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Exercise 5.2 in Elements of Statistical Learning

Goal is to show that an order $M$ B-Spline basis function is the density function of a convolution of $M$ uniform random variables. Although I feel the idea, I am looking for an elegant exhaustive solution. Below my attempts.

We denote $B_{i,m}(x)$ the $i$-th B-Spline basis function of order $m$ for the knot-sequence $\tau$, $m<M$, defined as

$$B_{i,1}(x)=\begin{cases} 1 & \text{if } \: \: \: \tau_{i} \leq x \leq \tau_{i+1}\\ 0 & \text{otherwise} \end{cases}$$

for $i=1,\dotsc ,K+2M-1$

$$B_{i,m}(x)= \frac{x-\tau_{i}}{\tau_{i+m-1}-\tau_{i}} B_{i,m-1}(x) + \frac{\tau_{i+m}-x}{\tau_{i+m}-\tau_{i+1}} B_{i+1,m-1}(x)$$

for $i-1, \dotsc ,K+2M-m$.

The distribution for the sum of $M$ uniform RVs is the convolution of density functions. I read the exercise as "show that the $i$-th B-Spline of order $M$ is the density function for the sum of $M$ uniform RVs".

Using the characteristic function, we can write

$$P_{ X_1 + \dotsb +X_M }(u)=\mathcal{F}^{-1}\!\!\left[ \left( \frac{i(1-e^{it}) }t \right)^M \right]\!\!(u)$$

( Elaboration on how to obtain this last result is welcome. ) After calculation,

$$P_{ X_1 + \dotsb +X_M }(u) = \frac{1}{2(n-1)!}\sum^{M}_{k=0}(-1)^{k}\binom{n}{k}(u-k)^{M-1} \mathrm{sgn}(u-k) \tag{*}$$

I propose to proceed by induction.

$$M=2: \qquad P_{X_{1}+X_{2}}(x)=\begin{cases} x & \text{if } \: \: \: 0 \leq x \leq 1\\ x-2 & \text{if } \: \: \: 1 \leq x \leq 2\\ 0 & \text{otherwise} \end{cases}$$

while

$$B_{i,2}(x)=\begin{cases} \dfrac{x-\tau_{i}}{\tau_{i+1}-\tau_{i}} & \text{if } \: \: \: \tau_{i} \leq x \leq \tau_{i+1}\\\\ \dfrac{\tau_{i+2}-x}{\tau_{i+2}-\tau_{i+1}} & \text{if } \: \: \: \tau_{i+1} \leq x \leq \tau_{i+2}\\\\ 0 & \text{otherwise} \end{cases}$$

Which is the same as $P_{X_{1}+X_{2}}$ up to a change in variable, approximately $X = \dfrac{x-\tau_{i}}{\tau_{i+1}-\tau_{i}}$. More rigorous elaboration on this change in variable or an argument to show the equivalence or both expressions is welcome.

Induction. We assume property true at order $M$.

$$B_{i,M+1}(x)= \frac{x-\tau_{i}}{\tau_{i+M}-\tau_{i}} B_{i,M}(x) + \frac{\tau_{i+M+1}-x}{\tau_{i+M+1}-\tau_{i+1}} B_{i+1,M}(x)$$

How to properly show that $B_{i,M+1}(x)$ expresses as $(*)$ where $M+1$ uniform RVs are added ?

One idea would be to express by induction the density of the sum of $M+1$ RVs in $(*)$ as a function of the density of the sum of $M$ RVs. How to write it properly ?

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