Strong law of the large numbers for totally monotone
capacities
Fabio Maccheronia Massimo Marinaccib
Istituto di Metodi Quantitativi and IGIER, Universita Bocconi
b
Dipartimento di Statistica e Matematica Applicata and ICER, Universita di Torino
a
February 2004
Abstract
We prove a strong law of the large numbers for totally monotone capacities, which
extends the results of Marinacci (JET, 1999). Specically, given a Polish space ,
a totally monotone capacity ν on its Borel σ-algebra B, and a sequence {Xn }n≥1 of
independent and identically distributed bounded random variables, then
(
)!
Pn
Pn
Z
Z
j =1 Xj (ω )
j =1 Xj (ω )
ν
ω ∈ : X1 dν ≤ limn
≤ limn
≤ − −X1 dν
= 1,
n
n
provided the Xn s are continuous or simple, or ν is continuous.
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