Example to show strong law of large numbers is stronger than weak law of large numbers

Question

I am aware there are already plenty of answers to this question by reasoning from definition of almost sure convergence versus convergence in probability, so I would like to see an example which satisfies WLLN but not SLLN (I am kind of a learn-by-example learner).

Answer 1

Suppose that $X_k$ are i.i.d. variables with density $$f(x)=\frac{c}{(1+x^2) \log(2+x^2)} $$ on $\mathbb R$, where $c$ is chosen to ensure that $\int_{ \mathbb R} f(x)\,dx=1$. Then $S_n:=\sum_{k=1}^n X_k$ satisfy $S_n/n \to 0$ in probability, but $ S_n/n$ does not converge almost surely. This can be inferred from the criteria in [1].

The sharp condition for the WLLN is given in Theorem 1, Section VII.7, page 235 of [1], while the sharp condition for the SLLN is integrability of $|X_k|$, see the discussion in [1] page 236 and Theorem 4 in Section VII.8 there.

[1] Feller, William. An introduction to probability theory and its applications, vol 2. John Wiley & Sons, 1971.