Economics PhD with minimal econ (but heavy math)

I'm interested in pursuing a PhD in Statistics, but my Statistics professors have told me to also look into programs that specialize in econometrics, since it will be doing essentially what I want to do, which is time series modeling, inference, and prediction. I'm interested also in financial modeling. I'm trying to do my homework and am going to enroll in some micro and hopefully macroeconomics courses. I understand this is less than ideal preparation. But...

I have a very heavy math background. Probability theory, mathematical Statistics, calculus, linear algebra, real analysis and measure theory. I'm also planning some graduate level Statistics and stochastics. My grades are "okay" with a 3.6 gpa.

Will economics programs consider me with a minimal economics background? Does it help if I do independent study in economics?

Edit: to be more clear, I'm primarily interested in becoming an academic, though industry is a possibility. Also, the reason I have some aversion to a finance PhD, is that I am worried I will not be trained to the level of rigor that I am used to and that I like. That is, higher level math on par with real analysis, and advanced probability & Statistics. Maybe this is just a bias, as finance programs are in "business schools", but I am not interested in applying to graduate programs that will not make use of the quantitative skills I've developed. I am looking at Statistics amd applied math programs, but I know some economics programs gave a very heavy quantitative emphasis, and that is what I'm also interested in.