Counterexamples for strong laws of large numbers

Question

Let $\{\xi_k\}_{k=1}^\infty$ be a sequence of random variables in $L^1(\Omega)$.

We have the following laws of large numbers:

Theorem (Strong law of large numbers 1)

Assume

1. $\xi_1, \xi_2, ...$ independent
2. $\sum_{n=1}^\infty \frac{\mathbf{Var}(\xi_n)}{n} < \infty$

Then $$\frac{1}{n}\sum_{k=1}^n(\xi_k-\mathbf{E}(\xi_k)) \to 0 \text{ a.e.}.$$

How can I prove that the result does not hold if we remove either assumption 1. or assumption 2.?

Theorem (Strong law of large numbers 2)

Assume

1. $\xi_1, \xi_2, ...$ independent
2. $\xi_1, \xi_2, ...$ identically distributed

Then $$\frac{1}{n}\sum_{k=1}^n(\xi_k-\mathbf{E}(\xi_k)) \to 0 \text{ a.e.}.$$

How can I prove that the result does not hold if we remove either assumption 1. or assumption 2.? How can I prove that if we remove either assumption 1. or assumption 2. we cannot even conclude that $\frac{1}{n}\sum_{k=1}^n(\xi_k-\mathbf{E}(\xi_k)) \to 0$ in probability(that is, weak law of large numbers)?

Answer 1

Hint: To show that you can't just drop 1.), consider the case where all $\xi_i$ are the same random variable (as in, exactly identical, not just identically distributed).