Expectation and convergence

[sanctus]

Hi all,

 

probably an easy question if a given series tends to a limit value is its expectation the limit value? I tend to think not because it depends how it converges.

 

 

The question arises from montecarlo method:

 

Start from

[math] E_f[h(x)]=\int h(x) f(x) dx[/math]

then for [math](x_1,x_2,...x_m)[/math] generated from the density f approximate the above expectation (MC-approximation) as:

[math]\bar{h}_m=\frac{1}{m}\sum_{j=1}^mh(x_j)[/math]

 

and then by the strong law of large numbers [math]\bar{h}_m\stackrel{a.s.}{\to} E_f[h(x)][/math]

 

somewhere in my notes I wrote then

[math] E[\bar{h}_m]=E_f[h(x)] [/math]

what i think is wrong in at least 2 ways:

1. with respect to which density would [math]E[\bar{h}_m][/math] be calculated anyway?
   
2. assuming that we calculate somehow [math]E[\bar{h}_m][/math], then the equal sign is still wrong, because [math]\bar{h}_m[/math] only tends to [math]E_f[h(x)][/math]. I.e for m big enough we could write [math] E[\bar{h}_m]\approx E_f[h(x)] [/math]
   

 

So my conclusion is that it is wrong and should be written as:

[math] \bar{h}_m\approx E_f[h(x)] [/math]

 

Thoughts?

[sanctus]

Wow 97 views and none of someone good in statistics...;-)

 

Or maybe it is just because I wrote unclearly...