Consider two sequences of (not necessarily independent) Bernoulli random variables $X_1, X_2, \ldots, X_n$ and $Y_1, Y_2, \ldots, Y_n$. Suppose that for any $i$, we have $\Pr[X_i = 1] = \Pr[Y_i = 1] = p_i$, but the actual value of $p_i$ is determined only after observing $\{X_1, \ldots, X_{i-1}, Y_1, \ldots, Y_{i-1} \}$. That is, each $p_i$ is also a random variable.

Intuitively it seems that the two sums $\sum_i X_i$ and $\sum_i Y_i$ should typically be very close to each other. Are any such bounds known? Is there e.g. a simple argument to show that with high probability $\sum_i X_i = \sum_i Y_i \pm O(\sqrt{n \log n})$?