proof of convergence of a sequence with finite upcrossings
then and the sequence converges to a limit if and only if .
We first show that if the sequence converges then is finite for . If then there is an such that for all . So, all upcrossings of must start before time , and we may conclude that is finite. On the other hand, if then and we can infer that for all and some . Again, this gives .
Conversely, suppose that the sequence does not converge, so that . Then choose in the interval . For any integer , there is then an such that and an with . This allows us to define infinite sequences by and
for . Clearly, and for all , so .
|Title||proof of convergence of a sequence with finite upcrossings|
|Date of creation||2013-03-22 18:49:39|
|Last modified on||2013-03-22 18:49:39|
|Last modified by||gel (22282)|