We suppose in the rest of this part that $\lambda > 1$ and we introduce the random row vector $$W_n = \lambda^{-n}\left(X_n - \|X_n\|_1 \pi\right).$$ (a) Show that the series $\displaystyle\sum_{n \geqslant 1} \left(\sum_{k=0}^{n-1} \lambda^{-k} \gamma^{2n-2k}\right)$ converges. (b) Let $w \in (\mathbb{R}_{+})^d$ and let $e_0 = (1,\ldots,1)$. Show that $$\left\langle w - \|w\|_1 \pi, \pi \right\rangle = \left\langle w, \pi - \langle \pi, \pi \rangle e_0 \right\rangle$$ and that the vector $\pi - \langle \pi, \pi \rangle e_0$ is orthogonal to $\pi$. (c) Show that the series $\displaystyle\sum_{n \geqslant 0} \mathbb{E}\left(\|W_n\|_2^2\right)$ is convergent. Deduce that the sequence $\left(\mathbb{E}\left(\|W_n\|_2^2\right)\right)_{n \geqslant 0}$ tends to $0$. (One may for example decompose $X_n$ in a well-chosen orthonormal basis of $\mathbb{R}^d$.) (d) Show that for all $\varepsilon > 0$, $$\lim_{n \rightarrow \infty} \mathbb{P}\left(\|W_n\|_2 \geqslant \varepsilon\right) = 0.$$
We suppose in the rest of this part that $\lambda > 1$ and we introduce the random row vector
$$W_n = \lambda^{-n}\left(X_n - \|X_n\|_1 \pi\right).$$
(a) Show that the series $\displaystyle\sum_{n \geqslant 1} \left(\sum_{k=0}^{n-1} \lambda^{-k} \gamma^{2n-2k}\right)$ converges.
(b) Let $w \in (\mathbb{R}_{+})^d$ and let $e_0 = (1,\ldots,1)$. Show that
$$\left\langle w - \|w\|_1 \pi, \pi \right\rangle = \left\langle w, \pi - \langle \pi, \pi \rangle e_0 \right\rangle$$
and that the vector $\pi - \langle \pi, \pi \rangle e_0$ is orthogonal to $\pi$.
(c) Show that the series $\displaystyle\sum_{n \geqslant 0} \mathbb{E}\left(\|W_n\|_2^2\right)$ is convergent. Deduce that the sequence $\left(\mathbb{E}\left(\|W_n\|_2^2\right)\right)_{n \geqslant 0}$ tends to $0$. (One may for example decompose $X_n$ in a well-chosen orthonormal basis of $\mathbb{R}^d$.)
(d) Show that for all $\varepsilon > 0$,
$$\lim_{n \rightarrow \infty} \mathbb{P}\left(\|W_n\|_2 \geqslant \varepsilon\right) = 0.$$