Upcrossing number of an interval

For a real sequence $x=(x_n{)}_{n\ge 0}$, reals $a<b$, and $N\ge 0$, define $U_N[a,b](x)=\max \{k:\mathrm{\exists }0\le s_1<t_1<\cdots <s_k<t_k\le N,x_{s_i}\le a,x_{t_i}\ge b\},$ where the empty tuple makes $0$ admissible, and put $U_{\mathrm{\infty }}[a,b](x)={\sup }_{N\ge 0}{U}_N[a,b](x)\mathrm{.}$ For a real process $X$, these definitions are applied pathwise.

For fixed $N$ there are finitely many candidate tuples. For each $k$, the event $\{U_N\ge k\}$ is a finite union of finite intersections of events $\{X_s\le a\}$ and $\{X_t\ge b\}$, hence is measurable. Thus the integer-valued $U_N$ is measurable. Since $U_N\uparrow U_{\mathrm{\infty }}$, measurability of the countable pointwise supremum follows from Sequential suprema, infima, limsup, liminf, and pointwise limits of measurable functions are measurable. No optimizing tuple is selected, so the definition is choice-free.

For example, take $a=0$, $b=1$, $N=6$, and $(x_0,\dots ,x_6)=(-1\mathrm{/}2,-1\mathrm{/}4,3\mathrm{/}2,3\mathrm{/}4,-1\mathrm{/}2,1\mathrm{/}2,3\mathrm{/}2)$. The marked pairs $(s_1,t_1)=(1,2)$ and $(s_2,t_2)=(4,6)$ are admissible. Here $U_6[0,1](x)=2$: these pairs give two upcrossings, and only the indices $2$ and $6$ can be terminal times at or above $b$. The maximizing tuple is not unique, since $s_1=0$ also works. Segments only guide the eye between the discrete observations; the definition counts index pairs.