Let $X$ be a smooth projective variety over a field $k$.

It is well-known that the sum of an ample divisor and an effective divisor is big (in fact this can be taken to be the definition of a big divisor). I'm looking for a weakening of this.

Is the sum of a big divisor and a pseudo-effective divisor itself a big divisor?

Recall that a divisor is called *pseudo-effective* if it lies in the closure of the cone of effective divisors.