# Average number of pieces of a random piecewise-linear function

Let $$I$$ be a (nonempty) compact interval in $$\mathbb R$$ and $$a_1,b_1,\ldots,a_L,b_L \in \mathbb R$$. Let $$\varphi$$ be a piecewise function with $$T \ge 2$$ pieces(for example $$T=2$$ for the choice $$\varphi(t) = \max(0, t)$$ for $$t\in \mathbb R$$). Consider the piecewise-linear function $$f:I \rightarrow \mathbb R$$ defined by $$f(x)=z_L$$, where $$\begin{split} z_0&=x,\\ z_l &= \varphi(a_l z_{l-1} + b_l),\;\forall l=1,2,\ldots,L. \end{split}$$

Let $$a_1,\ldots,a_L$$ be drawn i.i.d from some $$\mathcal N(0,\sigma_a^2)$$ and $$b_1,\ldots,b_L$$ be drawn i.i.d from some $$\mathcal N(0,\sigma_b^2)$$.

I'm interested in the distribution of the number of linear pieces of $$f$$.

# Questions

• What is a good estimate of the average number of linear pieces of $$\mathcal f$$ as a function of $$T$$, $$L$$, $$\sigma_a^2$$, $$\sigma_b^2$$, and the length of $$I$$ ?
• What kinds of techniques can be used for such problems ?
• Doesn't it just depend on the signs of a and b whether the number of linear pieces doubles or not? – Dirk Jun 13 at 17:41
• @Dirk Please could be more explicit on your remark ? :) – dohmatob Jun 13 at 17:54
• The next $z_l$ is always zero on some intervall. If the next $b$ is negative while $a$ is positive you'll have one more kink. (However, it occurs to me that the number of kinks may even double every layer, so probably I should stop thinking without a piece of paper.) – Dirk Jun 13 at 18:32
• @Dirk Thanks for the input. Such arguments can be used to obtain upper bounds on the max number of pieces, but I don't think they can go beyond this to give insight into means, etc. No ? I have found a result of type Kac-Rice, which from afar at least, should give me something. Indeed, see Theorem 4.1.1 of "Random Fields and their Geometry" pdfs.semanticscholar.org/cb22/…. I will write up what I arrive at, asap. – dohmatob Jun 24 at 10:41