  In this article, we prove that for a convergent sequence of residually
absolutely irreducible representations of the absolute Galois group of a number
field $F$ with coefficients in a domain finite over a power series ring over a
$p$-adic integer ring, the set of places of $F$ where some of the
representations ramifies has density zero. Using this, we extend a result of
Das--Rajan to such convergent sequences. We also establish a strong
multiplicity one theorem for big Galois representations.
