CLT in case the sum of variances grows as log N Consider the following (bounded) random variables $X_i$, which take outcome $1$ with probability $C/i$ and outcome $0$ with probability $1 - C/i$, where $C$ is a positive constant. Then one has,
$$E[X_i] = \frac{C}{i},$$
$$Var[X_i] = \frac{C}{i} - \frac{C^2}{i^2}.$$
Define $S_n =\sum_{i=1}^n X_i$. Both the variance and the expectation of $S_n$ grow as $\log(N)$.
Does the Central Limit Theorem hold? Is there a general formulation for bounded random variables to apply?
 A: I assume that the $X_i$'s are independent. Denote $E(X_i) \equiv \mu_i$, $Var[X_i]\equiv \sigma^2_i$, and $Var[S_n] \equiv \sigma^2_n$. Note that $C\le1$ for all probabilities to be well-defined. Then the variance of each $X_i$ is decreasing in $i$ for $i\ge 2$.
The variance of the sum $S_n$ is 
$$\sigma^2_n = C\sum_{i=1}^n\frac {1}{i} - C^2\sum_{i=1}^n\frac {1}{i^2}$$
The first sum is the harmonic series which diverges as $n\rightarrow \infty$, while the second sum converges. So $\sigma^2_n$ diverges.
Then we have  that
$$\max_{1\le i\le n} \frac {\sigma^2_i}{\sigma^2_n} \rightarrow 0\;\; \text{as}\;\; n\rightarrow \infty$$
This result makes the "Lindeberg condition" necessary and sufficient for the CLT to hold. This condition can be written as
$$\lim_{n \to \infty} \frac{1}{\sigma^2_n}\sum_{i = 1}^{n} \operatorname{E}\big[(X_i - \mu_i)^2 \cdot \mathbf{1}_{\{ | X_i - \mu_i | > \varepsilon \sigma_n \}}  \big] = 0\;\; \forall \varepsilon >0$$
Since $\sigma^2_n$ diverges, so is $\sigma_n$. Then the indicator function in the above expression will eventually take always the value zero after some finite value of $n$ . Then the sum will be bounded as $n$ increases, while the denominator diverges. So the Lindeberg condition holds and the CLT holds
$$\frac {S_n - E(S_n)}{\sigma_n} \rightarrow_d N(0,1) $$
A: Yes, the central limit theorem holds in this case.
This is worked out in Example 3.4.6 (page 111) of 
the fourth edition of Probability: Theory and Examples
by Richard Durrett. You can get a copy of this book
at http://www.math.duke.edu/~rtd/PTE/PTE4_1.pdf
