ABSTRACT
Most cmphasis has previoi~sly bee11 placed on asyrl~ptot,ics in which rrt ant1 .n tend to irlfinity in slicli a way that both log~n,/ log 11 ;~nt l X - X,,,, converge to finitr:. nonzero lirnits. Using (3.3.12) ant1 (3.2.13), wc can niakc more precise sta.ternerlt,s abo~l t how well the distril-mlion of 14" is being approxim:itcd> and urlclcr less rcstriclivc coiitlitiorls. For in and 72 given, 112 5 1-1: set
k - X. ,, : l o g ( ~ n / ~ ) / log(l/p) r , for anv = c,,,,, > 0, noting that then IEIIr i - /jp'. Floiri (3.2.13), we imrriediately have
which is small so long as log?,, << 7n and r is not too large. 111 asymptotic terms, as m,, 71 -+ (X) for fixed p, (me would rcquirc that log n - o ( m ) a.nd irl addition that (-,,,,, < G log{~n/ log I / ) / log(l/p) for some 0 < 1. Previoi~s asyrnptotics have rnost,l,y assurnet1 that c is bolrildcd above, so that this last condi t io~~ would then be a~toimt~ical ly satisfied.
