Slightly technical, but not bad: If h is a function with bounded variation on [a,b] and g is continuous on [a,b], then the Riemann-stieltjes integral \int_a^b g(x) dh(x) can be defined by approximating with step functions. Then, applying integration by parts, you can define the integral \int h(x) dg(x) as h(x)g(x)|_a^b - \int g(x) dh(x). Since W is almost surely continuous, and (T-t) has bounded variation, you apply this fact omega by omega to g(t) = W(t,\omega). See pages 12 and 13 in Kuo's book Introduction to Stochastic Integration for more detail.

Unfortunately, the case I'm interested in is 1/2 < alpha < 1, so the series converges pretty slowly. Thank you for posting these links and this great review paper, I will try this.

of course you're right - I should require that c_n be monotone. I'll look into your suggestion, thank you.