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We investigate weak convergence of finite-dimensional distributions of a renewal shot noise process ( Y ( t ) ) t ≥ 0 with deterministic response function h and the shots occurring at the times 0 = S 0 < S 1 < S 2 < … , where ( S n ) is a random walk with i.i.d. jumps. There has been an outbreak of recent activity around this topic. We are interested in one out of few cases which remained open: h is regularly varying at ∞ of index − 1 / 2 and the integral of h 2 is infinite. Assuming that S 1 has a moment of order r > 2 we use a strong approximation argument to show that the random fluctuations of Y ( s ) occur on the scale s = t + g ( t , u ) for u ∈ [ 0 , 1 ] , as t → ∞ , and, on the level of finite-dimensional distributions, are well approximated by the sum of a Brownian motion and a Gaussian process with independent values (the two processes being independent). The scaling function g above depends on the slowly varying factor of h . If, for instance, lim t → ∞ t 1 / 2 h ( t ) ∈ ( 0 , ∞ ) , then g ( t , u ) = t u . [ABSTRACT FROM AUTHOR]