We suppose that $\lambda > 1$ and we use the random row vector $W_n = \lambda^{-n}\left(X_n - \|X_n\|_1 \pi\right)$.
Show that the event $\left\{\lim_{n \rightarrow +\infty} W_n = 0_{\mathbb{R}^d}\right\}$ is almost surely true. (One may begin by computing the probability of the event
$$\left\{ \forall m \geqslant 0, \exists k \geqslant m \mid \|W_k\|_2 \geqslant \varepsilon \right\}$$
for all $\varepsilon > 0$.)