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Explosion of an upward-biased walk with geometrically increasing rates

Suppose a transient nearest-neighbour jump chain on

N

has uniformly bounded expected visits to each state and the holding rate at

i

grows geometrically. Then

E \sum_{n} \frac{1}{q _{Y_{n}}} = \sum_{i} \frac{E V _{i}}{q _{i}} < \infty,

so the continuous-time chain explodes almost surely.

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