 Small-ball conditioning for a Poisson random measure

ID: small-ball-conditioning-for-a-poisson-random-measure

R^{d}

with Lebesgue measure intensity and a nonnegative compactly supported continuous function

f

, put

m_{r} = μ (B (0, r))

and

a_{r} = \int_{B (0, r)} e^{- f} d μ

. Continuity gives

a_{r} / m_{r} \to e^{- f (0)}

. With

L = E e^{- M (f)}

, the count-weighted Laplace functional of a Poisson random measure and independent inside/outside counts give respectively

\frac{E [ M ( B ( 0 , r )) e ^{- M (f)} ]}{P ( M ( B ( 0 , r )) \geq 1 )} = L \frac{a _{r}}{1 - e ^{- m_{r}}}, E [e^{- M (f)} ∣ M (B (0, r)) \geq 1] = L \frac{1 - e ^{- a_{r}}}{1 - e ^{- m_{r}}} .

Both converge to the displayed value. The conditioning is meaningful for every

r > 0

; it is not conditioning on a positive-probability point at the origin.

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