structural:reformatting, quality:correction
Expectations under convergence of random variables

In general, it is not the case that E → E {\displaystyle \operatorname {E} \to \operatorname {E} } even if X n → X {\displaystyle X_{n}\to X} pointwise. Thus, one cannot interchange limits and expectation, without additional conditions on the random variables. To see this, let U {\displaystyle U} be a random variable distributed uniformly on. {\displaystyle.} For n ≥ 1, {\displaystyle n\geq 1,} define a sequence of random variables X n = n ⋅ 1 { U ∈ (0, 1 n) }, {\displaystyle X_{n}=n\cdot \mathbf {1} \left\{U\in \left(0,{\tfrac {1}{n}}\right)\right\},} with 1 { A } {\displaystyle \mathbf {1} \{A\}} being the indicator function of the event A. {\displaystyle A.} Then, it follows that X n → 0 {\displaystyle X_{n}\to 0} pointwise. But, E = n ⋅ Pr (U ∈) = n ⋅ 1 n = 1 {\displaystyle \operatorname {E} =n\cdot \Pr \left(U\in \left\right)=n\cdot {\tfrac {1}{n}}=1} for each n. {\displaystyle n.} Hence, lim n → ∞ E = 1 ≠ 0 = E. {\displaystyle \lim _{n\to \infty }\operatorname {E} =1\neq 0=\operatorname {E} \left.}

Analogously, for general sequence of random variables { Y n: n ≥ 0 }, {\displaystyle \{Y_{n}:n\geq 0\},} the expected value operator is not σ {\displaystyle \sigma } -additive, i.e. E ≠ ∑ n = 0 ∞ E. {\displaystyle \operatorname {E} \left\neq \sum _{n=0}^{\infty }\operatorname {E}.}

An example is easily obtained by setting Y 0 = X 1 {\displaystyle Y_{0}=X_{1}} and Y n = X n + 1 − X n {\displaystyle Y_{n}=X_{n+1}-X_{n}} for n ≥ 1, {\displaystyle n\geq 1,} where X n {\displaystyle X_{n}} 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.