{"id":5831,"date":"2013-03-23T18:41:21","date_gmt":"2013-03-23T17:41:21","guid":{"rendered":"http:\/\/djalil.chafai.net\/blog\/?p=5831"},"modified":"2013-03-28T17:23:29","modified_gmt":"2013-03-28T16:23:29","slug":"the-bernstein-theorem-on-completely-monotone-functions","status":"publish","type":"post","link":"https:\/\/djalil.chafai.net\/blog\/2013\/03\/23\/the-bernstein-theorem-on-completely-monotone-functions\/","title":{"rendered":"The Bernstein theorem on completely monotone functions"},"content":{"rendered":"<figure id=\"attachment_5836\" aria-describedby=\"caption-attachment-5836\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><a href=\"http:\/\/en.wikipedia.org\/wiki\/Potemkin_Stairs\" rel=\"attachment wp-att-5836\"><img loading=\"lazy\" class=\"size-medium wp-image-5836   \" alt=\"Potemkin stairs in Odessa, Ukraine.\" src=\"\/blog\/wp-content\/uploads\/2013\/03\/Potemkinstairs-300x219.jpg\" width=\"300\" height=\"219\" srcset=\"https:\/\/djalil.chafai.net\/blog\/wp-content\/uploads\/2013\/03\/Potemkinstairs-300x219.jpg 300w, https:\/\/djalil.chafai.net\/blog\/wp-content\/uploads\/2013\/03\/Potemkinstairs.jpg 350w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/a><figcaption id=\"caption-attachment-5836\" class=\"wp-caption-text\">Potemkin stairs in Odessa, Ukraine.<\/figcaption><\/figure>\n<p style=\"text-align: justify;\">This post is devoted to a simple proof of the Bernstein theorem on completely monotone functions. We have already mentioned this theorem in a <a href= \"\/blog\/2013\/02\/09\/a-probabilistic-proof-of-the-schoenberg-theorem\/\"> previous post on the Schoenberg theorem on positive definite functions<\/a>. Let \\( {g:[0,\\infty)\\rightarrow[0,\\infty)} \\) be a continuous function which is additionally in \\( {\\mathcal{C}^\\infty((0,\\infty))} \\). We say that \\( {g} \\) is <em>completely monotone<\/em> when<\/p>\n<p style=\"text-align: center;\">\\[ \\forall n\\geq0, \\forall t&gt;0, \\quad (-1)^ng^{(n)}(t)\\geq0. \\]<\/p>\n<p style=\"text-align: justify;\">This means that \\( {g^{(0)}:=g\\geq0} \\) on \\( {[0,\\infty)} \\), \\( {g^{(1)}:=g'\\leq0} \\) on \\( {(0,\\infty)} \\), \\( {g^{(2)}:=g''\\geq0} \\) on \\( {(0,\\infty)} \\), etc. In particular, \\( {g(+\\infty):=\\lim_{t\\rightarrow+\\infty}g(t)=0} \\) exists and \\( {g} \\) is bounded. The famous <b>Bernstein theorem on completely monotone functions<\/b> states that the following are equivalent:<\/p>\n<ul>\n<li>(k) \\( {g} \\) is completely monotone:\n<p style=\"text-align: center;\">\\[ \\forall n\\geq0, \\forall t&gt;0, \\quad (-1)^ng^{(n)}(t)\\geq0; \\]<\/p>\n<\/li>\n<li>(kk) \\( {g} \\) is the Laplace transform of a finite Borel measure \\( {\\mu} \\) on \\( {\\mathbb{R}_+} \\):\n<p style=\"text-align: center;\">\\[ \\forall x\\in\\mathbb{R}_+, \\quad g(x)=\\int_0^\\infty\\!e^{-xt}\\,d\\mu(t). \\]<\/p>\n<\/li>\n<\/ul>\n<p style=\"text-align: justify;\"><b>Short proof of the Bernstein theorem.<\/b> The fact that (kk) implies (k) follows from the fact that \\( {g^{(n)}(t)=(-1)^n\\int_0^\\infty\\!x^ne^{-tx}\\,d\\mu(x)} \\). Let us show that (k) implies (kk). Indeed, since \\( {(-1)^ng^{(n)}} \\) is non-negative and non-increasing, we have for any \\( {t&gt;0} \\) and \\( {n\\geq1} \\),<\/p>\n<p style=\"text-align: center;\">\\[ |g^{(n)}(t)| =(-1)^ng^{(n)}(t)\\leq\\frac{2}{t}\\int_{t\/2}^t\\!(-1)^ng^{(n)}(u)\\,du =\\frac{2}{t}|g^{(n-1)}(t)-g^{(n-1)}(t\/2)|. \\]<\/p>\n<p style=\"text-align: justify;\">By induction, it follows then that<\/p>\n<p style=\"text-align: center;\">\\[ \\forall n\\geq1,\\quad g^{(n)}(t)=o_{t\\rightarrow+\\infty}(t^{-n}). \\]<\/p>\n<p style=\"text-align: justify;\">By integration by parts (the boundary terms vanish thanks to the former result),<\/p>\n<p style=\"text-align: center;\">\\[ \\begin{array}{rcl} g(x)-g(+\\infty) &=&-\\int_x^\\infty\\!g'(t)\\,dt\\\\ &\\vdots&\\\\ &=&\\frac{(-1)^{n+1}}{n!}\\int_x^\\infty\\!(t-x)^ng^{(n+1)}(t)\\,dt\\\\ &=&\\frac{(-1)^{n+1}n}{n!}\\int_{x\/n}^{\\infty}\\!(1-x\/(tn))^n(nt)^ng^{(n+1)}(nt)\\,dt\\\\ &=&\\int_0^\\infty\\!\\varphi_n(x\/t)d\\left(\\frac{(-1)^{n+1}n}{n!}\\int_0^t\\!(nt)^ng^{(n+1)}(nt)\\,dt\\right)\\\\ &=&\\int_0^\\infty\\!\\varphi_n(xt)\\,d\\sigma_n(t). \\end{array} \\]<\/p>\n<p style=\"text-align: justify;\">where \\( {\\varphi_n(x):=(1-x\/n)^n\\mathbf{1}_{[0,n]}(x)} \\) and where<\/p>\n<p style=\"text-align: center;\">\\[ \\sigma_n(t):=\\frac{1}{n!}\\int_{1\/t}^\\infty\\!(-1)^{n+1}n(nt)^ng^{(n+1)}(nt)\\,dt. \\]<\/p>\n<p style=\"text-align: justify;\">By integration by parts again,<\/p>\n<p style=\"text-align: center;\">\\[ \\begin{array}{rcl} \\frac{1}{n!}\\int_0^\\infty\\!x^n|g^{(n+1)}|(x)\\,dx &=&\\frac{(-1)^{n}}{(n-1)!}\\int_0^\\infty\\!x^{n-1}g^{(n)}(x)\\,dx\\\\ &\\vdots&\\\\ &=&-\\int_0^\\infty\\!g'(x)\\,dx=g(0)-g(+\\infty). \\end{array} \\]<\/p>\n<p style=\"text-align: justify;\">Therefore, the total variation of \\( {\\sigma_n} \\) on \\( {[0,+\\infty)} \\) is<\/p>\n<p style=\"text-align: center;\">\\[ \\frac{1}{n!}\\int_0^\\infty\\!n(nt)^n|g^{(n+1)}(nt)|\\,dt=g(0)-g(+\\infty). \\]<\/p>\n<p style=\"text-align: justify;\">By the Helly selection theorem, there exists a sub-sequence \\( {\\sigma_{n_k}(t)} \\) that converges almost everywhere to a bounded non-negative non-decreasing function \\( {\\sigma(t)} \\) on \\( {[0,+\\infty)} \\). Since \\( {\\varphi_n(x)\\rightarrow e^{-x}} \\) uniformly on \\( {[0,\\infty)} \\) as \\( {n\\rightarrow\\infty} \\), it follows that<\/p>\n<p style=\"text-align: center;\">\\[ g(x)-g(+\\infty)=\\int_0^\\infty\\!e^{-tx}\\,d\\sigma(t). \\]<\/p>\n<p style=\"text-align: justify;\">Finally, if we set \\( {\\mu:=\\sigma+g(+\\infty)\\delta_0} \\) we get, then for all \\( {x\\in[0,+\\infty)} \\),<\/p>\n<p style=\"text-align: center;\">\\[ g(x)=\\int_0^\\infty\\!e^{-tx}\\,d\\mu(t) \\]<\/p>\n<p style=\"text-align: justify;\">This ends the proof of the Bernstein theorem.<\/p>\n<p style=\"text-align: justify;\"><b>Additional notes and further reading.<\/b> The Bernstein theorem is also known as the (Hausdorff-)Bernstein(-Widder) theorem and can be seen as a characterization of the Laplace transforms of finite Borel measures on \\( {\\mathbb{R}_+} \\). <a href= \"http:\/\/en.wikipedia.org\/wiki\/Sergei_Natanovich_Bernstein\">Sergei Natarovich Bernstein<\/a> (1880 - 1968) proved his theorem in <a href=\"http:\/\/dx.doi.org\/10.1007\/BF02592679\">a paper published in 1928 in Acta Mathematica<\/a>, see also theorem 12a in the book <a href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=5923\">The Laplace Transform<\/a> by Widder. The short proof that we gave is taken from the second section of <a href= \"http:\/\/www.ams.org\/mathscinet-getitem?mr=1118827\">a paper by Korenblum et al<\/a>. The Bernstein theorem is at the heart of the famous <a href= \"http:\/\/www.ams.org\/mathscinet-getitem?mr=416396\">Bessis-Moussa-Villani conjecture<\/a> (BMV), which states that for every \\( {n\\times n} \\) Hermitian matrices \\( {A} \\) and \\( {B} \\) with \\( {B} \\) semidefinite positive, the function<\/p>\n<p style=\"text-align: center;\">\\[ t\\geq0\\mapsto\\mathrm{Tr}(\\exp(A-tB)) \\]<\/p>\n<p style=\"text-align: justify;\">is the Laplace transform of a positive measure on \\( {[0,\\infty[} \\). It seems that Matteo Villani has nothing to do with C&eacute;dric Villani. The BMV conjecture comes from mathematical physics and was solved a couple of years ago by <a href=\"http:\/\/arXiv.org\/abs\/1107.4875\">Herbert Stahl<\/a>. For more informations on this conjecture, one may read <a href= \"http:\/\/arXiv.org\/abs\/1206.0460\">Lieb and Seiringer<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>This post is devoted to a simple proof of the Bernstein theorem on completely monotone functions. We have already mentioned this theorem in a previous&#8230;<\/p>\n<div class=\"more-link-wrapper\"><a class=\"more-link\" href=\"https:\/\/djalil.chafai.net\/blog\/2013\/03\/23\/the-bernstein-theorem-on-completely-monotone-functions\/\">Continue reading<span class=\"screen-reader-text\">The Bernstein theorem on completely monotone functions<\/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":2764},"categories":[1],"tags":[],"_links":{"self":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts\/5831"}],"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=5831"}],"version-history":[{"count":12,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts\/5831\/revisions"}],"predecessor-version":[{"id":5848,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts\/5831\/revisions\/5848"}],"wp:attachment":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/media?parent=5831"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/categories?post=5831"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/tags?post=5831"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}