2016/11/22 by Pinsky, Ross G.
#60F05 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1611.07207
Let \Bk\k=1^∞, \Xk\k=1^∞ all be independent random variables. Assume that \Bk\k=1^∞ are \0,1\-valued Bernoulli random variables satisfying Bk\stackreldist=Ber(pk), with ∑k=1^∞ pk=∞, and assume that \Xk\k=1^∞ satisfy: Xk>0, μk≡ EXk0 and let \beginaligned amp;μn∼ cμna0∏j=1Jμ(log(j)n)aj, amp;pn∼ cp(nb0∏j=1Jp(log(j)n)bj)-1, bJp≠0. \endaligned If \beginaligned amp;i. Jp≤ Jμ; amp;ii. bj=1, 0≤ j≤ Jp; amp;iii. aj=0, 0≤ j≤ Jp-1, and aJpgt;0, \endaligned then limn→∞Wn\stackreldist=\frac1θGD(θ), where θ=\fraccpaJp. Otherwise, limn→∞Wn\stackreldist=c, for some c∈[0,1]. We also give an application to the statistics of the number of inversions in certain shuffling schemes.