>X<\/mml:mi><mml:mi>j<\/mml:mi><\/mml:msub><\/mml:mrow><mml:mo>)<\/mml:mo><\/mml:mrow><\/mml:math>,\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E32\"><mml:mrow><mml:mo>{<\/mml:mo><mml:msub><mml:mover accent=\"true\"><mml:mi>p<\/mml:mi><mml:mo stretchy=\"true\">\u02c7<\/mml:mo><\/mml:mover><mml:mi>j<\/mml:mi><\/mml:msub><mml:mo>}<\/mml:mo><\/mml:mrow><\/mml:math>\nis the stationary distribution for the embedded jump process\nof the MRP, <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E33\"><mml:msub><mml:mi>\u210b<\/mml:mi><mml:mi>j<\/mml:mi><\/mml:msub><mml:mrow><mml:mo>(<\/mml:mo><mml:mi>t<\/mml:mi><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mo>=<\/mml:mo><mml:msubsup><mml:mi>\u03bc<\/mml:mi><mml:mi>i<\/mml:mi><mml:mrow><mml:mo>\u2212<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msubsup><mml:mstyle displaystyle=\"true\"><mml:mrow><mml:msubsup><mml:mo>\u222b<\/mml:mo><mml:mn>0<\/mml:mn><mml:mi>\u221e<\/mml:mi><\/mml:msubsup><mml:mrow><mml:mo>min<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:mstyle><mml:mrow><mml:mo>(<\/mml:mo><mml:mrow><mml:mi>u<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>t<\/mml:mi><\/mml:mrow><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mo>[<\/mml:mo><mml:mrow><mml:mn>1<\/mml:mn><mml:mo>\u2212<\/mml:mo><mml:msub><mml:mi>H<\/mml:mi><mml:mi>j<\/mml:mi><\/mml:msub><mml:mrow><mml:mo>(<\/mml:mo><mml:mi>u<\/mml:mi><mml:mo>)<\/mml:mo><\/mml:mrow><\/mml:mrow><mml:mo>]<\/mml:mo><\/mml:mrow><mml:mtext>d<\/mml:mtext><mml:mi>u<\/mml:mi><\/mml:math>, and\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E34\"><mml:mi>\ud835\udca2<\/mml:mi><mml:mrow><mml:mo>(<\/mml:mo><mml:mi>t<\/mml:mi><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mo>\u223c<\/mml:mo><mml:mstyle displaystyle=\"true\"><mml:mrow><mml:msubsup><mml:mo>\u222b<\/mml:mo><mml:mn>0<\/mml:mn><mml:mi>t<\/mml:mi><\/mml:msubsup><mml:mrow><mml:mo>min<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:mstyle><mml:mrow><mml:mo>(<\/mml:mo><mml:mrow><mml:mi>u<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>t<\/mml:mi><\/mml:mrow><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mo>[<\/mml:mo><mml:mrow><mml:mn>1<\/mml:mn><mml:mo>\u2212<\/mml:mo><mml:msub><mml:mi>G<\/mml:mi><mml:mrow><mml:mi>j<\/mml:mi><mml:mi>j<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mrow><mml:mo>(<\/mml:mo><mml:mi>u<\/mml:mi><mml:mo>)<\/mml:mo><\/mml:mrow><\/mml:mrow><mml:mo>]<\/mml:mo><\/mml:mrow><mml:mrow><mml:mrow><mml:mtext>d<\/mml:mtext><mml:mi>u<\/mml:mi><\/mml:mrow><mml:mo>\/<\/mml:mo><mml:mrow><mml:msub><mml:mi>m<\/mml:mi><mml:mrow><mml:mi>j<\/mml:mi><mml:mi>j<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:mrow><mml:mo>\u223c<\/mml:mo><mml:mstyle displaystyle=\"true\"><mml:msub><mml:mo>\u2211<\/mml:mo><mml:mi>i<\/mml:mi><\/mml:msub><mml:mrow><mml:msub><mml:mi>\u03d6<\/mml:mi><mml:mi>i<\/mml:mi><\/mml:msub><mml:msub><mml:mi>\u210b<\/mml:mi><mml:mi>i<\/mml:mi><\/mml:msub><\/mml:mrow><\/mml:mstyle><mml:mrow><mml:mo>(<\/mml:mo><mml:mi>t<\/mml:mi><mml:mo>)<\/mml:mo><\/mml:mrow><\/mml:math>\nwhere <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E35\"><mml:mrow><mml:msub><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>jj<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math> is the\nDF and <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E36\"><mml:msub><mml:mi>m<\/mml:mi><mml:mrow><mml:mi>j<\/mml:mi><mml:mi>j<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:math> the mean of the generic return time <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E37\"><mml:msub><mml:mi>Y<\/mml:mi><mml:mrow><mml:mi>j<\/mml:mi><mml:mi>j<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:math> of the MRP\nbetween successive\nentries to the state <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E38\"><mml:mi>j<\/mml:mi><\/mml:math>.  These two variances are of similar order\nfor <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E39\"><mml:mi>t<\/mml:mi><mml:mo>\u2192<\/mml:mo><mml:mi>\u221e<\/mml:mi><\/mml:math> only when each <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E40\"><mml:mrow><mml:msub><mml:mi>\u210b<\/mml:mi><mml:mi>i<\/mml:mi><\/mml:msub><mml:mrow><mml:mrow><mml:mrow><mml:mo>(<\/mml:mo><mml:mi>t<\/mml:mi><mml:mo>)<\/mml:mo><\/mml:mrow><\/mml:mrow><mml:mo>\/<\/mml:mo><mml:mi>\ud835\udca2<\/mml:mi><\/mml:mrow><mml:mrow><mml:mo>(<\/mml:mo><mml:mi>t<\/mml:mi><mml:mo>)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math> converges to some\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E41\"><mml:mrow><mml:mo>[<\/mml:mo><mml:mrow><mml:mn>0<\/mml:mn><mml:mo>,<\/mml:mo><mml:mi>\u221e<\/mml:mi><\/mml:mrow><mml:mo>]<\/mml:mo><\/mml:mrow><\/mml:math>-valued constant, say, <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E42\"><mml:msub><mml:mi>\u03b3<\/mml:mi><mml:mi>i<\/mml:mi><\/mml:msub><\/mml:math>, for <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E43\"><mml:mi>t<\/mml:mi><mml:mo>\u2192<\/mml:mo><mml:mi>\u221e<\/mml:mi><\/mml:math>.<\/jats:p>","DOI":"10.1155\/2007\/83852","type":"journal-article","created":{"date-parts":[[2007,12,13]],"date-time":"2007-12-13T18:41:26Z","timestamp":1197571286000},"page":"1-15","source":"Crossref","is-referenced-by-count":1,"title":["Long-Range Dependence in a Cox Process Directed by a Markov Renewal Process"],"prefix":"10.1080","volume":"2007","author":[{"given":"D. J.","family":"Daley","sequence":"first","affiliation":[]},{"given":"T.","family":"Rolski","sequence":"additional","affiliation":[]},{"given":"R.","family":"Vesilo","sequence":"additional","affiliation":[]}],"member":"301","reference":[{"key":"1"},{"key":"2","doi-asserted-by":"publisher","DOI":"10.2307\/1426217"},{"issue":"5","key":"3","first-page":"178","volume":"22","year":"1981","journal-title":"Sibirskii Matematicheskii Zhurnal"},{"first-page":"xxii+469","year":"2003","series-title":"Probability and Its Applications","key":"4"},{"year":"1975","key":"5"},{"first-page":"xiv+619","year":"1995","series-title":"Texts in Statistical Science Series","key":"6"},{"key":"7","doi-asserted-by":"publisher","DOI":"10.2307\/3215067"}],"container-title":["Journal of Applied Mathematics and Decision Sciences"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/archive\/2007\/083852.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/archive\/2007\/083852.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,8,8]],"date-time":"2024-08-08T14:23:35Z","timestamp":1723127015000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.hindawi.com\/journals\/ads\/2007\/083852\/abs\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,11,22]]},"references-count":7,"alternative-id":["083852","83852"],"URL":"https:\/\/doi.org\/10.1155\/2007\/83852","relation":{},"ISSN":["1173-9126","1532-7612"],"issn-type":[{"type":"print","value":"1173-9126"},{"type":"electronic","value":"1532-7612"}],"subject":[],"published":{"date-parts":[[2007,11,22]]}}}