Asymptotic Distribution/Examples/Arbitrary Example 1

Example of Asymptotic Distribution
Let $X_1, X_2, \ldots, X_n$ be $n$ independent observations from a probability distribution with expectation $\mu$ and finite variance $\sigma_2$.

Then $Y_n = \ds \sum_{k \mathop = 1}^n X_k$ has expectation $n \mu$ and variance $n \sigma_2$.

Both of these tend to infinity, unless $\mu = 0$.