Forr1 and eachnr, letMnr be therth largest ofX1,X2, ...,Xn, where {Xn,n1} is an i.i.d. sequence. Necessary and sufficient conditions are presented for the convergence of
for all >0 and some –1, where {an} is a real sequence. Furthermore, it is shown that this series converges for all >–1, allr1 and all >0 if it converges for some >–1, somer1 and all >0. 相似文献
The mutagen, 2-aminonaphthalene, which is the dominant aromatic amine species in several fossil fuels, may be isolated readily by using a sequential high-performance liquid chromatographic procedure. An initial isolation is achieved on a semi-preparative scale aminosilane column and the final separation is completed with a nonpolar stationary phase column on an analytical scale with a pH-adjusted solvent. The use of fluorescence detection permits discrimination (at least 100-fold) against the nonmutagenic isomer, 1-aminonaphthalene. The overall recovery of 2-aminonaphthalene, as determined by scintillation counting of a radioactive 1-aminonaphthalene tracer, typically exceeds 50% and can reach 92% depending on both the sample matrix and the solubility of the isolate in acetonitrile. The minimum detectable quantity is 0.06 μg g-1 of crude sample. 相似文献
Let X,Xn;n1 be a sequence of real-valued i.i.d. random variables with E(X)=0. Assume B(u) is positive, strictly increasing and regularly-varying at infinity with index 1/2<1. Set bn=B(n),n1. If
and
for some [0,), then it is shown that
and
for every real triangular array (an,k;1kn,n1) and every array of bounded real-valued i.i.d. random variables W,Wn,k;1kn,n1`` independent of {X,Xn;n1}, where (W)=(E(W–E(W))2)1/2. An analogous law of the iterated logarithm for the unweighted sums nk=1Xk;n1} is also given, along with some illustrative examples. 相似文献