Multiplicative Chernoff Bound for coins We flip 1000 fair coins and mark the results with $X_1, . . . , X_{1000}$.  A pair of adjacent coins are $X_i,X_{i+1}$ coins - i count $X_{1000}$ and $X_1$ as adjacent. We refer to the number of pairs of adjacent coins showing both heads as X. 
I want to use the multiplicative Chernoff Bound to estimate $Pr[X  \geq 300] $. But then $X$ must be the sum of independent bernoulli distributed random variable. Apparently $X_1$, for example, is not independent of $X_2$. Can I somehow work around this problem and use the multiplicative Chernoff Bound.

Let $Y$ be the number of pairs of consecutive heads of the form $(X_{2i},X_{2i+1})$, and let $Z$ be the number of pairs of the form $(X_{2i-1},X_{2i})$. Note $X=Y+Z$, so
$$
P(X\ge 300)\le P(Y\ge 150)+P(Z\ge 150)
$$
You can then apply the Chernoff bound to each of $Y$ and $Z$.
