Central Limit Theorem for Wilcoxon signed rank tests Let $X_i, i=1,2,...,n$ be a set of iid observations, assumed symmetric about $\mu$. Let $R_i$ be the rank of the absolute deviations from some $\mu_0$, i.e. $R_i=\text{rank}(|X_i-\mu_0|)$. Let $Z_i=\text{sign}(X_i-\mu_0)$. Under the null hypothesis $\mu_0=\mu$.
If I pick up only one sample each time, let X be the random variable, the expected average value E(X) is 0.5, isn't it? the variance of X is 0, isn't it? so how 
how can I apply CLT for signed rank tests?
 A: I suppose that perhaps what you're expected to do is treat the set of ranks $\{R_i\}$ as fixed and the $Z_i$ as independent Bernoulli. Which is to say you have a set of scaled Bernoulli variates. 
(e.g. one could relabel so that $R_i=i$ without changing the distribution of the sum $\sum_i R_iZ_i$. But rather than deal with $\sum_i iZ_i$ let's generalize a little.)
Forget about them being ranks for the moment and imagine instead you have a set of constants, $a_i$ (which obey certain conditions relating to how big they get as $n$ increases so that we can apply the limit theorems we need later) and a set of independent Bernoulli$(\frac12)$ variates and you want the mean and variance of $\sum_i a_iZ_i$.
To start with individual terms:
$\text{E}(a_iZ_i)$
$\text{Var}(a_iZ_i)$
These are easy to calculate! 
You can then progress to the mean and variance of the sum. Of course for the variance of the sum you also need to worry about covariance terms. If you do that right you get exactly the mean and variance you need.
Now if we treat our $a_i$ values as constant, the $a_iZ_i$ are independent but not identically distributed --  but there are certainly versions of the CLT for non identically distributed variates e.g. 1, e.g.2. If you check their conditions you might be able to apply one of them.
(Alternatively, if we were to regard the  $a_i$ values as randomly selected without replacement from the set of ranks, then we have identically distributed $a_iZ_i$ but they're no longer independent. There's also limit theorems which can deal with that dependence. Indeed there's extensive literature on the asymptotic distribution of rank-statistics)
A: One way to show the asymptotic normality of the Wilcoxon signed rank statistic is through the use of Hoeffding's U-Statistics Theorem. For this statistic you create a kernel based on Walsh averages (these are the ($X_i$ + $X_j$)/2 averages). U-Statistics theory allows you to obtain asymptotics results even for dependent summands, and central to this result is the notion of projections which approximates the sum of the dependent variables by a sum of independent random variables and on which you could obtain the usual CLT. For details see, for instance, the book by Ronald Randles and Douglas Wolfe on Theory of Nonparametric Statistics.,
A: To add to the above answers (+1 to both), here's a good resource on deriving a CLT for a U-statistic when the ranks are assumed to be random (when they are nonrandom you can use Lyapunov's CLT--this is what @Glen_b is referring to). They mention the projections that @Michael R. Chernick is referring to. You can see it is involved.
Here's the thing I wanted to add: the signed-rank test isn't exactly a U-statistic, though--there's still a bit of a jump. The answer in this thread shows you can write the signed-rank test as a sum of "Walsh averages," however notice that the sum runs over indexes where $i \le j$. In a U-Statistic the sum runs over all $i < j$. Even though the signed-rank test has sum extra sums corresponding to where $i=j$, you can show that they still (after having been properly rescaled and recentered) have the same asymptotic normal distribution. This is exercise 1.7 in Chapter 6 of this book.
