91影视

$Zn,n鈮�1,$be a sequence of random variables and $c$a constant such that for each

$蔚>0,P|Zn鈭�c|>蔚鈫�0$as$n鈫�q$.

We are given that$g$is a continuous function, which means that for every $c鈭�\mathrm{鈩�}$and$系>0$there exists role="math" localid="1649851182951" $未>0$such that

$|x-c|鈮�未鈬�\left|g\right(x)-g(c\left)\right|鈮�系$

Secondly, we are given that$g$is a bounded function which means that there exists$B>0$such that

$\left|g\right(x\left)\right|鈮�B$

for every$x鈭�\mathrm{鈩�}$. Using the theorem about the expected value of a function of a random variable, we have that

$E\left[g\left(Z_n\right)\right]={鈭�}_{\mathrm{鈩�}}g\left(x\right)d{P}_{Z_n}={鈭�}_{|x-c|鈮�未}g\left(x\right)d{P}_{Z_n}+{鈭�}_{|x-c|>未}g\left(x\right)d{P}_{Z_n}$

for some fixed$c鈭�\mathrm{鈩�}$. Now, for $x$such that $|x-c|鈮�未$we have that $g\left(x\right)鈮�g\left(c\right)+系$and for$x$such that$|x-c|>未$we have that$g\left(x\right)鈮�B$. Therefore

$E\left[g\left(Z_n\right)\right]鈮�\left(g\right(c)+系){鈭�}_{|x-c|鈮�未}d{P}_{Z_n}+B{鈭�}_{|x-c|>未}d{P}_{Z_n}$$=\left(g\right(c)+系)P\left(\left|Z_n-c\right|鈮�未\right)+BP\left(\left|Z_n-c\right|>未\right)$

Now, apply lim sup$n$to both sides and use the assumption that $P\left(\left|Z_n-c\right|>未\right)鈫�0$as$n鈫�鈭�$. It yields that $P\left(\left|Z_n-c\right|鈮�系\right)鈫�1$as$n鈫�鈭�$. Therefore, we end up with

$\underset{n}{\mathrm{limsup}}E\left[g\left(Z_n\right)\right]鈮�g\left(c\right)+系$

On the other hand, we can make lower boundary on$E\left[g\left(Z_n\right)\right]$. Observe that for that same $系$we have that for$|x-c|鈮�未$holds $g\left(x\right)鈮�g\left(c\right)-系$Also, for every $x$we havelocalid="1649852924731" $-B鈮�$$g\left(x\right)$. Therefore, similarly to previous, we have that

$E\left[g\left(Z_n\right)\right]鈮�\left(g\right(c)-系){鈭�}_{|x-c|鈮�未}d{P}_{Z_n}-B{鈭�}_{|x-c|>未}d{P}_{Z_n}$

$=\left(g\right(c)-系)P\left(\left|Z_n-c\right|鈮�未\right)-BP\left(\left|Z_n-c\right|>未\right)$

Applying limit inferior to both sides, we have that

$\underset{n}{\mathrm{liminf}}E\left[g\left(Z_n\right)\right]鈮�g\left(c\right)-系$

because of$P\left(\left|Z_n-c\right|>未\right)鈫�0$as$n鈫�鈭�$. Finally, we have that

$g\left(c\right)-系鈮�\underset{n}{\mathrm{liminf}}E\left[g\left(Z_n\right)\right]鈮�\underset{n}{\mathrm{limsup}}E\left[g\left(Z_n\right)\right]鈮�g\left(c\right)+系$

and since$系>0$was arbitrary, we can get that $系鈫�0$to finally obtain that

$\underset{n}{\lim }E\left[g\left(Z_n\right)\right]=g\left(c\right)$

so we have proved the claim.