{"id":8355,"date":"2015-04-14T19:19:21","date_gmt":"2015-04-14T17:19:21","guid":{"rendered":"http:\/\/djalil.chafai.net\/blog\/?p=8355"},"modified":"2022-08-29T10:24:27","modified_gmt":"2022-08-29T08:24:27","slug":"tightness","status":"publish","type":"post","link":"https:\/\/djalil.chafai.net\/blog\/2015\/04\/14\/tightness\/","title":{"rendered":"Tightness"},"content":{"rendered":"
\"Yuri<\/a>
Yuri V. Prokhorov (1929 - 2013)<\/figcaption><\/figure>\n

This post is about a basic fact which is not very well known. Let \\( {{(\\mu_i)}_{i\\in I}} \\) be a family of probability measures on a topological space \\( {E} \\) equipped with its Borel sigma-field. The following two properties are equivalent, and when they hold we say that \\( {{(\\mu_i)}_{i\\in I}} \\) is tight<\/b>.<\/p>\n

    \n
  1. for any \\( {\\varepsilon>0} \\) there exists a compact set<\/b> \\( {K_\\varepsilon\\subset E} \\) such that\n

    \\[ \\sup_{i\\in I}\\mu_i(K_\\varepsilon^c)\\leq\\varepsilon; \\]<\/p>\n<\/li>\n

  2. there exists a measurable \\( {f:E\\rightarrow[0,\\infty]} \\) with compact sub-level-sets<\/b> such that\n

    \\[ \\sup_{i\\in I}\\int\\!f\\,d\\mu_i<\\infty. \\]<\/p>\n<\/li>\n<\/ol>\n

    Recall that the sub-level-sets of \\( {f} \\) are the sets \\( {\\{x\\in E:f(x)\\leq r\\}} \\), \\( {r\\geq0} \\). If \\( {E} \\) is not compact then it cannot be a sub-level-set of \\( {f} \\) and thus \\( {f} \\) cannot be bounded, while if \\( {E} \\) is compact then the property holds trivially with \\( {f} \\) constant.<\/p>\n

    To deduce 1. from 2. we write, using the Markov inequality, for any \\( {r>0} \\),<\/p>\n

    \\[ \\sup_{i\\in I}\\mu_i(\\{x\\in E:f(x)\\leq r\\}^c) \\leq \\frac{1}{r}\\sup_{i\\in I}\\int\\!f\\,d\\mu_i=\\frac{C}{r}, \\]<\/p>\n

    which leads to take \\( {r=r_\\varepsilon=C\/\\varepsilon} \\) and \\( {K_\\varepsilon=\\{x\\in E:f(x)\\leq r_\\varepsilon\\}} \\), for any \\( {\\varepsilon>0} \\).<\/p>\n

    To deduce 2. from 1. we first extract from 1. a sequence of compact subsets \\( {{(K_{1\/n^2})}_{n\\geq1}} \\). We can assume without loss of generality that it grows: \\( {K_{1\/n^2}\\subset K_{1\/m^2}} \\) if \\( {n\\leq m} \\). Now, for any \\( {x\\in F=\\cup_{n}K_n} \\), there exists \\( {n_x} \\) such that \\( {x\\in K_{1\/m^2}} \\) for any \\( {m\\geq n_x} \\), and thus \\( {\\sum_n \\mathbf{1}_{K_{1\/n^2}^c}(x)=n_x-1<\\infty} \\). As a consequence, if one defines<\/p>\n

    \\[ f:=\\sum_n\\mathbf{1}_{K_{1\/n^2}^c} \\]<\/p>\n

    then \\( {f<\\infty} \\) on \\( {F} \\) while \\( {f=\\infty} \\) on \\( {F^c} \\), and \\( {f} \\) has compact sub-level-sets since \\( {\\{x\\in E:f(x)\\leq n-1\\}=K_{1\/n^2}} \\) for any \\( {n\\geq1} \\). On the other hand, by definition of \\( {K_\\varepsilon} \\),<\/p>\n

    \\[ \\sup_{i\\in I}\\int\\!f\\,d\\mu_i \\leq\\sum_n\\frac{1}{n^2}<\\infty. \\]<\/p>\n

    Tightness is an important concept of probability theory. A famous theorem of Prokhorov<\/b> states that a family of probability measures is tight if and only if it is relatively compact for the topology of narrow convergence.<\/p>\n

    Taking \\( {X_i\\sim\\mu_i} \\) for every \\( {i\\in I} \\), the second property reads<\/p>\n

    \\[ \\sup_{i\\in I}\\mathbb{E}(f(X_i))<\\infty. \\]<\/p>\n

    It plays for tightness the role played by the famous de la Vall\u00e9e Poussin criterion<\/b> for uniform integrability<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"

    This post is about a basic fact which is not very well known. Let \\( {{(\\mu_i)}_{i\\in I}} \\) be a family of probability measures on…<\/p>\n

    Continue readingTightness<\/span><\/a><\/div>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":224},"categories":[1],"tags":[],"_links":{"self":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts\/8355"}],"collection":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/comments?post=8355"}],"version-history":[{"count":14,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts\/8355\/revisions"}],"predecessor-version":[{"id":16256,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts\/8355\/revisions\/16256"}],"wp:attachment":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/media?parent=8355"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/categories?post=8355"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/tags?post=8355"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}