# Concentration inequality of joint event over time of a submartingale

Consider a discrete time submartingale $X_n$ with bounded difference $|X_n-X_{n-1}|\leq c$. With Azuma inequality we have the concentration of a single time event as

$$P(X_t-X_0 \leq -t) \leq exp\left( -{\frac{t}{2c^2}} \right)\tag{1}$$

Now, is there anyway that we can bound this probability $$P(X_n-X_0 \leq -n, \ \ \forall n\geq t ) \tag{2}$$ other than applying the union bound?

I am expecting (1) and (2) to have close value.

In other words, I'm looking for a way to tighten the union bound for events with large overlap, in the setting of this multiple joint tail event probability.

Any reference is appreciated. Thanks.

Yes, but with the price of additional assumptions on variances $\sigma^2$ of samples $X_t$([1]'s proof can be extended to non-identical variance $\sigma ^2_i$). For example, [1] Theorem 2 strengthens the result to (28). These results can also be proved using the methods of types as mentioned in [1] part IV. Consider the case of martingale first, for submartingale in OP, see part III E., Theorem 2 stated that $$P(|X_n-X_0|>\alpha n)\leq 2 \exp \left(n\cdot \operatorname{KLdiv}\left(\frac{\frac{\alpha}{c}+\frac{\sigma^{2}}{c^{2}}}{1+\frac{\sigma^{2}}{c^{2}}},\frac{\frac{\sigma^{2}}{c^{2}}}{1+\frac{\sigma^{2}}{c^{2}}}\right) \right)$$ Specifically when $\sigma^2=c^2$, 1 showed this special case $$P(|X_n-X_0|>\alpha n)\leq 2 \exp \left(\frac{n\alpha^2}{2c^2}\right)$$ for some constant $d>0$. You can refine it further using (34). [1] Theorem 3 yields an even more complicated(but looser bound) Lambert-functional bound.