In order to test the robustness of the Jurkevich method and to investigate intermediate time-scale periods, we exclude Miller's data and use the observational data from 1972 to 1991. During this period, Mkn 421 was more extensively monitored and thus has sufficient data for our analysis to give reliable results. The result of the analysis, with m=5, is shown in Fig. 3 (click here).
Although the time interval considered in this case is less than that covered in
Fig. 2 (click here) (only 19 years), a minimum at 
years is
quite significant and broad, and consistent with the results given in
Fig. 2 (click here). This confirms the period of 15.3 years and test the
robustness of the Jurkevich method.
In addition to the minimum at 13.7 years, a second minimum at period
P = 6.0 years with 
and f=0.649 is found to be
significant and
broad. Its relative depth, however, is only about 8 times the nearby noise.
Although the period of P
= 6.0 years is about half the period of P = 13.7 years, we cannot be
sure of the reality of the former, as it does not appear in Fig. 2 (click here).
In addition to the possible periods of 22.8, 15.3 and 6.0 years, the plots also show minima at P = 1.1, 2.2 and 3.4 years in Figs. 2 (click here) and 3 (click here) with relatively less significance. A one-year period was also found in the light curves of ON 231 (Liu et al. 1995) and 3C 120 (Jurkevich et al. 1971). A one year period is doubtful as the astronomical cycle is of one year. In order to check whether the period is a spurious result of the Jurkevich method, we did the following test.
Figure 4: The plot of normalized 
vs. trial period
for a test object with random variations. No
significant minima corresponding to trial periods of one year and
multiple are found in the plot
Figure 5: Same as Fig. 4 for a test object with sinusoidal
variation of a period of 12.5 years. In addition to the minima at P =
12.5 years and multiple, the minima corresponding to a period of one
year and multiple are significant and are artifacts of the method used
To test the method, we take an object with only random variations
with an amplitude of 4 magnitudes. To mimic real
observations, we make a further assumption that the object can been
observed only from the beginning of January to the end of March every
year and that the available data
covers a one hundred year range. We also assume that, for moon light
reasons for example, it can been observed only
for 10, 20 or 30 days a month. Under these assumptions, the number of
data points (one point a day) would be 3000, 6000 and 9000. The result
of the Jurkevich analysis for the 20-day case is shown in Fig. 4 (click here). The results
for the others are similar. No significant minima are found at one
year and multiple. When we change the assumption from three months to
four months and do the test again, the conclusion is unchanged. Now,
we assume that the source varies sinusoidally with a period
of 12.5 years and we keep all the other assumptions. The result of
the analysis for the 20-day case is shown in Fig. 5 (click here). In addition to the
minima at 12.5 years and multiple, the minima at one year and multiple
become very significant. If we assume we could observe 12 months a year,
the minimum
corresponding to a period of one year does not exist any more in the
plot. We conclude that the Jurkevich method
does not give a spurious period of one year for a randomly variable
source, but if there exists a long term period in the light curve
of the source, a spurious period of around one year will
appear. This can probably be understood as follows: when the trial period is
slightly different from half a year, and one year and its multiple, some
of the ten groups (m) contain very few observational points which
lead to a very small variance 
and therefore small 
(cf. Fig. 5 (click here)). So the minima at about one year and multiple might
be taken as another signal of the existence of a long time-scale period
in the light curve.
The Jurkevich analyses of the observational data with Miller's and without Miller's data provide similar results. These results are independent of the parameter m. Our analysis shows that probably two periods exist in the light curve of the BL Lac object Mkn 421: one of around 15 years and another of around 23 years. However, the current data set covers only four times the possible period of 23 years, so more data is needed to confirm this 23 years period.