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Is there a theorem/reference that states if $M(t)$ is a martingale, then under certain mild conditions, $M(t)/t\to 0$ a.s. as $t\to\infty$?

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There is Theorem 9 p.142 in Liptser and Shiryaev's "Theory of Martingales" of which a particular case is the following :

Let $M = (M_t)_{t \geq 0}$ be a local martingale and define the process $B = (B_t)_{t \geq 0}$ by : $$B_t = \sum_{0 < s \leq t}\frac{(\Delta M_s / (1+s))^2}{1 + |\Delta M_s / (1 + s)|},$$ where $\Delta M_t$ is the jump of $M$ at time $t \geq 0$. Let $\tilde{B} = (\tilde{B}_t)_{t \geq 0}$ be the compensator of $B$ and denote by $\langle M^c \rangle = (\langle M^c \rangle_t)_{t \geq 0}$ the quadratic variation of the continuous martingale part of $M$. Then, if $$\int_0^{\infty}(1+s)^{-2}d\langle M^c \rangle_s + \tilde{B}_{\infty} < +\infty \hspace{2mm} ({\bf P}-a.s.),$$ we have $\frac{M_t}{t} \to 0$ $({\bf P}-a.s.)$, as $t \to \infty$.

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