Let X1,X2,… be a sequence of independent random variables, uniformly distributed on [0,1] . Define Nn to be the smallest k such that X1+X2+⋯+Xk exceeds cn=n2+n12−−−√ , namely,

Question

Nn = min{k≥1:X1+X2+⋯+Xk>cn}
Does the limit

limn→∞P(Nn>n)

exist? If yes, enter its numerical value. If not, enter −999 .

Answer 1

correction:
cn = (n/2) + sqrt(n/12)