For a symmetric stable process X (t, ω) with index α ∈ (1, 2], f ∈ Lp [0, 2 π], p ≥ α, an = frac(1, 2 π) ∫02 π e- i n t f (t) d t and An (ω) = frac(1, 2 π) ∫02 π e- i n t d X (t, ω), we establish that the random Fourier-Stieltjes (RFS) series ∑n = - ∞′ ∞ frac(an An (ω) ei n t, (i n)β) converges in the mean to the stochastic integral frac(1, 2 π) ∫02 π fβ (t - u) d X (u, ω), where fβ is the fractional integral of order β of the function f for frac(1, p) < β < 1 + frac(1, p). Further it is proved that the RFS series ∑n = - ∞′ ∞ frac(an An (ω) ei n t, (i n)β) is Abel summable to frac(1, 2 π) ∫02 π fβ (t - u) d X (u, ω). Also we define fractional derivative of the sum ∑n = - ∞∞ an An (ω) ei n t of order β for an, An (ω) as above and frac(1, p) < 1 - β < 1 + frac(1, p). We have shown that the formal fractional derivative of the series ∑n = - ∞∞ an An (ω) ei n t of order β exists in the sense of mean. © 2007 Elsevier Inc. All rights reserved.