I don't currently have the time to look at everything or be very careful, but I think that you're on the right track.

The way I interpret condition (b) is that is the average over the , and you want the average to go to zero. The first thing I considered is that if all of your are positive, the only way to do this is to make all of the terms go to zero. But I don't think that this is quite correct. For example, you can take the sequence 1, 1/3, 1, 1/5, 1/6, 1/7, 1,... (i.e. it is the harmonic series with every power of 2 replaced with a 1). It seems to me that the averages will still go to zero even though the sequence is not convergent. (I haven't checked this, but I think that it seems reasonable, especially if the 1's are very sparse.)

How strict of a condition are you looking for? It seems that there are many possibilities.