In a society there might be a random number N of speakers of a certain accent. Then there is the question: How many people will take this number of people as reference for theit own accent? This can be described in the following way:
Let 
be a series of independent random variables taking values from 
and having the same distribution. Those 
will symbolise a number of people who directly derive their accent from one specific person. For this purpose here one may suppose that these random variables are discrete, that means in this context that the whole society is homogeneous in the way in which it gives prestige to a certain accent 
.
The random variables share the same distribution and N -- the number of speakers of one accent -- is an independent random variable also taking values from . The total number of speakers taking the accent as their reference is described by 
where -- as usual -- the empty sum equals 0. The random variables 
share the same generating function named f. g is the generating function of N.
The generating function of 
is 
.
This generating function is needed to investigate the following Galton-Watson-Process.
A society is investigated at discrete times 
. At the time n the number of speakers of a certain accent is 
; of course is an independent random variable with values from . Every speaker of this accent serves as reference to another speaker, or does not use this accent any more in the next time-interval. This happens for every person individually with the same probability, which is the measurement value of the prestige of the accent. The number of people using the accent of a person i as reference is called 
at the time
n. 
, if nobody takes i as reference.
The number 
of people at the time (n+1) follows with
and by
As already mentioned the random variables 
are independent of each other and all share the same distribution. Let q be the generating function of and 
the generating function of . The combination of those generating functions results in
The series 
of random variables is called Galton-Watson-Chain or
-Process. Here 
is the number of speakers of one accent at the beginning, which might be random. 
is the probability that an accent disappears before time n.
is the probability of extinction. To calculate 
one might view the case of 
first. As an abbreviation 
is the number of persons using the accent of the first person as their reference.
is one solution of the equation y=q(y) in the interval [0,1].
There are just few matching solutions; y=1 is one of them. But there might be one more:
For all 
it is true that:
So q is monotonic, growing and convex. If 
, then q is strictly convex. In this case besides y=1 there might be one more solution of the equation y=q(y) in the interval [0,1].
A second solution exist if and only if E(X)>1 or 
.
To illustrate this for a 
and 
where P(X=m)=p and
P(X=0)=1-p.
Then 
for all 
, and 
is true if and only if
is true. If mp>1, then is the only solution of 
in the interval [0,1).
For instance be m=3 -- that means three persons taking one person as their reference -- and p>1/3. The equation 
has the solution
in [0,1).
For this example let us assume that the probability of X to be 3 is 1/2 . This gives for the value of X of approx. 0.6180 . This means that under this condition with the probability of 
the accent will become ``extinct'' at some time in the future.
Labov's approach can result in a way to measure a certain prestige and the number of speakers. By the model of Galton-Waston Processes it becomes possible to anticipate a possible development of one specific accent in its social context.