How to prove this martingale's jumps are not summable?

Question

I'm working on a problem from Stochastic Integration and Differential Equations by Protter and am not sure how to proceed:

Let $(N_t^i)_{t \ge 0}$ be an iid sequence of Poisson processes, each with parameter $\lambda_i = 1$. Define for each $i$:

$$M_t^i = \frac{1}{i}(N_t^i-t)$$ $$ M_t = \sum_{i=1}^{\infty}M_t^i$$

How do I prove that for any $t > 0$: $\sum_{s \le t} \Delta M_s = \infty$, where $\Delta M_s = M_s - M_{s_{-}}$.

Any help would be appreciated

Answer 1

First note that $M$ is finite a.s. (e.g. by Kolmogorov's one series theorem).

But if the sum of jumps were finite with positive probability, then we would have $$M_t = \text{something finite} - \sum_{n=1}^\infty \frac{t}{n} =-\infty$$ with positive probability, a contradiction.