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Kolmogorov's regularity criterion, that a version of G exists with almost surely continuous paths. The version is thus a d-valued random element. We now show that for any interval I 0; T , condition (5.10) of Theorem 5.3.2 is met. Let E be xed as in Lemma 5.3.3. Let t1 t pt2 lie in I . If t2 t1 E, Condition (5.10) is met with p q 2 and F x dx, in view of (5.13). It remains to nd a suitable bound on P N t1 ; t; t2 when t2 t1 > E. Using the basic inequality 2jabj a2 b2 in conjunction with the Cauchy Schwarz inequality, (5.12) entails that P N t1 ; t; t2 This bound is in turn less than 3fEL t1 ; t 4 EL t; t2 4 l t t1 4 l t2 t 4 g: In order to bound the means in the last display expression, note that, by Lemma 5.3.3, both L t1 ; t and L t; t2 are less than the sum of dT =Ee random variables, each of which is less than a Poisson random variable with mean mE. Let m4 denote the fourth moment of such a Poisson random variable. In view of the straightforward P P inequality k ai n k n 1 k jai jn, each of the means in the preceding bound is i 1 i 1 less than dT =Ee dT =Ee4 m4 dT =Ee5 m4 . One thus obtains P N t1 ; t; t2 6fdT =Ee5 m4 lT 4 g: 3 Ex4 EZ4 :

Since RCF is a ratio between two forces, it has no units; however, it is customary to follow the numerical value of RCF with the g symbol. In the centrifuge tube, the

Client Buffer Requirement (MB)

Denoting the latter upper bound by K, t2 t1 ! E then entails that P N t1 ; t; t2 K t t1 2 : E2 2

Thus condition (5.10) of Theorem 5.3.2 is met, with p q 2 and F x p d k=E2 x. j 5.3.2 CLT for Rate and Input Processes

BS (PRT) BS (FEC)

Figure 14.5. Calibration plot of Kav against log(MW).

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The propositions to follow state the functional CLT for the two sequences of processes fL N g and fWt N g introduced in Section 5.2. t Proposition 5.3.5. Let A B denote the total duration of two successive on and off periods of rate process flt g and assume that the distribution of A B has a bounded density in the neighborhood of 0. The sequence of processes Y N de ned by Yt N L N N n t p ; N t ! 0; 5:14

SSS (PRT) BS (non-FT) SSS (PRT & RSS) SSS (FEC)

then converges weakly in the space d toward a zero-mean stationary Gaussian process Y with almost surely continuous paths.

diffusion within the column contributes to band broadening, but this is a relatively small effect. The effect of ow rate has been examined as a factor that in uences bandwidth, and therefore resolution. Figure 14.6 shows a typical plot of band width as a function of ow rate. At low ow rates, longitudinal diffusion is signi cant, and leads to larger bandwidths than at the optimum. At high ow rates, the kinetics of partitioning becomes the dominant factor that leads to increased bandwidths and decreased resolution. The contribution of eddy diffusion is independent of ow rate. The size of the particles used in the stationary phase has also been shown to exert a signi cant effect on resolution. Small particles result in fast equilibration of solutes, as well as smaller contributions of eddy diffusion to band broadening. There is a practical limitation in the use of small particles, however. The smaller

5:15

SSS (non-FT)

where the flj;t g are i.i.d. copies of the process flt g. In view of the ordinary CLT, the nite-dimensional distributions of the process Y N converge weakly to limiting Gaussian distributions. Weak convergence of the sequence fY N gN >0 will then hold provided it is tight. Since L N is the superposition of N i.i.d. on=off rate processes lj with peak rate 1, Eq. (5.15) entails that

the particle, the higher is the resistance to the ow of mobile phase Thus, small particles require higher pressures to elute solute at a given ow rate The gels used in SEC stationary phases are usually soft, or elastic, so that higher pressures can result in compression and column blockage The separation temperature also affects resolution, in that diffusion coef cients increase with temperature, so that partitioning equilibria are faster at higher T This equilibration time is the factor that usually limits resolution The length of the column is important, because the number of theoretical plates (N) increases with the square root of column length; however, practical limitations are imposed on column length Components eluted from the column are diluted with respect to their initial concentrations, and reasonable separations yield dilution factors of $2, for standard column lengths (20 cm).

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