# If $P(\max_{n \leq i \leq j \leq k \leq m} \min \{|S_i - S_j|, |S_k - S_j|\} \geq \lambda)$ is bounded appropriately, does $\sum \xi_j$ converge a.s.?

Suppose that

$$E\left[|S_j - S_i|^\gamma |S_k - S_j|^\gamma \right] \leq \left(\sum_{i < l \leq j} u_l\right)^\alpha \left(\sum_{j < l \leq k} u_l\right)^\alpha$$

for $$0 \leq i \leq j \leq k \leq m$$ with $$S_k = \sum_{i \leq k} \xi_i$$ ($$S_0 = 0$$) and $$\gamma, \alpha$$ postitive, and $$u_l \geq 0$$ for every $$l$$.

It follows from a theorem in Billingsley's book Convergence of Probability Measures (first edition) that

$$P\left(\max_{n \leq i \leq j \leq k \leq m} \min \{|S_i - S_j|, |S_k - S_j|\} \geq \lambda\right) \leq \frac{K}{\lambda^{2\gamma}} \left(u_{n+1} + \ldots + u_{m}\right)^{2\alpha} \times \min_{n + 1 \leq h \leq m}\left(1 - \frac{u_h}{u_{n + 1} + \ldots + u_m}\right)$$

where $$K$$ depends only on $$\lambda$$ and $$\gamma$$.

Assume additionally that $$\sum u_l < \infty$$. Can we use the above bounded probability to show that $$\sum \xi_i < \infty$$ almost surely?

I think we can simplify things by saying that since $$\min_{n + 1 \leq h \leq m}\left(1 - \frac{u_h}{u_{n + 1} + \ldots + u_m}\right) \leq 1$$ we actually only need to work with

$$P\left(\max_{n \leq i \leq j \leq k \leq m} \min \{|S_i - S_j|, |S_k - S_j|\} \geq \lambda\right) \leq \frac{K}{\lambda^{2\gamma}} \left(u_{n+1} + \ldots + u_{m}\right)^{2\alpha}$$

This question is related to a question I asked before and I think a similar approach to the solution given there could be used here, but I don't know how to do it. I want to relate $$\max_{n \leq i, j \leq m} |S_i - S_j|$$ to $$\max_{n \leq i \leq j \leq k \leq m} \min \{|S_i - S_j|, |S_k - S_j|\}$$ but I don't know if I can do so in a useful way.

• Bump because bounty is about to expire tomorrow. Even an incomplete answer with a great idea could be rewarded. Commented Jan 18, 2019 at 20:48
• Even a comment such as this cannot be done. This is one of Billingsley's problems, but he left it pretty open ended. Commented Jan 18, 2019 at 20:49