Given a sequence of $n$ instances of Bernoulli random variables $x_1,\ldots,x_n$, I am interested in testing whether $x_i$ is independent of $x_{i-1}$ for $i=2,\ldots,n$ (i.e. $p(x_i=0~|~x_{i-1})=p(x_i=0)$). Unfortunately, unlike the scenario in, say, this question, $x_i$'s are NOT drawn from an identical Bernoulli distribution--that is, the probability of success $p_i$ may be different from for each $i$. Does such test exist? What if we assume only a small drift in $p_i$'s (i.e. $p_{i+1}=p_i+\delta$ where $\delta\approx 0$)?
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