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Products of random matrix products of $\mathrm{SL}(2,\mathbb{R})$,corresponding to transfer matrices for the one-dimensional Schr\"odingerequation with a random potential $V$, are studied. I consider both the casewhere the potential has a finite second moment $\langle V^2\rangle<\infty$ andthe case where its distribution presents a power law tail$p(V)\sim|V|^{-1-\alpha}$ for $0<\alpha<2$. I study the generalized Lyapunovexponent of the random matrix product (i.e. the cumulant generating function ofthe logarithm of the wave function). In the high energy/weak disorder limit, itis shown to be given by a universal formula controlled by a unique scale(single parameter scaling). For $\langle V^2\rangle<\infty$, one recoversGaussian fluctuations with the variance equal to the mean value:$\gamma_2\simeq\gamma_1$. For $\langle V^2\rangle=\infty$, one finds$\gamma_2\simeq(2/\alpha)\,\gamma_1$ and non Gaussian large deviations, relatedto the universal limiting behaviour of the conductance distribution $W(g)\simg^{-1+\alpha/2}$ for $g\to0$.

Generalized Lyapunov exponent of random matrices and universality classes for SPS in 1D Anderson localisation