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If I have $Z=o_p(1)$ where $o_p$ is the little-o in probability. I'm interested in find some properties about $E(Z)$.

My first idea was

$E(Z)=E(Z (1_{Z>\varepsilon} + 1_{Z\leq\varepsilon}) ) \leq E(Z^2)P(Z>\varepsilon) +\varepsilon P(Z\leq\varepsilon)$,

for some $\varepsilon > 0$.

As you can see, it's required that $E(Z^2)<\infty$ and it don't seems like an appropriate condition.

So my philosophical question is: Can we give to $E(Z)$ any sense?


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closed as off topic by Dan Petersen, Mark Meckes, Kevin O'Bryant, Bill Johnson, Did Jan 29 '12 at 19:53


Incomprehensible homework, voting to close. –  Igor Rivin Jan 11 '12 at 13:44
To avoid closure, you need to rewrite it more carefully. But probably it is not a "research level" question even so, and should be at another site, not here. –  Gerald Edgar Jan 11 '12 at 13:50

1 Answer 1

It may help to consider the indicators of the intervals [0,n] on the probability space [0,1] with Lebesgue measure.

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