In this post we show that the joint distribution of two iid random variables is spherically symmetric iff the marginal distribution is Gaussian.<\/p>\n\n\n\n
Proof:<\/strong> If the marginal distribution is Gaussian, the joint distribution is clearly spherical. Below we show the converse: if the joint distribution is spherical, the marginal distribution is Gaussian. <\/p>\n\n\n\n Let the marginal distribution be $g(x)$, so $p(x,y) = g(x)g(y)$. Spherical means the gradient of the joint distribution is proportional to $(x,y)$. That is $$\\nabla p(x,y) = \\begin{bmatrix} g'(x) g(y) \\\\ g(x) g'(y)\\end{bmatrix} = c(x,y) \\begin{bmatrix}x \\\\ y \\end{bmatrix}.$$ <\/p>\n\n\n\n To eliminate the constant of proportionality $c(x,y)$, we divide the first term in the gradient by the second, to get $$ {g'(x) g(y) \\over g(x) g'(y)} = {x \\over y}.$$ Recognizing $g'(x)\/g(x) = d\\ln g(x)\/dx$, we get $$ {d \\ln g(x)\/ dx \\over d \\ln g(y)\/dy } = {x \\over y}.$$<\/p>\n\n\n\n This motivates expressing $g(x) = e^{\\phi(x)}$, for some unknown $\\phi(x)$. Then $d \\ln g(x)\/dx = \\phi'(x),$ and we get $${\\phi'(x) \\over \\phi'(y)} = {x \\over y}.$$ <\/p>\n\n\n\n We then set $y$ to 1, and get $$ {d \\phi(x) \\over dx} = \\phi'(1) x \\implies d\\phi(x) = \\phi'(1) x dx \\implies \\phi(x) = {\\phi'(1) \\over 2}x^2 + c,$$ which then implies that $$ g(x) = e^{\\phi(x)} \\propto e^{\\phi'(1) {x^2 \\over 2}},$$ and the proof is complete.<\/p>\n\n\n\n $$\\blacksquare$$<\/p>\n","protected":false},"excerpt":{"rendered":" In this post we show that the joint distribution of two iid random variables is spherically symmetric iff the marginal distribution is Gaussian.<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[1,152],"tags":[118,120,119],"class_list":["post-4231","post","type-post","status-publish","format-standard","hentry","category-blog","category-post","tag-iid","tag-proof","tag-spherical"],"acf":[],"_links":{"self":[{"href":"https:\/\/sinatootoonian.com\/index.php\/wp-json\/wp\/v2\/posts\/4231","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sinatootoonian.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sinatootoonian.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sinatootoonian.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sinatootoonian.com\/index.php\/wp-json\/wp\/v2\/comments?post=4231"}],"version-history":[{"count":15,"href":"https:\/\/sinatootoonian.com\/index.php\/wp-json\/wp\/v2\/posts\/4231\/revisions"}],"predecessor-version":[{"id":5729,"href":"https:\/\/sinatootoonian.com\/index.php\/wp-json\/wp\/v2\/posts\/4231\/revisions\/5729"}],"wp:attachment":[{"href":"https:\/\/sinatootoonian.com\/index.php\/wp-json\/wp\/v2\/media?parent=4231"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sinatootoonian.com\/index.php\/wp-json\/wp\/v2\/categories?post=4231"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sinatootoonian.com\/index.php\/wp-json\/wp\/v2\/tags?post=4231"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}