# Central Limit Theorem for Functions of Uniform Random Variables on the Sphere

For each $n$, let $\boldsymbol{X}_1$ and $\boldsymbol{X}_2$ be random length-$n$ sequences, distributed uniformly on the surface of the spheres of radii $nP_1$ and $nP_2$. Let $\boldsymbol{Y}$ = $\boldsymbol{X}_1$ + $\boldsymbol{X}_2$ + $\boldsymbol{Z}$, where $\boldsymbol{Z}$ is an i.i.d. sequence with each entry being distributed as $N(0,1)$.

Let $X_{1,i}$ be the $i$-th entry of $\boldsymbol{X}\_{1}$, and similarly for $X_{2,i}$ and $Y_i$. Let $f(x_1,x_2,y)$ be a function of the form

$$f(x_1,x_2,y) = ax_1^2 + bx_2^2 + cy^2 + dx_1y + ex_2y + fx_1x_2.$$

Does a central limit theorem (CLT) hold for the quantity $S_n=\frac{1}{n}\sum_{i=1}^n f(X_{1,i},X_{2,i},Y_i)$ ? (in the sense that its distribution, after subtracting the mean and dividing by the standard deviation, converges to $N(0,1)$). If so, does a multdimensional CLT also hold for vectors of the form

$$\boldsymbol{f}(x_1,x_2,y)=\left[\begin{array}{c} f_{1}(x_{1},x_{2},y) \\\ f_{2}(x_{1},x_{2},y) \end{array}\right],$$

where $f_1$ and $f_2$ each have a similar form to $f$?

Ideally, I would also like to know the rate of convergence (i.e. a Berry-Esseen Theorem). If $\boldsymbol{X}_1$ and $\boldsymbol{X}_2$ were i.i.d. Gaussian instead of uniform on the sphere, we would certainly have a CLT with $\frac{1}{\sqrt{n}}$ convergence.

Some related results from the paper "Limit theorems for uniform distributions on spheres in high-dimensional euclidean spaces" (Stam, 1982):

• Each entry of $\boldsymbol{X}_1$ and $\boldsymbol{X}_2$ is asymptotically Gaussian (see Theorem 3), and similarly for the joint distribution of $k$ entries (where $k$ does not grow with $n$)

• If quantities $a$ to $e$ above are zero (i.e. only $f$ is non-zero) then a CLT holds (see Theorem 4)

It may also be useful to know that the uniform distribution on the sphere can be obtained by taking an i.i.d. Gaussian sequence and normalizing it (e.g. with the above definition, $\boldsymbol{Z}/\|\boldsymbol{Z}\|_2$ would be uniformly distributed on the sphere of radius 1)

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