The Principle of Contingent Causation: The Impossibility of Infinite Regress.

[GrandizerII]

Again, the BGV theorem doesn't say that any universe must have a beginning. There is a condition to keep in mind here for the BGV to apply. The universe must, on average, be expanding.

So even if this observable universe is expanding and thus has a beginning (per the theorem), there could be a "higher-level" universe out there that exists but is not expanding and which doesn't have a beginning.

And the article you share here is Vilenkin's view on what universes are possible/plausible, not just what the BGV theorem states. So he is going beyond what the theorem itself says. Vilenkin is not the sole authority on this matter though. Alan Guth, one of the two other founders of the theorem, disagrees with Vilenkin and thinks it is possible for there to be a universe that is eternal. The BGV theorem itself doesn't make a statement about who between them is correct.

Also IN THIS THREAD I've already proven him wrong. "Modern Science" proves no such thing. 

"Theoretical cosmologist Sean M. Carroll argues that the theorem only applies to classical spacetime, and may not hold under consideration of a complete theory of quantum gravity. He added that Alan Guth, one of the co-authors of the theorem, disagrees with Vilenkin and believes that the universe had no beginning."
 

Turns out Vilenkin doesn't either. See below. Craig LIED about the theorem and what it implied.
"At 49:00, (in their debate) Dr. Carroll explains why Craig's argument misrepresents the BGV theorem." ... Dr. Sean Carroll (Cal Tech) schools WLC why his shit is so wrong, in their debate.

"The Borde-Guth-Vilenkin theorem has been used by William Lane Craig to argue that the universe itself had to have a beginning.
We saw that cosmologists I contacted, including Vilenkin, Carroll, and Aguirre, all of whom have published works on the subject, agreed that no such conclusion is warranted."
... Stenger, Victor J. The Fallacy of Fine-Tuning: Why the Universe is not Designed for Us. Amherst, NY: Prometheus, 2011. p. 145

The article Nishant linked to makes it clear Vilenkin didn't consider the universe being eternal to be plausible, though. I think Stenger was just saying that Vilenkin didn't believe the BGV theorem itself warranted that the universe had to have a beginning, but it doesn't mean Vilenkin thought the universe being eternal was plausible.