# Problem 10.4 in Convergence of Probability Measures by Billingsley

Let $$\mu_i$$ be random variables (independent or not) and define $$S_k=\mu_1+\cdots+\mu_k$$.

The hint in the question suggests using the following theorem:

Suppose that $$\alpha>1/2$$ and $$\beta\geq 0$$ and that $$m_1,\dots m_n$$ are nonnegative numbers such that

$$$$P(|S_j-S_i|\geq\lambda)\leq \frac{1}{\lambda^{4\beta}}\Big(\sum_{i for $$\lambda>0$$. Then $$$$P(\max_{k\leq n}|S_k|\geq\lambda)\leq \frac{K}{\lambda^{4\beta}}\Big(\sum_{0

The question is to prove that $$S_i$$ converges with probability 1 if $$S_i$$ satisfies (1) in the theorem.

Although not mentioned in the question, I assumed that $$(\sum_{l=1}^\infty m_l)^{2\alpha}<\infty$$. For some $$\sigma>0$$, I tried to define $$E_n=\{|S_i-S_j|>\sigma,for\,some\,i,j\geq n\}$$ and it suffice to prove that $$\sum_{i=1}^\infty P(E_i)<\infty$$, but I don't know how to proceed.

• I'm confused, are you trying to prove (2) or the convergence in probability? Also, am I correct to assume that $K$ doesn't depend on $n$? Jul 3, 2021 at 20:53
• the question is to prove converge a.e. (with probability 1). Yes K only depend on $\alpha$ and $\beta$ Jul 3, 2021 at 21:02

I am going to assume that $$\sum_k m_k$$ converges and property (2) -- how to go from (1) to (2) is already proved in the Theorem 10.2 of the book you mention.
Consider $$Z_n=\sup_{k,l\ge n} |S_k-S_l|$$. We want to prove that a.s. $$Z_n\to 0$$. From (2),
$$\mathsf{P}(Z_n\ge \lambda)\le 2\mathsf{P}\left(\sup_{k\ge n} |S_k-S_n|\ge \lambda\right)\le \frac{2K}{\lambda^{4\beta}}\left(\sum_{k\ge n} m_k\right)^{2\alpha}.$$
Since $$\sum_k m_k$$ converges, for all $$r\ge 1$$ one can find $$n_r\ge 1$$ such that $$\mathsf{P}(Z_{n_r}\ge 1/r)\le 1/r^2$$. By Borel-Cantelli lemma, almost surely it holds that $$Z_{n_r}< 1/r$$ for all but finitely many $$r$$. Finally $$(Z_n)$$ is nonincreasing, so that $$Z_n\to 0$$ almost surely.