Expected value Warning: You are not logged in. Your IP address will be publicly visible if you make any edits. If you log in or create an account, your edits will be attributed to your username, along with other benefits.Anti-spam check. Do not fill this in! ===Expectations under convergence of random variables=== In general, it is not the case that <math>\operatorname{E}[X_n] \to \operatorname{E}[X]</math> even if <math>X_n\to X</math> pointwise. Thus, one cannot interchange limits and expectation, without additional conditions on the random variables. To see this, let <math>U</math> be a random variable distributed uniformly on <math>[0,1].</math> For <math>n\geq 1,</math> define a sequence of random variables :<math>X_n = n \cdot \mathbf{1}\left\{ U \in \left(0,\tfrac{1}{n}\right)\right\},</math> with <math>{\mathbf 1}\{A\}</math> being the indicator function of the event <math>A.</math> Then, it follows that <math>X_n \to 0</math> pointwise. But, <math>\operatorname{E}[X_n] = n \cdot \operatorname{P}\left(U \in \left[ 0, \tfrac{1}{n}\right] \right) = n \cdot \tfrac{1}{n} = 1</math> for each <math>n.</math> Hence, <math>\lim_{n \to \infty} \operatorname{E}[X_n] = 1 \neq 0 = \operatorname{E}\left[ \lim_{n \to \infty} X_n \right].</math> Analogously, for general sequence of random variables <math>\{ Y_n : n \geq 0\},</math> the expected value operator is not <math>\sigma</math>-additive, i.e. :<math>\operatorname{E}\left[\sum^\infty_{n=0} Y_n\right] \neq \sum^\infty_{n=0}\operatorname{E}[Y_n].</math> An example is easily obtained by setting <math>Y_0 = X_1</math> and <math>Y_n = X_{n+1} - X_n</math> for <math>n \geq 1,</math> where <math>X_n</math> is as in the previous example. A number of convergence results specify exact conditions which allow one to interchange limits and expectations, as specified below. * [[Monotone convergence theorem]]: Let <math>\{X_n : n \geq 0\}</math> be a sequence of random variables, with <math>0 \leq X_n \leq X_{n+1}</math> (a.s) for each <math>n \geq 0.</math> Furthermore, let <math>X_n \to X</math> pointwise. Then, the monotone convergence theorem states that <math>\lim_n\operatorname{E}[X_n]=\operatorname{E}[X].</math> {{pb}} Using the monotone convergence theorem, one can show that expectation indeed satisfies countable additivity for non-negative random variables. In particular, let <math>\{X_i\}^\infty_{i=0}</math> be non-negative random variables. It follows from [[#Monotone convergence theorem|monotone convergence theorem]] that <math display="block"> \operatorname{E}\left[\sum^\infty_{i=0}X_i\right] = \sum^\infty_{i=0}\operatorname{E}[X_i]. </math> * [[Fatou's lemma]]: Let <math>\{ X_n \geq 0 : n \geq 0\}</math> be a sequence of non-negative random variables. Fatou's lemma states that <math display="block">\operatorname{E}[\liminf_n X_n] \leq \liminf_n \operatorname{E}[X_n].</math> {{pb}} '''Corollary.''' Let <math>X_n \geq 0</math> with <math>\operatorname{E}[X_n] \leq C</math> for all <math>n \geq 0.</math> If <math>X_n \to X</math> (a.s), then <math>\operatorname{E}[X] \leq C.</math> {{pb}} '''Proof''' is by observing that <math display="inline"> X = \liminf_n X_n</math> (a.s.) and applying Fatou's lemma. * [[Dominated convergence theorem]]: Let <math>\{X_n : n \geq 0 \}</math> be a sequence of random variables. If <math>X_n\to X</math> [[pointwise convergence|pointwise]] (a.s.), <math>|X_n|\leq Y \leq +\infty</math> (a.s.), and <math>\operatorname{E}[Y]<\infty.</math> Then, according to the dominated convergence theorem, ** <math>\operatorname{E}|X| \leq \operatorname{E}[Y] <\infty</math>; ** <math>\lim_n\operatorname{E}[X_n]=\operatorname{E}[X]</math> ** <math>\lim_n\operatorname{E}|X_n - X| = 0.</math> * [[Uniform integrability]]: In some cases, the equality <math>\lim_n\operatorname{E}[X_n]=\operatorname{E}[\lim_n X_n]</math> holds when the sequence <math>\{X_n\}</math> is ''uniformly integrable.'' Summary: Please note that all contributions to Christianpedia may be edited, altered, or removed by other contributors. If you do not want your writing to be edited mercilessly, then do not submit it here. You are also promising us that you wrote this yourself, or copied it from a public domain or similar free resource (see Christianpedia:Copyrights for details). Do not submit copyrighted work without permission! Cancel Editing help (opens in new window) Discuss this page