Variance of FGN

Mandelbrot and Van Ness (p. 425)

\(TraditionalForm\`V\_H = \((Γ(H + 1/2))\)\^\(\(-1\)/2\)\)\(TraditionalForm\`{∫\_\(-∞\)\%0\(\((\((1 - s)\)\^\(H - 1/2\) - \((\(-s\))\)\^\(H - 1/2\))\)\^2\) ds + 
          1\/\(2  H\)}\)

Mandelbrot, B.B. and Van Ness, J. (1968), Fractional Brownian motions, fractional noises and applications.  SIAM Review, 10, 422-437.

VH[H_] := (((Gamma[H + 0.5]))^(-1/
              2)) (NIntegrate[(((1 - s))^(H - 1/2)) - (((-s))^(H - 1/2)), {s, -1, 0}] + 
            NIntegrate[(((1 - s))^(H - 1/2)) - (((-s))^(H - 1/2)), {s, -(10^10), -1}] + 
            NIntegrate[(((1 - s))^(H - 1/2)) - (((-s))^(H - 1/2)), {s, -(10^15), -(10^10)}] + (((2  H))^(-1)))

H = 0.7;

Integrate[(((1 - s))^(H - 1/2)) - (((-s))^(H - 1/2)), {s, -∞, 0}]

VH[H_] := (((Gamma[H + 0.5]))^(-1/
              2)) (NIntegrate[(((1 - s))^(H - 1/2)) - (((-s))^(H - 1/2)) // Log, {s, -∞, 
                0}] + (((2  H))^(-1)))

VH[0.7]

?NIntegrate

H = 0.7; 
    Plot[((((1 - s))^(H - 1/2)) - (((-s))^(H - 1/2))) // 
        Log, {s, -(10^3), 0}]