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The Fatou lemma makes every time value of a nonnegative discrete-time local martingale integrable, assuming the usual integrable initial value. The integrable discrete-time result then makes it a true martingale.

A nonnegative martingale takes nonnegative values almost surely at each time. In discrete time its expectations are constant, so it is bounded in $L^1$ and the Martingale convergence theorem gives a finite integrable almost sure limit. Its expectations need not converge to the expectation of that limit; uniform integrability is needed for convergence in L1.

For a centered square-integrable martingale with $v=\mathbb{E}{M}_n^2$ and $\lambda >0$, apply the Doob maximal inequality for a nonnegative submartingale to $(M_k+c{)}^2$ for $c\ge 0$. Conditional Jensen inequality gives the submartingale property, and crossing $\lambda$ forces its square above $(\lambda +c{)}^2$. The bound $(v+c^2)/(\lambda +c{)}^2$ is minimized by $c=v/\lambda$. This extends the Cantelli inequality from one random variable to a martingale maximum, without requiring independent increments.

Every parameter derivativeis a martingale. Differentiation passes through conditional expectation because Gaussian exponential moments dominate every derivative locally uniformly in $\lambda$. At $\lambda =0$ these derivatives yield the time-space Hermite martingales, beginning with $B_t$, $B_t^2-t$, and $B_t^3-3t{B}_t$.