Problem 10.4 in Convergence of Probability Measures by Billingsley

Question

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

\begin{equation} P(|S_j-S_i|\geq\lambda)\leq \frac{1}{\lambda^{4\beta}}\Big(\sum_{i<l\leq j}m_l\Big)^{2\alpha},0\leq i\leq j\leq n,\qquad \qquad\qquad(1) \end{equation} for $\lambda>0$. Then \begin{equation} P(\max_{k\leq n}|S_k|\geq\lambda)\leq \frac{K}{\lambda^{4\beta}}\Big(\sum_{0<l\leq n}m_l\Big)^{2\alpha}\qquad \qquad\qquad(2) \end{equation}

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.

Answer 1

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.

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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.