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SUMMARY:Finding the distribution of random multiplicative functions in sho
 rt intervals - Adam Harper (University of Warwick)
DTSTART:20260211T133000Z
DTEND:20260211T143000Z
UID:TALK242035@talks.cam.ac.uk
CONTACT:Julia Wolf
DESCRIPTION:Let f be a random multiplicative function\, and consider the s
 um of f(n) over a short interval x ≤ n ≤ x+y (where y = o(x) as x tend
 s to infinity). Thanks to work of Chatterjee-Soundararajan and Soundararaj
 an-Xu\, it is known that these sums have a Gaussian limiting distribution 
 when rescaled by their standard deviation\, provided x/y is at least a cer
 tain power of log x. On the other hand\, work of Harper and of Caich impli
 es that these sums will converge to zero when rescaled by their standard d
 eviation\, if y is "close'' to x. I will report on joint work (in preparat
 ion) of myself\, Soundararajan and Xu on this problem. We find that on the
  full range y = o(x)\, the sums have a Gaussian limiting distribution when
  rescaled properly\, but the correct scaling factor changes as y approache
 s x. In contrast\, when y ~ x there is no rescaling under which the sums h
 ave a (non-degenerate) Gaussian limit.
LOCATION:MR4\, CMS
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