Evaluate the limit of $\int\dots\int_{x_1^2+x_2^2+\dots+x_n^2 \leq r^2} f(x_1) f(x_2)\dots f(x_n) \, dx_1 \, dx_2 \dots \, dx_n$ when $n\to\infty$

Question

The following question is taken from here problem $2.5:$

Exercise $2.5:$ Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous function such that $f(x) \geq 0$ for all $x$ and $$\int_{-\infty}^\infty f(x) \, dx = 1.$$ For $r\geq0,$ let $$I_n(x) = \int\dots\int_{x_1^2+x_2^2+\dots+x_n^2 \leq r^2} f(x_1) f(x_2)\dots f(x_n) \, dx_1 \, dx_2 \dots \, dx_n.$$ Find $\lim_{n\to\infty}I_n(x)$ for a fixed $r.$

The answer given is $0.$ I think I need to use $n$-dimensional spherical substitution to reduce the problem over an $n$-dimension sphere.

However, I have no idea how to use it. Any hint would be appreciated.

Answer 1

$f$ satisfies definition for probability density function. So the problem can be reformulated as given $X_i \overset{iid}{\sim} X$, \begin{align*} I_n &= E[I(X_1^2+\dotsb, X_n^2 \le r^2) ]\\ &= P(X_1^2+\dotsb, X_n^2 \le r^2)\\ &= P(nX^2 \le r^2)\\ &= P(X^2 \le \frac{r^2}{n})\\ \lim_{n\to\infty} I_n &= P(X^2 \le 0)\\ &= 0 \end{align*} The last few lines are a bit hand wavy, but I hope this helps.

Answer 2

Hint: More generally, suppose $f$ is continuous and nonnegative on $[-1,1],$ with $f\le M$ there. Then $I_n$ is bounded above by $M^n$ times the $n$-dimensional volume of $B(0,r).$ That volume is $r^n$ times the $n$-dimensional volume of $B(0,1).$ So everything depends on finding a good estimate of $V_n(B(0,1)).$