Full text: The Personal Distribution of Income

continues to age or it rejuvenates and starts afresh from 
age zero.The successive passages through the zero state 
represent a recurrent event. The probability that the 
recurrence time equals k is p^ ■ L q. 
We are interested in the question:How many years have 
passed,or rather how many steps in the hierarchy have been 
mounted,since the last rejuvenation? This is the "spent 
waiting time" of a renewal process. Choosing an arbitrary 
starting point we can say that in the year n the system 
will be in state if and only if the last re-juvenation 
occurred in year n-k. Letting n-k increase we obtain in 
the limit the steady state probability of the "spent 
waiting time" (Feller 1968 Chapter V ). It is 
proportionate to the tail of the recurrence time 
• • • • lc 
distribution i.e. to p . 
More directly the vector of steady state probabilities u^ 
can be derived from the following two conditions: 
u k = P u k-l* 
Uq = qu Q + qu x + qu 2 + (1) 
The first condition ensures the invariance of the steady 
state;the second ensures that entries to and exits from 
the population balance. 
It follows that 
u k = P ku o*

Note to user

Dear user,

In response to current developments in the web technology used by the Goobi viewer, the software no longer supports your browser.

Please use one of the following browsers to display this page correctly.

Thank you.