{"id":1667,"date":"2010-03-13T17:11:40","date_gmt":"2010-03-13T15:11:40","guid":{"rendered":"https:\/\/wpethzprd.ethz.ch\/kowalski\/?p=1667"},"modified":"2024-02-20T13:40:53","modified_gmt":"2024-02-20T11:40:53","slug":"more-cauchy-distribution","status":"publish","type":"post","link":"https:\/\/blogs.ethz.ch\/kowalski\/2010\/03\/13\/more-cauchy-distribution\/","title":{"rendered":"More Cauchy distribution"},"content":{"rendered":"

I’ve mentioned earlier<\/a> the first cases I had found of (almost) mod-Cauchy convergence (see this post for the definition). <\/p>\n

Yesterday, following from a citation in Sarnak’s note to one in a paper of Guivarc’h and Le Jan<\/a>, I ended up looking at this paper of F. Spitzer<\/a>, which implicitly proves a true mod-Cauchy convergence result. Morever, this result is very interesting from the probabilistic point of view, since it concerns that most beautiful of objects, the (complex) Brownian motion<\/a>.<\/p>\n

Here is the result: consider such a complex Brownian motion<\/p>\n

$latex W(t)=W_1(t)+iW_2(t),$<\/p>\n

for non-negative t<\/i>, where the real and imaginary parts are themselves independent standard Brownian motions on the real line, except that the real part is started at <\/p>\n

$latex W_1(0)=R,$<\/p>\n

for some fixed R> 0<\/i> (for instance R=1<\/i>).<\/p>\n

Because almost surely the Brownian motion is not zero, one can define a continuous process <\/p>\n

$latex \\theta_R(t)=\\mathrm{Im} \\log W(t),\\quad\\quad \\theta_R(0)=0,$<\/p>\n

which measures the argument of the Brownian motion and its winding around the origin. Spitzer’s result is then the fact that<\/p>\n

$latex \\frac{\\theta_R(t)}{\\log \\sqrt{t}}$<\/p>\n

converges in law, as t<\/i> goes to infinity, to a Cauchy distribution with parameter 1 (it had already been noticed by P. Lévy that the argument of the Brownian motion is not square-integrable).<\/p>\n

However, his argument is more precise: using relations with suitable PDE’s, he manages to compute exactly the Fourier transform of the law of the argument in terms of Bessel functions: for u≥0<\/i> (this characteristic function is even), we have<\/p>\n

$latex \\mathbf{E}(e^{iu\\theta_R(t)})=\\sqrt{\\frac{\\pi R^2}{8t}}e^{-R^2\/(4t)}\\Bigl\\{I_{(u-1)\/2}\\Bigl(\\frac{R^2}{4t}\\Bigr)+I_{(u+1)\/2}\\Bigl(\\frac{R^2}{4t}\\Bigr)\\Bigr\\}$<\/p>\n

where Iν<\/sub>(x)<\/i> is the I-Bessel function<\/a> of index ν<\/i>.<\/p>\n

Since the first term in the power series expansion (at 0) of the Bessel function is given by<\/p>\n

$latex I_{\\nu}(x)=\\frac{(x\/2)^{\\nu}}{\\Gamma(\\nu+1)}+O(x^{\\nu+1}),$<\/p>\n

it follows easily that we have a mod-Cauchy convergence:<\/p>\n

$latex \\exp(A(t)|u|) \\mathbf{E}(e^{iu \\theta_R(t)})\\longrightarrow \\Phi(u),$<\/p>\n

as t<\/i> goes to infinity, with parameters and limiting function given by<\/p>\n

$latex A(t)=\\log\\frac{\\sqrt{8t}}{R},\\quad\\quad \\Phi(u)=\\frac{\\sqrt{\\pi}}{\\Gamma((|u|+1)\/2)}=\\frac{\\Gamma(1\/2)}{\\Gamma((|u|+1)\/2)}.$<\/p>\n","protected":false},"excerpt":{"rendered":"

I’ve mentioned earlier the first cases I had found of (almost) mod-Cauchy convergence (see this post for the definition). Yesterday, following from a citation in Sarnak’s note to one in a paper of Guivarc’h and Le Jan, I ended up looking at this paper of F. Spitzer, which implicitly proves a true mod-Cauchy convergence result. … Continue reading More Cauchy distribution<\/span><\/a><\/p>\n","protected":false},"author":625,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[],"class_list":["post-1667","post","type-post","status-publish","format-standard","hentry","category-blogroll"],"_links":{"self":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/posts\/1667","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/users\/625"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/comments?post=1667"}],"version-history":[{"count":0,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/posts\/1667\/revisions"}],"wp:attachment":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/media?parent=1667"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/categories?post=1667"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/tags?post=1667"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}