It is known that given a pair of real seq
uences
urce" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si1.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=c00fdb7b599c8631f42bd9b4f2084704">
urce" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16304188-si1.gif">, with
urce" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si101.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=89538fa935d65967b5b42744b77fc88f">
urce" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16304188-si101.gif"> a positive chain seq
uence, we can associate a
uniq
ue nontrivial probability meas
ure
μ on the
unit circle. Precisely, the meas
ure is s
uch that the corresponding Verbl
unsky coefficients
urce" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si3.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=b36b296f671647cf0cfcf34f32ecb9fd">
urce" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16304188-si3.gif"> are given by the relation
where
ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si5.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=5b23b5ebcf9d4f3365601b2db8f1772d" title="Click to view the MathML source">ρub>0ub>=1,
urce" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si6.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=83d94d4f92f3e7f2d6cb8b526d62fa6d">
urce" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16304188-si6.gif">,
ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si490.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=76424479c0a197fe721e22aed714cdf2" title="Click to view the MathML source">n≥1 and
urce" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si394.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=bb132d6a41ce02c224f0c250c51e6f61">
urce" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16304188-si394.gif"> is the minimal parameter seq
uence of
urce" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si101.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=89538fa935d65967b5b42744b77fc88f">
urce" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16304188-si101.gif">. In this paper we consider the space, denoted by
ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si10.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=ac4f18dcb4053c26213c45e712bcc6d4" title="Click to view the MathML source">Nub>pub>, of all nontrivial probability meas
ures s
uch that the associated real seq
uences
urce" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si11.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=b2926c0e11ab3162da16324f1a061ee1">
urce" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16304188-si11.gif"> and
urce" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si12.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=fa9636849e18783c1e7c2e4e0bbef0c8">
urce" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16304188-si12.gif"> are periodic with period
p , for
ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si13.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=38b82d41dd2132bfb22bea267bc698b3" title="Click to view the MathML source">p∈N. By ass
uming an appropriate metric on the space of all nontrivial probability meas
ures on the
unit circle, we show that there exists a homeomorphism
ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si14.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=ff550e036d8eb17f1d631c1b44cf8ee0" title="Click to view the MathML source">gub>pub> between the metric s
ubspaces
ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si10.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=ac4f18dcb4053c26213c45e712bcc6d4" title="Click to view the MathML source">Nub>pub> and
ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si15.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=dbaa018d8f3fc632afdc589385f16d5e" title="Click to view the MathML source">Vub>pub>, where
ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si15.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=dbaa018d8f3fc632afdc589385f16d5e" title="Click to view the MathML source">Vub>pub> denotes the space of nontrivial probability meas
ures with associated
p -periodic Verbl
unsky coefficients. Moreover, it is shown that the set
ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si16.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=8838092044eda4abc463630ffd841d45" title="Click to view the MathML source">Fub>pub> of fixed points of
ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si14.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=ff550e036d8eb17f1d631c1b44cf8ee0" title="Click to view the MathML source">gub>pub> is exactly
ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si17.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=d00c29731cc554f4abc6356216db67a1" title="Click to view the MathML source">Vub>pub>∩Nub>pub> and this set is characterized by a
ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si18.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=aab7e30a4d21ea1cc37d71d6c8fcd01b" title="Click to view the MathML source">(p−1)-dimensional s
ubmanifold of
ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si19.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=af1316156a55e2691d427b2ce8c5a9e1" title="Click to view the MathML source">Rup>pup>. We also prove that the st
udy of probability meas
ures in
ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si10.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=ac4f18dcb4053c26213c45e712bcc6d4" title="Click to view the MathML source">Nub>pub> is eq
uivalent to the st
udy of probability meas
ures in
ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si15.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=dbaa018d8f3fc632afdc589385f16d5e" title="Click to view the MathML source">Vub>pub>. F
urthermore, it is shown that the p
ure points of meas
ures in
ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si10.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=ac4f18dcb4053c26213c45e712bcc6d4" title="Click to view the MathML source">Nub>pub> are, in fact, zeros of associated para-orthogonal polynomials of degree
p . We also look at the essential s
upport of probability meas
ures in the limit periodic case, i.e., when the seq
uences
urce" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si11.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=b2926c0e11ab3162da16324f1a061ee1">
urce" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16304188-si11.gif"> and
urce" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304188&_mathId=si12.gif&_user=111111111&_pii=S0022247X16304188&_rdoc=1&_issn=0022247X&md5=fa9636849e18783c1e7c2e4e0bbef0c8">
urce" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16304188-si12.gif"> are limit periodic with period
p. Finally, we give some examples to ill
ustrate the res
ults obtained.