Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Assume $\displaystyle\lim_{t\to0}X_t=\gamma\hspace{3pt}a.s.$ where $X_t\geq 0$. I would like to show that $\displaystyle\lim_{t\to0}E[X_t]=E[\lim_{t\to0}X_t]=\gamma$, i.e. that it's possible to interchange the order of limit and expectation.

It would be sufficient to show $\displaystyle\lim_{t\to0}E[|X_t-\gamma|]=0$. I can't find a way to use the dominated/monotone convergence theorems, so I am wondering whether the following argument is valid:

$\displaystyle\lim_{t\to0}E[|X_t-\gamma|] =\lim_{t\to0}E[|X_t-\gamma|{\bf 1}\{|X_t-\gamma|<M\}] +\lim_{t\to0}E[|X_t-\gamma|{\bf 1}\{|X_t-\gamma|>M\}]$

For the first term we have by using the dominated convergence theorem and the fact that $X_t\to\gamma$ with probability one: \begin{equation} \begin{aligned} & \displaystyle\lim_{t\to0}E[|X_t-\gamma|{\bf 1}\{|X_t-\gamma|<M\}]\\ & =E[\lim_{t\to0}|X_t-\gamma|{\bf 1}\{|X_t-\gamma|<M\}]\\ & =0 \end{aligned} \end{equation}

For the second term we have by the inverse of Fatou's lemma and the fact that $X_t\to\gamma$ with probability one: \begin{equation} \begin{aligned} & \lim_{t\to0}E[|X_t-\gamma|{\bf 1}\{|X_t-\gamma|>M\}]\\ & \leq\limsup_{t\to0}E[|X_t-\gamma|{\bf 1}\{|X_t-\gamma|>M\}]\\ & \leq E[\limsup_{t\to0}|X_t-\gamma|{\bf 1}\{|X_t-\gamma|>M\}]\\ & \leq E[\lim_{t\to0}|X_t-\gamma|{\bf 1}\{|X_t-\gamma|>M\}]\\ & =E[0]\\ & =0 \end{aligned} \end{equation}

Combining we have $\displaystyle\lim_{t\to0}E[|X_t-\gamma|]=0$

Is this a valid solution or am I cheating somewhere?

share|cite|improve this question
up vote 1 down vote accepted

The limsup version of Fatou's lemma only holds if there exists an integrable function $g$, such that $f_n < g$ for all n. In that case, your proof is valid; if not, there are counter-examples (e.g. $f_n = \gamma + n * 1_{[0,1/n]}$ )

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.