Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 28K / a / i

Table of contents
- Solution i

Solution

Let

E_{n}

be the holding time in state

n

. The variables are independent with

E_{n} \sim Exp (nλ)

, and the explosion time would be

T_{\infty} = \sum_{n \geq 1} E_{n} .

For

s > 0

,

E e^{- s T_{\infty}} = \prod_{n \geq 1} \frac{nλ}{nλ + s} .

This product is zero because

\sum_{n} lo g (1 + s / (nλ)) = \infty

. If

T_{\infty}

were finite with positive probability, the nonnegative variable

e^{- s T_{\infty}}

would have positive expectation. Therefore

T_{\infty} = \infty

almost surely, and the Yule process is nonexplosive.

 Ancestors (11)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version