count_0_k1[i,1] counts the number of transitions from zero to zero in row i,
```
12 Air Pollution

order of the chain.

**Acknowledgements**

stay at those departments.

estimating the necessary quantities.

**Appendix**

way the empirical probability of having an exceedance in a given day behaves. That can be seen in Figure 1 where, as an illustration, the proportion of exceedances of the threshold 0.11ppm is presented for each region. The values correspond to the proportion of years in which in a fixed day the threshold was exceeded. As we vary the days, we have the behaviour throughout the 366 days. In that figure we have in the horizontal axis the days

the proportion of years with an exceedance in the *<sup>t</sup>*th day, *<sup>t</sup>* = 1, 2, . . . , *<sup>T</sup>*. The notation *<sup>Y</sup>*(*i*)

The plots in Figure 1 reflect well the fact that in region SW, in most of the days of the year, there are exceedances of the threshold 0.11. We may also see the influence of the seasons of the year. The hill between days 100 and 200 appearing in every plot, corresponds to measurements taken between April and June. Higher values occur during the days corresponding to approximately mid April to mid May. Those months are in the middle of Spring. During this season it does rain much in Mexico City. Additionally, there is a lot of sunlight. Hence, the ozone concentration is bound to be high, and as a consequence, the proportion of years in which exceedances occur at that period is large. The values decrease

If we consider the threshold values 0.15ppm and 0.17ppm, the behaviour of the proportion of years where in a given day exceedances occurred is similar to the case of 0.11ppm. The difference is that the values of the proportions are smaller. It is possible to see that the proportion of exceedances may vary according to the seasons of the year, and that, within a given season, changes are, in general, not drastic. Therefore, it is possible that measurements from more than a few days may have an influence on the behaviour of future measurements, and with that, make the estimation method considered here to produce high values for the

The authors thank the Editor for sending comments that helped to improve the presentation of the results. The authors also thank Peter Guttorp for providing a copy of his works related to applications of non-homogeneous Markov chains. ERR and MHT were partially funded by the project PAPIIT-IN102713-3 of the Dirección General de Apoyo al Personal Académico de la Universidad Nacional Autónoma de México (DGAPA-UNAM), Mexico. ERR also received funds from a sabbatical year grant from DGAPA-UNAM. ERR is grateful to the Departments of Statistics of the Universidade Estadual Paulista "Júlio de Mesquita Filho" – Campus Presidente Prudente, Brazil, and of the University of Oxford, UK, where parts of this work were developed, for all the support and hospitality received during her

In this Appendix we present the code of the programme in R used to estimate the order of the non-homogeneous Markov chain as well as its transition probabilities. The code it not optimal and can be highly improved, but in its present form it provides elements for

*<sup>i</sup>*=<sup>1</sup> *<sup>Y</sup>*(*i*)

*<sup>t</sup>* , which represents

*<sup>t</sup>* is

and in the vertical axis we have the values of *prop*(*t*)=(1/*N*) ∑*<sup>N</sup>*

used to indicate the variable *Yt* defined in (1) on the *i*th year.

when the raining season starts (around the beginning of June).

```