# Non-negative, integrable random variables which converge in probability and whose expected values have a finite limit

Suppose we have a sequence $X_1, X_2,...$ of non-negative, real random variables (not necessarily increasing) in $L^1$ which converge in probability to an integrable, non-negative random variable $X \in L^1$. Moreover, let's assume

\begin{equation} E(X_n) \rightarrow E(X). \end{equation}

Since the sequence converges in probability, there exists a subsequence converging to $X$ a.s.. This, together with $E(X_n) \rightarrow E(X)$, implies that this subsequence converges in $L^1$ to $X$.

But how can one prove that $X_n \rightarrow X$ in $L^1$?

• Show that every subsequence of $(X_n)$ has a further subsequence that converges to $X$ in $L_1$. – David Mitra Jul 20 '16 at 20:05

Suppose that we do not have the convergence in $\mathbb L^1$. Then there exists a positive $\delta$ and an increasing sequence of integers $\left(n_j\right)_{j\geqslant 1}$ such that for any $j\geqslant 1$, $$\lVert X_{n_j}-X\rVert_1\gt \delta.$$ Now define $Y_j :=X_{n_j}$. We have for any $j\geqslant 1$, $$\lVert Y_j-X\rVert_1\gt \delta.$$ Moreover, the sequence $\left( Y_j\right)_{j\geqslant 1}$ converges to $X$ in probability, hence we can find a subsequence $\left( Y_{j_l} \right)_{l\geqslant 1}$ which converges almost surely to $X$. The fact that $$\lVert Y_{j_l} -X\rVert_1\gt \delta$$ for any $l$ together with the case mentioned in the opening post gives a contradiction.