Let $X=(X_1,\ldots,X_d)$ be uniformly-distributed on the sphere of radius $\sqrt{d}$ in $\mathbb R^d$. It is well-known that in the limit $d \to \infty$, the marginal distribution of $X_1$ converges weakly to the standard normal distribution $N(0,1)$.

Question. Let $k \ge 2$ be a fixed integer. Is it true that marginal distribution of $(X_1,\ldots,X_k)$ converges (say weakly) to $N(0,I_k)$ in the limit $d \to \infty $?

Note. From this post https://mathoverflow.net/a/359747/78539, we know that the marginal distribution of $(X_1/\sqrt{d},\ldots,X_k/\sqrt{d})$ has density $$ p_k(x_1,\ldots,x_k) \propto \begin{cases}(1-\sum_{j=1}^k x_j^2)^{(d-k)/2},&\mbox{ if }\sum_{j=1}^k x_j^2 < 1,\\ 0,&\mbox{ else.} \end{cases} $$

Let $X=(X_1,\ldots,X_d)$ be uniformly-distributed on the sphere of radius $\sqrt{d}$ in $\mathbb R^d$. It is well-known that in the limit $d \to \infty$, the marginal distribution of $X_1$ converges weakly to the standard normal distribution $N(0,1)$.

Question. Let $k \ge 2$ be a fixed integer. Is it true that marginal distribution of $(X_1,\ldots,X_k)$ converges (say weakly) to $N(0,I_k)$ in the limit $d \to \infty $?

Note. From this post https://mathoverflow.net/a/359747/78539, we know that the marginal distribution of $(X_1/\sqrt{d},\ldots,X_k/\sqrt{d})$ has density $$ p_k(x_1,\ldots,x_k) \propto \begin{cases}(1-\sum_{j=1}^k x_j^2)^{(d-k)/2},&\mbox{ if }\sum_{j=1}^k x_j^2 < 1,\\ 0,&\mbox{ else.} \end{cases} $$

Let $X=(X_1,\ldots,X_n)$ be uniformly-distributed on the sphere of radius $\sqrt{n}$ in $\mathbb R^n$. It is well-known that in the limit $n \to \infty$, the marginal distribution of $X_1$ converges weakly to the standard normal distribution $N(0,1)$.

Question. Let $k \ge 2$ be a fixed integer. Is it true that marginal distribution of $(X_1,\ldots,X_k)$ converges (say weakly) to $N(0,I_k)$ in the limit $n \to \infty $?

Let $X=(X_1,\ldots,X_d)$ be uniformly-distributed on the sphere of radius $\sqrt{d}$ in $\mathbb R^d$. It is well-known that in the limit $d \to \infty$, the marginal distribution of $X_1$ converges weakly to the standard normal distribution $N(0,1)$.

Question. Let $k \ge 2$ be a fixed integer. Is it true that marginal distribution of $(X_1,\ldots,X_k)$ converges (say weakly) to $N(0,I_k)$ in the limit $d \to \infty $?

Note. From this post https://mathoverflow.net/a/359747/78539, we know that the marginal distribution of $(X_1/\sqrt{d},\ldots,X_k/\sqrt{d})$ has density $$ p_k(x_1,\ldots,x_k) \propto \begin{cases}(1-\sum_{j=1}^k x_j^2)^{(d-k)/2},&\mbox{ if }\sum_{j=1}^k x_j^2 < 1,\\ 0,&\mbox{ else.} \end{cases} $$

Let $X=(X_1,\ldots,X_n)$ be uniformly-distributed on the sphere of radius $\sqrt{n}$ in $\mathbb R^n$. It is well known that in the limit $n \to \infty$, the marginal distribution of $X_1$ converges weakly to the $N(0,1)$. Let $k \ge 2$ be a fixed integer.

Question. Is it true that marginal distribution of $(X_1,\ldots,X_k)$ converges (say weakly) to $N(0,I_k)$ in the limit $n \to \infty $?

Let $X=(X_1,\ldots,X_n)$ be uniformly-distributed on the sphere of radius $\sqrt{n}$ in $\mathbb R^n$. It is well-known that in the limit $n \to \infty$, the marginal distribution of $X_1$ converges weakly to the standard normal distribution $N(0,1)$.

Question. Let $k \ge 2$ be a fixed integer. Is it true that marginal distribution of $(X_1,\ldots,X_k)$ converges (say weakly) to $N(0,I_k)$ in the limit $n \to \infty $?