Sequential likelihood-ratio tests

By [F8] with $g=1$, $\int \lambda d{Q}_0=1$, so $N=\{\lambda =\mathrm{\infty }\}$ is $Q_0$-null and hence $Q_1$-null. Set $\lambda =0$ on $N$; this preserves its density class and gives a finite nonnegative representative. Now fix $t\ge 1$ and $j\in \{0,1\}$. The vector $Y_t=(X_1,\dots ,X_t)$ is measurable into the finite product sigma-algebra because preimages of all-coordinate rectangles are finite intersections of coordinate events, and those rectangles generate that sigma-algebra by [F3]. Its law ${\mu }_{j,t}$ is a probability measure by [F5]. On every rectangle $R={\prod }_{i=1}^t{B}_i$, [F1] and [F2] give ${\mu }_{j,t}(R)={\prod }_i{Q}_j(B_i)$. Repeated application of [F4] constructs $Q_j^{\otimes t}$ with those same rectangle masses. The all-coordinate rectangles form a pi-system by [F3] and [F6], generate the product sigma-algebra, and have a finite exhaustion consisting of the whole space, whose mass is one for both measures. Hence [F6] gives ${\mu }_{j,t}=Q_j^{\otimes t}$.

Define ${\mathrm{\Lambda }}_t(x_1,\dots ,x_t)=(\lambda (x_1),\dots ,\lambda (x_t))$. Preimages of product rectangles are finite intersections of measurable coordinate preimages, so [F3] makes ${\mathrm{\Lambda }}_t$ measurable. The finite multiplication map $m_t:{\mathbb{R}}^t\to \mathbb{R}$, $m_t(y)={\prod }_{i=1}^t{y}_i$, is continuous: the two-coordinate multiplication map is continuous by $\mathrm{\mid }ab-a^{\mathrm{\prime }}{b}^{\mathrm{\prime }}\mathrm{\mid }\le \mathrm{\mid }a\mathrm{\mid }\mathrm{\mid }b-b^{\mathrm{\prime }}\mathrm{\mid }+\mathrm{\mid }{b}^{\mathrm{\prime }}\mathrm{\mid }\mathrm{\mid }a-a^{\mathrm{\prime }}\mathrm{\mid }$, and finite products follow by induction. Thus [F12] makes $m_t$ Borel and [F13] makes $h_t=m_t\circ {\mathrm{\Lambda }}_t$ measurable. By [F9], ${\nu }_t(B):={\int }_B{h}_{t}d{Q}_0^{\otimes t}$ is a measure. For each rectangle $R={\prod }_i{B}_i$, repeated Tonelli integration, justified by induction on the finite number $t$ of factors, gives ${\nu }_t(R)={\prod }_{i=1}^t{\int }_{B_i}\lambda d{Q}_0={\prod }_{i=1}^t{Q}_1(B_i)$, where each factor uses [F8]. This is the rectangle formula for $Q_1^{\otimes t}$ by [F4]. The two measures agree on the generating pi-system and both have total mass one, so [F6] gives ${\nu }_t=Q_1^{\otimes t}$.

If $A\in {\mathcal{F}}_t=\sigma (Y_t)$, then $A=Y_t^{-1}(B)$ for some product-measurable $B$: the family of such preimages is a sigma-algebra and is exactly the sigma-algebra generated by $Y_t$. The definition of law [F5] gives the integral change-of-variables identity first for indicators and nonnegative simple functions; increasing simple approximation and [F11] extend it to $h_t{\mathbf{1}}_B$. Thus, using steps 1.1 and 2.1, $P_1(A)=Q_1^{\otimes t}(B)={\int }_B{h}_{t}d{Q}_0^{\otimes t}={\int }_A{h}_t(Y_t)d{P}_0$. This is the Radon–Nikodym characterization of $L_t=h_t(Y_t)$.

Each $L_t$ is nonnegative by [F3] and ${\mathcal{F}}_t$-measurable by [F2]. Applying [F4] with $g=1$ gives ${\mathbb{E}}_{P_0}{L}_t=P_1(\mathrm{\Omega })=1$ for every $t\ge 0$, so $\{L_t=\mathrm{\infty }\}$ is $P_0$-null. Define $M_t=L_t$ where $L_t<\mathrm{\infty }$ and $M_t=0$ otherwise. This modification on an ${\mathcal{F}}_t$-measurable null set preserves the density class. Thus $M$ is a real-valued nonnegative adapted process with ${\mathbb{E}}_{P_0}{M}_t=1$, and each $M_t$ has the same density event integrals as $L_t$.

For $n\ge 1$, $\{\tau \le n\}={\bigcup }_{t=1}^n(\{L_t\ge 1\mathrm{/}\alpha \}\cup \{L_t\le \beta \})\in {\mathcal{F}}_n$, because each $L_t$ is ${\mathcal{F}}_t$-measurable and the filtration is increasing; $\{\tau \le 0\}=\mathrm{\varnothing }$. Hence [F12] makes $\tau$ a stopping time.

Fix $t\ge 1$ and $A\in {\mathcal{F}}_{t-1}$. Since $A\in {\mathcal{F}}_t$, [F4] applied to the density representatives $M_t$ and $M_{t-1}$ gives ${\mathbb{E}}_{P_0}[{\mathbf{1}}_A{M}_t]=P_1(A)={\mathbb{E}}_{P_0}[{\mathbf{1}}_A{M}_{t-1}]$. By [F7], choose a version $Y_t$ of ${\mathbb{E}}_{P_0}[M_t\mid {\mathcal{F}}_{t-1}]$. Its defining event-integral identities [F6] show that $Y_t$ and $M_{t-1}$ are versions of that conditional expectation; [F8] makes them equal almost surely. Thus $M$ is a nonnegative $P_0$-martingale by [F9].

For each $t\ge 0$, step 1.1 and equivalence of the restrictions give $P_0(L_t=\mathrm{\infty })=P_1(L_t=\mathrm{\infty })=0$. Moreover, [F4] gives $P_1(L_t=0)={\int }_{\{L_t=0\}}{L}_{t}d{P}_0=0$, so equivalence gives $P_0(L_t=0)=0$. Thus $N_t=\{L_t=0\}\cup \{L_t=\mathrm{\infty }\}\in {\mathcal{F}}_t$ is null under both laws. Define the real nonnegative $R_t=1\mathrm{/}{L}_t$ where $0<L_t<\mathrm{\infty }$ and set it to $0$ on $N_t$. It is ${\mathcal{F}}_t$-measurable, and $R_t{M}_t=1$ off $N_t$. Applying [F4] with density $M_t$ and $g={\mathbf{1}}_A{R}_t$ for $A\in {\mathcal{F}}_t$ gives ${\mathbb{E}}_{P_1}[{\mathbf{1}}_A{R}_t]={\mathbb{E}}_{P_0}[{\mathbf{1}}_A{R}_t{M}_t]=P_0(A)$, so $R_t$ is integrable with mean one.

For $t\ge 1$ and $A\in {\mathcal{F}}_{t-1}$, step 2.2 at times $t$ and $t-1$ gives ${\mathbb{E}}_{P_1}[{\mathbf{1}}_A{R}_t]=P_0(A)={\mathbb{E}}_{P_1}[{\mathbf{1}}_A{R}_{t-1}]$. Existence [F7], the defining conditional-expectation identity [F6], and uniqueness [F8] identify ${\mathbb{E}}_{P_1}[R_t\mid {\mathcal{F}}_{t-1}]=R_{t-1}$ almost surely. Thus $R$ is a nonnegative $P_1$-martingale by [F9].

By steps 1.1 and 2.1, $M$ is a real nonnegative martingale under $P_0$ with ${\mathbb{E}}_{P_0}{M}_0=1$; by [F10] it is a nonnegative test supermartingale for the singleton null $\{P_0\}$. By steps 2.2 and 3.1, $R$ is a real nonnegative test supermartingale for $\{P_1\}$. Applying [F11] gives $P_0({\sup }_{t\ge 0}{M}_t\ge 1\mathrm{/}\alpha )\le \alpha$ and $P_1({\sup }_{t\ge 0}{R}_t\ge 1\mathrm{/}\beta )\le \beta$.

The measurable set $N={\bigcup }_{t\ge 0}{N}_t$ is null under both laws by countable subadditivity. Outside $N$, simultaneously for every $t$, $M_t=L_t$ and $R_t=1\mathrm{/}{L}_t$. Hence the terminal $H_1$ event is contained in $N\cup \{{\sup }_{t\ge 0}{M}_t\ge 1\mathrm{/}\alpha \}$, and the terminal $H_0$ event is contained in $N\cup \{{\sup }_{t\ge 0}{R}_t\ge 1\mathrm{/}\beta \}$. These terminal events are measurable countable unions of $\{\tau =t\}$ intersected with the corresponding boundary events. Since $P_0(N)=P_1(N)=0$, step 4.1 proves both terminal-error inequalities.

Let $r$ be any nonnegative RN representative of $Q_1$ with respect to $Q_0$. Since $\int rd{Q}_0=1$, the set where $r=\mathrm{\infty }$ is $Q_0$-null and hence $Q_1$-null. On $A=\{r=0\}$ the change-of-measure identity [F5] gives $Q_1(A)={\int }_{A}rd{Q}_0=0$; equivalence $Q_0\ll Q_1$ then makes $A$ $Q_0$-null too. Replacing $r$ by $1$ on these common null sets yields a finite strictly positive measurable representative $\lambda$ of the same density class. Its range lies in $(0,\mathrm{\infty })$, so it is measurable into that subspace with its Borel sigma-algebra $\{B\cap (0,\mathrm{\infty }):B\in \mathcal{B}(\mathbb{R})\}$. Although the logarithm is continuous on this subspace, [F8] only covers Euclidean domains. For every open $U\subseteq \mathbb{R}$, ${\log }^{-1}(U)$ is open in $(0,\mathrm{\infty })$ and therefore Borel there; since open sets generate $\mathcal{B}(\mathbb{R})$, $\log :(0,\mathrm{\infty })\to \mathbb{R}$ is Borel measurable. Thus [F9] makes $Z=\log \circ \lambda$ real-valued and measurable.

For each $j$, the variables $Z_i:=Z(X_i)$ are IID under $P_j$: measurability gives the same pushforward law for all coordinates, and [F2] preserves their independence. The change-of-variables identity [F10] gives ${\mathbb{E}}_{P_j}\mathrm{\mid }{Z}_i\mathrm{\mid }={\mathbb{E}}_{Q_j}\mathrm{\mid }Z\mathrm{\mid }<\mathrm{\infty }$ and ${\mathbb{E}}_{P_j}{Z}_i={\mu }_j:={\mathbb{E}}_{Q_j}Z$. Thus [F3] gives $t^{-1}{\sum }_{i=1}^t{Z}_i\to {\mu }_j$ almost surely under $P_j$.

Put $G_t={\prod }_{i=1}^t\lambda (X_i)$. By [F6], $G_t$ is a version of the finite-time density, so the algorithm's chosen representative $L_t$ equals $G_t$ $P_0$-almost surely by uniqueness of Radon–Nikodym derivatives. The representative $\lambda$ is finite and strictly positive $Q_0$-almost surely; equivalence of $Q_0,Q_1$ gives the same under $Q_1$. Thus $G_t$ is finite and strictly positive $P_0$-almost surely, and for $A\in {\mathcal{F}}_t$, $P_1(A)={\int }_A{G}_{t}d{P}_0$ implies $P_1(A)=0$ exactly when $P_0(A)=0$. The finite-time restrictions are therefore equivalent, so $L_t=G_t$ also $P_1$-almost surely. Intersecting these probability-one events over countably many $t$ gives one event under each law on which the equalities hold for every $t$. Positivity of $\lambda$ there yields $\log L_t={\sum }_{i=1}^t{Z}_i$.

Under $P_0$, steps 1.2 and 2.1 give $\log L_t\mathrm{/}t\to {\mu }_0<0$, hence $\log L_t\to -\mathrm{\infty }$. Since $\log \beta$ is finite, some finite $t$ satisfies $L_t\le \beta$, so the lower boundary has been crossed and the algorithm has $\tau <\mathrm{\infty }$.

Under $P_1$, steps 1.2 and 2.1 give $\log L_t\mathrm{/}t\to {\mu }_1>0$, hence $\log L_t\to +\mathrm{\infty }$. Since $\log (1\mathrm{/}\alpha )$ is finite, some finite $t$ satisfies $L_t\ge 1\mathrm{/}\alpha$, so the upper boundary has been crossed and $\tau <\mathrm{\infty }$. This proves almost-sure termination under both laws, without a claim about ${\mathbb{E}}_{P_j}\tau$.