Piece wise Sequence Convergence in $L^p$ Proof

Question

Let Z be an integrable RV, we have the following,

$ Z_i= \begin{cases} -i&\text{if}\, Z < -i\\ Z &\text{if}\, -i\leq Z \leq i\\ i &\text{if}\, Z>i \end{cases} $

Show whether $Z_1,Z_2,Z_3,...$ converge almost surely, in L1.

I think it converges almost surely if we fix an event $\omega$, then as we approach $Z_i(\omega)$ as i approaches infinity, we have Z_i=Z with probability equal to $\int_{-\infty}^{\infty} f_z dz=1$

How about $L_1$? I know its the same as expectation of Zi but how to proceed?

Answer 1

So long as you are happy that the sequence converges almost surely, $Z_i \overset{a.s.}{\longrightarrow} Z$, then we are in a position to use the Dominated Convergence Theorem (DCT).

From the definition you provide we have have that for all $\omega \in \Omega$

$$|Z_i(\omega)| \leq |Z(\omega)|,$$

moreover $Z$ is given to be integrable, $\mathbf E[|Z|] < \infty$, and since we have almost sure convergence then by the DCT:

$$ \lim_{i \rightarrow \infty} \mathbf E[ |Z_i - Z| ] = 0,$$ which is exactly to say that $Z_i \overset{L^1}{\longrightarrow} Z$.