You're being serious? I don't mean to be overly hostile, but the implications would be world-shattering and the "mundane" explanation is quite sufficient. Also lottery-winning psychics wouldn't get caught.
Btw try this: close your nose and mouth and think about something, you'll realize that you're kind of talking/transferring your thoughts in the air.
Here is a good link "Scientists Transmit Thoughts from one Brain to Another": http://abcnews.go.com/Technology/scientists-transmit-thought...
For the Powerball, two ways to play: pick your own numbers or randomly generate them. Would a winning psychic choose one over the other more often? An ability to sense and know the future means they would have to either know the winning numbers and pick them, know the machine that produces the numbers (but not necessarily the numbers themselves), or somehow manipulate chance or psychics so that the lottery machine produces the psychic's number during the draw.
Psychic is kind of loosely defined right now by your question so for convenience let's rule out the last one since that's more telekinetic/magic than psychic.
With these conditions, we should see some pattern from the winner: they would either always pick their own and get the winning numbers or they would always happen to go to the right machine at the right time.
Both should show pretty consistent behavior on the part of a psychic, as we can see that the bar for big winnings is pretty low ($600) and multiple statistically improbable winnings would likely be investigated. (per the atriclr, only 10 states do not investigate frequent winners).
So to summarize so far:
Frequent winners very likely investigated unless they are in the 10 states
Psychic will likely pick their own numbers
Psychic will go to a specific machine and have erratic pattern of purchase
If psychic are playing the lottery, my assumptions lead me to think that it makes no sense to frequently win, instead banking on a single explainable big win. This of course would be only if the psychic did not want to be found out as a psychic, which for the purpose of the lottery I think would be a valid want as I believe that there are rules about additional knowledge, etc meant to catch people defrauding the lottery. Future-visions would most definitely qualify, so probably would require that the psychic forfeit their winnings.
The mundane explanations do explain a lot, both from the investigations and admissions from the frequent winners. To me anyways, the psychics, if they exist, would more likely be the one time big winners, not the frequent ones.
In that case, they might win more often, but not every time. Given that so many things in nature follow a power curve or a bell curve, I think your presumed psychic abilities, we can assume, follow the same curves, and if most of your psychics then have "some ability" to predict these numbers, but not a certainty, then I think they are more likely to be frequent smaller winners.
For example, if you have an ability that let's you see a 4/8 of the future numbers, with 33% certainty, then one time out of three, you will win 4/8 of the numbers, and sometimes more than 4/8 if you're lucky.
TL;DR -- your presumed psychic abilities probably exist on a spectrum ( bell / power curve ) and likely most of your psychics do not have "perfect" future information.
PS - I have a feeling that psychics would prefer to play the stock market, than lottery. I just sense their expected payoff is higher in that case, probably because there are a lot more factors that affect a stock price, so even partial information across a lot of these seems to provide a better advantage, than partial information on 1 single list of numbers.
I still think you'd see a pretty clear pattern emerge from winners however, at least should the individual grasp the impact of their ability, even if the picture is incomplete. I also agree stocks or sports betting would be a better choice as partial information seems more valuable in those cases.
But I still feel confident you'd see the same general truths from psychics with partial clarity; still picking numbers and never using random and still an erratic purchase pattern as they chase locations and time. I think the former would likely show more purchases with the same constant numbers from the futuresight and then a spread of other numbers; such behavior may stand out even more as there would be a history of them always getting a small subset right and guessing the rest.
Psychic powers would impede apprehension by authorities.
See The Golden Man by Philip K. Dick.
And you'd get some weird paradoxes too, or weird implications for information theory anyway. See also Harry Potter and the Methods of Rationality.
The other option is that these frequent winners are surrogates for others who can or would not hand in their lottery ticket on their own.
What follows the razor?
While I doubt it's psychics, if I were to assert that as fact I'd have to disprove that hypothesis.