Strong law of the large numbers for totally monotone capacities

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.
1