probability and statistics: Does having little correlation imply independence? Suppose there are two correlated random variable and having very small correlation coefficient (order of 10-1). Is it valid to approximate it as  independent random variables? 

Independence of random variables implies that they are uncorrelated, but the reverse is not true. Hence no, such an approximation is not valid (given that information alone).
A clear description is given on this Wikipedia page:

If X and Y are independent, then they
  are uncorrelated. However, not all
  uncorrelated variables are
  independent. For example, if X is a
  continuous random variable uniformly
  distributed on [−1, 1] and Y = X²,
  then X and Y are uncorrelated even
  though X determines Y and a particular
  value of Y can be produced by only one
  or two values of X.

