I'm not sure if I know the precise definition, but I think I remember how it's used.
weakly random:
let X a random variable over {0,1}^n. That is, X is a distribution over bit sequences of length n. Distribution means I assign each of the 2^n bit sequences a nonnegative probability that sums to 1.
If X were uniform (true random), then each sequence in the 2^n such possible sequences would have probability 2^-n. Unfortunately, we have no such sequence generators, or life would be easy.
Let the min entropy of X be the largest natural k such that P[X=x] <= 2^-k for every x. ie roughly how likely are such sequences to take their most likely value. Or how biased is our RV.
eg: uniform random over sequences of length n would have min-entropy n.
Again, the problem is we have no uniform random sequences available to us. Instead, we have biased sequences. So there is a long history of asking how can we take a biased random sequence and turn it into a random sequence. You do this with an extractor.
For example, suppose we had a sequence which produced independent bits 1 with probability p and 0 with probability q=1-p. However, we don't know p. What we could do (an extractor!) is use two sequence bits to produce one output bit, as so: sequence 0,1 has probability p x q, and sequence 1,0 has probability q x p. Map 0,1 to 0 and 1,0 to 1, and ignore 0,0 and 1,1. Now we can produce 0 with probability 1/2 and 1 with probability 1/2, ie we have uniform random from an input sequence of unknown bias (but perfectly uncorrelated, which also doesn't exist in nature. Life is hard.)
We measure the quality of extractors by calculating how bad (lower entropy) a sequence they can take and how close (statistical distance, which has a technical definition) they can output a distribution close to uniform.
Anyway, I got distracted, but in general weakly random means distributions or sequences which are biased. The less biased they are the more they're called "almost random." In general, you want to build extractors that run on sequences with unknown bias.
weakly random:
let X a random variable over {0,1}^n. That is, X is a distribution over bit sequences of length n. Distribution means I assign each of the 2^n bit sequences a nonnegative probability that sums to 1.
If X were uniform (true random), then each sequence in the 2^n such possible sequences would have probability 2^-n. Unfortunately, we have no such sequence generators, or life would be easy.
Let the min entropy of X be the largest natural k such that P[X=x] <= 2^-k for every x. ie roughly how likely are such sequences to take their most likely value. Or how biased is our RV.
eg: uniform random over sequences of length n would have min-entropy n.
Again, the problem is we have no uniform random sequences available to us. Instead, we have biased sequences. So there is a long history of asking how can we take a biased random sequence and turn it into a random sequence. You do this with an extractor.
For example, suppose we had a sequence which produced independent bits 1 with probability p and 0 with probability q=1-p. However, we don't know p. What we could do (an extractor!) is use two sequence bits to produce one output bit, as so: sequence 0,1 has probability p x q, and sequence 1,0 has probability q x p. Map 0,1 to 0 and 1,0 to 1, and ignore 0,0 and 1,1. Now we can produce 0 with probability 1/2 and 1 with probability 1/2, ie we have uniform random from an input sequence of unknown bias (but perfectly uncorrelated, which also doesn't exist in nature. Life is hard.)
We measure the quality of extractors by calculating how bad (lower entropy) a sequence they can take and how close (statistical distance, which has a technical definition) they can output a distribution close to uniform.
Anyway, I got distracted, but in general weakly random means distributions or sequences which are biased. The less biased they are the more they're called "almost random." In general, you want to build extractors that run on sequences with unknown bias.
hth