1965 CORRESPONDENCE 2 133

A Test for Regularity of Markov Chains

As is well known the theory of Markov chains is applied [l] , [2] in many areas. If P = bi,(1)] is the transition probability matrix (n Xn) of a stationary Markov chain and P'= [p i j") ] , r positive integer, then the chain is said to be regular [3] provided, for some r = Y O , p i j ( ' 0 ) > O for all i, J. Also. if Y O is the smallest [4] positive integer possible, then p$) > O for I 210. One obvious way of checking a Markov chain for regularity is to compute directly P?, P4, and so on. The ob- ject of the present correspondence is to dis- cuss an alternate method.

First it is to be noted that

In (1) the expression under the summation signs represents one way of reaching state Si from state Si in r steps. From this it follows that, for P$) to be positive for all i, j , it is necessary and sufficient that there is at least one way of reaching every Si from every S; in Y steps with nonzero probability. On the basis of this it can be stated further that, for a Markov chain to be regular, it is necessary and sufficient that there is a posi- tive integral number I = Y O of steps in which every Si can be reached from every Si with nonzero probability. In other words the problem of checking a Markov chain for regularity reduces to one of finding whether such an Y O exists. The latter problem can be tackled with the help of state diagrams. The following examples will illustrate the pre- ceding discussion. In the examples, all nonzero pij 's are represented by x as their individual values are not pertinent to the discussion.

ILLVSTRATIVE EXAMPLES Emmple 1

trix given by Consider the transition probability ma-

sj

- 0 ~ 0 0 0 ~ 0 0 0 x O r O x O O O O O r O x O O O O O o o r o x o o o x

P = s ; o x o x o s o x o , ( 2 ) x o o o x o x o o 1 o o o o o x o x o o o o o x o x o x

~ o o o x o o o ~ o J

where the x's represent nonzero entries, and the corresponding state diagram shown in Fig. 1. To begin with let us consider the transition form S I to SI. Inspection of Fig. 1 shows that SI+SI can take place with non- zero probability only in 2, 4, 6, * e steps. Next i t is seen that SI+S? can happen with

Manuscript received October 25. 1%5.

nonzero probability only in 1 ,3 ,5 , . . . steps. This means that the condition that, for regularity, there should be some r = r o of steps in which Si+S, is possible for all i, j with nonzero probability, is already violated. Hence P of (2) is not regular.

Fig. 1 . State diagram relevant to equation (2).

Example 2

trix Consider the transition probability ma-

sj

and the associated state diagram shown in Fig. 2. Examination of Fig. 2 shows that transition is possible for all i, j in three steps with nonzero probability. There- fore P of (3) is regular.

n

Fig. 2. State diagram relevant to equation (3).

CONCLUDISG REMARKS A "state diagram" test for regularity

based on the fact that, for a Markov chain to be regular, i t is necessary and sufficient that there is a common number of steps in which transition Si+Si is possible for all i , j with nonzero probability has been discussed. This conclusion may be contrasted with ergodicity which also demands the possibil- i ty of transition S;+S, for all i, j with non-

zero probability in a finite number of steps, but does not require this number to be the same for all i, j .

S. G. S. SHIVA Dept. of Elec. Engrg. University of Ottawa

Ottawa, Ontario Canada

REFERENCES [I] A. T. Bharucha-Reid. Elements of the Theory of

Mwkm Processes and Their Applications. New Yo&: McGraw-Hill. 1960.

[Z] J. G. Kemeny and J. L. Snell. Finite Morkoo Chains. Princeton. N. 1.: Van Nostrand. 1960.

131 F. M. Rem. A n Inlroduition lo Information Theorv. .~ ~

[4] F. R. Gantmacher. Applicafions of t h e Theory of New York:'McGraw-Hill. 1961, i p . 376-377.

Mahicrc. New York: Interscience, 1959. PP. 1 0 6 118: particularly p. 112.

Orientation Dependence of Built-In Surface Charge on Thermally Oxidized Silicon

The rate of thermal oxidation of silicon at atmospheric pressure is a function of the exposed crystal face orientation. Ligenza and Spitzer [ l ] mention that the oxidation rate is "very slightly" dependent on crystal orientation. Deal [2] reports that for dry as well as for wet oxygen, and steam, the growth rates at 1OOO" or lower on (111) or (1 10) surfaces are approximately equal, but that those on (100)- surfaces are definitely lower. He suggests a mixed linear-paraholic law for the oxide growth in this temperature region and thinks that the reaction may be partially controlled a t the oxide-silicon inter- face. W. A. Pliskin [3] finds that it is the linear part of the growth mechanism which exhibits the orientation dependence. I t is, therefore, reasonable to expect that the electrical surface properties also show an orientation dependence. A dependence of this nature was indeed reported recently by Delord et al. [4] for the built-in surface charge (1.2 X lo'*, 6 X loll, and 4 X 10" elec- tron charges per cm* for ( 1 1 1 1 , {110]., and (loo] surfaces, respectively).

In a series of experiments on silicon samples that were oriented in the ( I l l ) , (110), and (100) directions, and processed simultaneously, the present authors have confirmed Deal's general findings on the rate behavior for dr). oxygen and steam oxida- tion. In addition, they have established that, even though a t 1100"C, no difference could be detected, the rates a t 975°C were higher for both procedures on the (111). than on the (110) surfaces. 4 preliminary evalua- tion of the electrical surface characteristics was made from MOS capacitance measure- ments with evaporated aluminum electrodes. The data qualitatively agree with those of Delord et al. [4]; both for oxygen and for steam grown oxides the amount of built-in charge decreases in the order ( l l l ) , (110],

Some characteristic results that were ob- tained on 100-n cm p-type float-zone silicon

{loo:.

Manuscript received October 4, 1%5.