# Statistical Distance Growth Given K Independent Copies

Let $$X$$ and $$Y$$ be distributions with statistical distance (total variation distance) at most $$d$$. What is the best upper bound you can give on the statistical distance between $$k$$ independent copies of $$X$$ and $$k$$ independent copies of $$Y$$?
I can show using a "hybrid argument" that it is at most $$k\cdot d$$, and I am looking for something better (maybe not always, but at least in some cases).

For example if $$d=1/2$$ and $$k=2$$ this bound is $$1$$ which is meaningless and it cannot be reached by actual distributions because $$X_1,X_2$$ and $$Y_1,Y_2$$ have statistical distance $$1$$ iff their supports are disjoint which would imply that the supports of $$X$$ and $$Y$$ are also disjoint in contradiction to $$X$$ and $$Y$$ having statistical distance at most $$1/2$$. The best example I could find was $$X\equiv1$$ and $$Y$$ being a uniformly distributed bit so the distance between $$X$$ and $$Y$$ is $$1/2$$ and the distance between $$X_1,X_2$$ and $$Y_1,Y_2$$ is $$3/4$$.

For general $$k,d$$ I'd like to show that the distance between $$k$$ samples of $$X$$ and $$k$$ samples of $$Y$$ is at most $$1-\left(1-d\right)^k$$. (For sanity check: $$\left(1-d\right)^k\geq 1-d\cdot k$$ for $$k\geq 1,d\in\left[0,1\right]$$ can be easily shown using basic calculus).
Can you do better (give a counter example), or can you prove that it is the best?

See the "inclusion-exclusion" Lemma 2.2 here https://www.cs.bgu.ac.il/~karyeh/mark-conc2.pdf . For distributions $$p,q,p',q'$$, we have $$||p\otimes q-p'\otimes q'|| \le ||p-p'|| + ||q-q'|| - ||p-p'|| \cdot ||q-q'||.$$
From here, it immediately follows that if $$p=p_1=p_2=\cdots=p_k$$ and $$q=q_1=p_2=\cdots=q_k$$ and furthermore $$||p-q||=d$$, then the TV between the corresponding products is upper-bounded by $$1-(1-d)^k$$. This is proved explicitly in Lemma 4.2 in the linked paper.
Another (weaker) bound, along with a lower bound, both easy to obtain: using Hellinger distance as a proxy (and its relation to total variation distance), you get $$1-(1-d_{\rm TV}(p, q)^2 )^{k/2} \leq d_{\rm TV}(p^{\otimes k}, q^{\otimes k}) \leq \sqrt{1-(1-d_{\rm TV}(p, q) )^{2k}}$$ See e.g., Fact C.2.3 from my survey on distribution testing.