A hypothesis is when someone has an idea about what is happening, but no actual proof.
In my hypothesis, the traffic comes from another domain, which redirects to the one that is for sale. That allows the vendor to show the purchaser completely authentic traffic and revenue data for the domain which is for sale.
When the sale is complete, the vendor can redirect the traffic-generating domain to yet another site, at which point traffic to the purchased site will dry up, and the new site will be making money from day 1.
A check for backlinks will not usually show or include the redirected domain, but sometimes an Alexa search will show it as an associated domain.
If the new site has only been running for a month, an Alexa search for a domain shown as associated might reveal a previous site, ie. the last "mugged purchaser".
In reality, a good link in from a very powerful and respected site might produce the same effect of kick-starting the site from day one. From what you've said, I don't think that is how this site is being "jump-started".
A week would not be enough to tell. The only way to be certain would be to avoid completing until you had your money back from the site, and that timescale is not practical - the vendor might as well keep the site and get his money from that revenue.