{"id":159,"date":"2018-12-20T10:20:28","date_gmt":"2018-12-20T10:20:28","guid":{"rendered":"http:\/\/statp02.epgpbooks.inflibnet.ac.in\/?post_type=chapter&#038;p=159"},"modified":"2018-12-20T10:27:50","modified_gmt":"2018-12-20T10:27:50","slug":"kolmogorovs-convergence-theorem","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/statp02\/chapter\/kolmogorovs-convergence-theorem\/","title":{"rendered":"Kolmogorov\u2019s Convergence Theorem"},"content":{"raw":"<div><span style=\"float: right\"><a href=\"https:\/\/youtu.be\/TO8_QIOYBq4\" target=\"_blank\" rel=\"noopener\"><img src=\"http:\/\/epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/2018\/11\/download.png\" alt=\"epgp books\" width=\"75px\" height=\"75px;\" \/><\/a>\r\n<\/span><\/div>\r\n<img class=\"aligncenter size-full wp-image-161\" src=\"http:\/\/statp02.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235299KolmogorovsConveregenceTheorem-1.jpg\" alt=\"\" width=\"1255\" height=\"1175\" \/> <img class=\"aligncenter size-full wp-image-162\" src=\"http:\/\/statp02.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235299KolmogorovsConveregenceTheorem-2.jpg\" alt=\"\" width=\"1271\" height=\"1533\" \/> <img class=\"aligncenter size-full wp-image-163\" src=\"http:\/\/statp02.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235299KolmogorovsConveregenceTheorem-3.jpg\" alt=\"\" width=\"1249\" height=\"1599\" \/> <img class=\"aligncenter size-full wp-image-164\" src=\"http:\/\/statp02.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235299KolmogorovsConveregenceTheorem-4.jpg\" alt=\"\" width=\"1271\" height=\"1553\" \/> <img class=\"aligncenter size-full wp-image-165\" src=\"http:\/\/statp02.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235299KolmogorovsConveregenceTheorem-5.jpg\" alt=\"\" width=\"1231\" height=\"201\" \/>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><strong>you can view video on Kolmogorov's Convergence Theorem<\/strong><\/td>\r\n<td><a href=\"https:\/\/youtu.be\/TO8_QIOYBq4\" target=\"_blank\" rel=\"noopener\"><img class=\"alignnone wp-image-120\" src=\"http:\/\/epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/2018\/11\/download.png\" alt=\"\" width=\"36\" height=\"36\" \/><\/a><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<img class=\"aligncenter size-full wp-image-169\" src=\"http:\/\/statp02.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235304KolmogorovsConveregenceTheorem_Ref-1.jpg\" alt=\"\" width=\"1177\" height=\"755\" \/>","rendered":"<div><span style=\"float: right\"><a href=\"https:\/\/youtu.be\/TO8_QIOYBq4\" target=\"_blank\" rel=\"noopener\"><img decoding=\"async\" src=\"http:\/\/epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/2018\/11\/download.png\" alt=\"epgp books\" width=\"75px\" height=\"75px;\" \/><\/a><br \/>\n<\/span><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-161\" src=\"http:\/\/statp02.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235299KolmogorovsConveregenceTheorem-1.jpg\" alt=\"\" width=\"1255\" height=\"1175\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/statp02\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235299KolmogorovsConveregenceTheorem-1.jpg 1255w, https:\/\/ebooks.inflibnet.ac.in\/statp02\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235299KolmogorovsConveregenceTheorem-1-300x281.jpg 300w, https:\/\/ebooks.inflibnet.ac.in\/statp02\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235299KolmogorovsConveregenceTheorem-1-768x719.jpg 768w, https:\/\/ebooks.inflibnet.ac.in\/statp02\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235299KolmogorovsConveregenceTheorem-1-1024x959.jpg 1024w, https:\/\/ebooks.inflibnet.ac.in\/statp02\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235299KolmogorovsConveregenceTheorem-1-65x61.jpg 65w, https:\/\/ebooks.inflibnet.ac.in\/statp02\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235299KolmogorovsConveregenceTheorem-1-225x211.jpg 225w, https:\/\/ebooks.inflibnet.ac.in\/statp02\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235299KolmogorovsConveregenceTheorem-1-350x328.jpg 350w\" sizes=\"auto, (max-width: 1255px) 100vw, 1255px\" \/> <img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-162\" src=\"http:\/\/statp02.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235299KolmogorovsConveregenceTheorem-2.jpg\" alt=\"\" width=\"1271\" height=\"1533\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/statp02\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235299KolmogorovsConveregenceTheorem-2.jpg 1271w, https:\/\/ebooks.inflibnet.ac.in\/statp02\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235299KolmogorovsConveregenceTheorem-2-249x300.jpg 249w, https:\/\/ebooks.inflibnet.ac.in\/statp02\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235299KolmogorovsConveregenceTheorem-2-768x926.jpg 768w, https:\/\/ebooks.inflibnet.ac.in\/statp02\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235299KolmogorovsConveregenceTheorem-2-849x1024.jpg 849w, https:\/\/ebooks.inflibnet.ac.in\/statp02\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235299KolmogorovsConveregenceTheorem-2-65x78.jpg 65w, https:\/\/ebooks.inflibnet.ac.in\/statp02\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235299KolmogorovsConveregenceTheorem-2-225x271.jpg 225w, https:\/\/ebooks.inflibnet.ac.in\/statp02\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235299KolmogorovsConveregenceTheorem-2-350x422.jpg 350w\" sizes=\"auto, (max-width: 1271px) 100vw, 1271px\" \/> <img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-163\" src=\"http:\/\/statp02.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235299KolmogorovsConveregenceTheorem-3.jpg\" alt=\"\" width=\"1249\" height=\"1599\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/statp02\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235299KolmogorovsConveregenceTheorem-3.jpg 1249w, https:\/\/ebooks.inflibnet.ac.in\/statp02\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235299KolmogorovsConveregenceTheorem-3-234x300.jpg 234w, https:\/\/ebooks.inflibnet.ac.in\/statp02\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235299KolmogorovsConveregenceTheorem-3-768x983.jpg 768w, https:\/\/ebooks.inflibnet.ac.in\/statp02\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235299KolmogorovsConveregenceTheorem-3-800x1024.jpg 800w, https:\/\/ebooks.inflibnet.ac.in\/statp02\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235299KolmogorovsConveregenceTheorem-3-65x83.jpg 65w, https:\/\/ebooks.inflibnet.ac.in\/statp02\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235299KolmogorovsConveregenceTheorem-3-225x288.jpg 225w, https:\/\/ebooks.inflibnet.ac.in\/statp02\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235299KolmogorovsConveregenceTheorem-3-350x448.jpg 350w\" sizes=\"auto, (max-width: 1249px) 100vw, 1249px\" \/> <img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-164\" src=\"http:\/\/statp02.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235299KolmogorovsConveregenceTheorem-4.jpg\" alt=\"\" width=\"1271\" height=\"1553\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/statp02\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235299KolmogorovsConveregenceTheorem-4.jpg 1271w, https:\/\/ebooks.inflibnet.ac.in\/statp02\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235299KolmogorovsConveregenceTheorem-4-246x300.jpg 246w, https:\/\/ebooks.inflibnet.ac.in\/statp02\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235299KolmogorovsConveregenceTheorem-4-768x938.jpg 768w, https:\/\/ebooks.inflibnet.ac.in\/statp02\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235299KolmogorovsConveregenceTheorem-4-838x1024.jpg 838w, https:\/\/ebooks.inflibnet.ac.in\/statp02\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235299KolmogorovsConveregenceTheorem-4-65x79.jpg 65w, https:\/\/ebooks.inflibnet.ac.in\/statp02\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235299KolmogorovsConveregenceTheorem-4-225x275.jpg 225w, https:\/\/ebooks.inflibnet.ac.in\/statp02\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235299KolmogorovsConveregenceTheorem-4-350x428.jpg 350w\" sizes=\"auto, (max-width: 1271px) 100vw, 1271px\" \/> <img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-165\" src=\"http:\/\/statp02.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235299KolmogorovsConveregenceTheorem-5.jpg\" alt=\"\" width=\"1231\" height=\"201\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/statp02\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235299KolmogorovsConveregenceTheorem-5.jpg 1231w, https:\/\/ebooks.inflibnet.ac.in\/statp02\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235299KolmogorovsConveregenceTheorem-5-300x49.jpg 300w, https:\/\/ebooks.inflibnet.ac.in\/statp02\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235299KolmogorovsConveregenceTheorem-5-768x125.jpg 768w, https:\/\/ebooks.inflibnet.ac.in\/statp02\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235299KolmogorovsConveregenceTheorem-5-1024x167.jpg 1024w, https:\/\/ebooks.inflibnet.ac.in\/statp02\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235299KolmogorovsConveregenceTheorem-5-65x11.jpg 65w, https:\/\/ebooks.inflibnet.ac.in\/statp02\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235299KolmogorovsConveregenceTheorem-5-225x37.jpg 225w, https:\/\/ebooks.inflibnet.ac.in\/statp02\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235299KolmogorovsConveregenceTheorem-5-350x57.jpg 350w\" sizes=\"auto, (max-width: 1231px) 100vw, 1231px\" \/><\/p>\n<table>\n<tbody>\n<tr>\n<td><strong>you can view video on Kolmogorov&#8217;s Convergence Theorem<\/strong><\/td>\n<td><a href=\"https:\/\/youtu.be\/TO8_QIOYBq4\" target=\"_blank\" rel=\"noopener\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-120\" src=\"http:\/\/epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/2018\/11\/download.png\" alt=\"\" width=\"36\" height=\"36\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-169\" src=\"http:\/\/statp02.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235304KolmogorovsConveregenceTheorem_Ref-1.jpg\" alt=\"\" width=\"1177\" height=\"755\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/statp02\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235304KolmogorovsConveregenceTheorem_Ref-1.jpg 1177w, https:\/\/ebooks.inflibnet.ac.in\/statp02\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235304KolmogorovsConveregenceTheorem_Ref-1-300x192.jpg 300w, https:\/\/ebooks.inflibnet.ac.in\/statp02\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235304KolmogorovsConveregenceTheorem_Ref-1-768x493.jpg 768w, https:\/\/ebooks.inflibnet.ac.in\/statp02\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235304KolmogorovsConveregenceTheorem_Ref-1-1024x657.jpg 1024w, https:\/\/ebooks.inflibnet.ac.in\/statp02\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235304KolmogorovsConveregenceTheorem_Ref-1-65x42.jpg 65w, https:\/\/ebooks.inflibnet.ac.in\/statp02\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235304KolmogorovsConveregenceTheorem_Ref-1-225x144.jpg 225w, https:\/\/ebooks.inflibnet.ac.in\/statp02\/wp-content\/uploads\/sites\/126\/2018\/12\/1522235304KolmogorovsConveregenceTheorem_Ref-1-350x225.jpg 350w\" sizes=\"auto, (max-width: 1177px) 100vw, 1177px\" 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