Asymptotically Optimal Quickest Change Detection in Multistream Data—Part 1: General Stochastic Models

Abstract

Assume that there are multiple data streams (channels, sensors) and in each stream the process of interest produces generally dependent and non-identically distributed observations. When the process is in a normal mode (in-control), the (pre-change) distribution is known, but when the process becomes abnormal there is a parametric uncertainty, i.e., the post-change (out-of-control) distribution is known only partially up to a parameter. Both the change point and the post-change parameter are unknown. Moreover, the change affects an unknown subset of streams, so that the number of affected streams and their location are unknown in advance. A good changepoint detection procedure should detect the change as soon as possible after its occurrence while controlling for a risk of false alarms. We consider a Bayesian setup with a given prior distribution of the change point and propose two sequential mixture-based change detection rules, one mixes a Shiryaev-type statistic over both the unknown subset of affected streams and the unknown post-change parameter and another mixes a Shiryaev–Roberts-type statistic. These rules generalize the mixture detection procedures studied by Tartakovsky (IEEE Trans Inf Theory 65(3):1413–1429, 2019) in a single-stream case. We provide sufficient conditions under which the proposed multistream change detection procedures are first-order asymptotically optimal with respect to moments of the delay to detection as the probability of false alarm approaches zero.

Appendix: An Auxiliary Lemma and Proofs

The following lemma is extensively used for obtaining upper bounds for the moments of the detection delay, which are needed for proving asymptotic optimality properties of the introduced detection procedures. In this lemma, P is a generic probability measure and E is a corresponding expectation.

Lemma A.1

Letτ(τ = 0,1,…)be a non-negative integer-valued random variable and let N (N ≥ 1)be an integer number. Then, for anyr ≥ 1,

$$ \mathsf{E}[\tau]^{r} \le N^{r} + r 2^{r-1} \sum\limits_{n=N}^{\infty} n^{r-1} \mathsf{P}({\tau > n}). $$

(A.1)

Proof

$$ \begin{array}{@{}rcl@{}} \mathsf{E}[{\tau}]^{r} = {\int}_{0}^{\infty} r t^{r-1} \mathsf{P}({\tau > t}) \mathrm{d} t \le N^{r} + \sum\limits_{n=0}^{\infty} {\int}_{N+n}^{N+n+1} r t^{r-1} \mathsf{P} ({\tau > t}) \mathrm{d} t \le N^{r} + \sum\limits_{n=0}^{\infty} {\int}_{N+n}^{N+n+1} r t^{r-1} \mathsf{P}({\tau > N+n}) \mathrm{d} t = N^{r} + \sum\limits_{n=0}^{\infty} [(N+n+1)^{r}- (N+n)^{r} ] \mathsf{P}({\tau > N+n}) = N^{r} + \sum\limits_{n=N}^{\infty} [(n+1)^{r}-n^{r}] \mathsf{P}({\tau > n}) \le N^{r} +\sum\limits_{n=N}^{\infty} r (n+1)^{r-1} \mathsf{P}({\tau > n}) \le N^{r} + r 2^{r-1} \sum\limits_{n=N}^{\infty} n^{r-1} \mathsf{P}({\tau > n}). \end{array} $$

□

1.1 Proof of Proposition 1

To prove asymptotic approximations (5.3) and (5.4) note first that by Eq. 5.1 the detection procedure ${T_{A}^{W}}$ belongs to class $\mathbb {C}(1/(A+1))$, so replacing α by 1/(A + 1) in the asymptotic lower bounds (4.3) and (4.4), we obtain that under the right-tail condition C₁ the following asymptotic lower bounds hold for all r > 0, $\mathcal {B}\in \mathcal {P}$, and 𝜃 ∈Θ:

$$ \begin{array}{@{}rcl@{}} \liminf_{A\to\infty} \frac{\mathcal{R}_{k, \mathcal{B},\theta}^{r}({T_{A}^{W}})}{(\log A)^{r}} & \ge& \frac{1}{(I_{\mathcal{B},\theta} +\mu)^{r}} , \quad k \in \mathbb{Z}_{+}, \end{array} $$

(A.2)

$$ \begin{array}{@{}rcl@{}} \liminf_{A\to\infty} \frac{\overline{\mathcal{R}}_{\mathcal{B},\theta}^{r}({T_{A}^{W}})}{(\log A)^{r}} & \ge& \frac{1}{(I_{\mathcal{B},\theta} +\mu)^{r}} . \end{array} $$

(A.3)

Therefore, to prove the assertions of the proposition it suffices to show that, under the left-tail condition C₂, for all 0 < m ≤ r, $\mathcal {B}\in \mathcal {P}$, and 𝜃 ∈Θ

$$ \begin{array}{@{}rcl@{}} \limsup_{A\to\infty} \frac{\mathcal{R}_{k, \mathcal{B},\theta}^{m}({T_{A}^{W}})}{(\log A)^{m}} & \le \frac{1}{(I_{\mathcal{B},\theta} +\mu)^{m}} , \quad k \in \mathbb{Z}_{+}, \end{array} $$

(A.4)

$$ \begin{array}{@{}rcl@{}} \limsup_{A\to\infty} \frac{\overline{\mathcal{R}}_{\mathcal{B},\theta}^{m}({T_{A}^{W}})}{(\log A)^{m}} & \le \frac{1}{(I_{\mathcal{B},\theta} +\mu)^{m}} . \end{array} $$

(A.5)

The proof of part (i). Let $\pi ^{A}=\{{\pi _{k}^{A}}\}$, ${\pi _{k}^{A}}=\pi _{k}^{\alpha }$ for α = α_A = 1/(1 + A), and define

$$ N_{A}=N_{A}(\varepsilon,\mathcal{B},\theta)=1+\left\lfloor \frac{\log (A/{\pi_{k}^{A}})}{I_{\mathcal{B},\theta}+\mu-\varepsilon} \right\rfloor . $$

Obviously, for any n ≥ 1,

$$ \begin{array}{@{}rcl@{}} \log S_{\textbf{p},W}^{\pi}(k+n) &\ge& \log {\Lambda}_{\textbf{p},W}(k,k+n) + \log \pi_{{k}^{A}} -\log {\Pi}_{k-1+n}^{A}\\ &\ge& \inf_{\vartheta \in {\Gamma}_{\delta,\theta}} \lambda_{\mathcal{B},\vartheta}(k, k+n) +\log W({\Gamma}_{\delta,\theta}) +\log p_{\mathcal{B}} +\log {\pi_{k}^{A}} -\log {\Pi}_{k-1+n}^{A} , \end{array} $$

where Γ_δ,𝜃 = {𝜗 ∈Θ : |𝜗 − 𝜃| < δ}, so that for any $\mathcal {B}\in \mathcal {P}$, 𝜃 ∈Θ, $k\in \mathbb {Z}_{+}$

$$ \begin{array}{@{}rcl@{}} \mathsf{P}_{k, \mathcal{B},\theta}\left( {{T_{A}^{W}}-k >n}\right) &\le& \mathsf{P}_{k, \mathcal{B},\theta}\left\{\frac{1}{n} \log S_{\textbf{p},W}^{\pi}(k+n) < \frac{1}{n} \log A \right\} \\ &\le& \mathsf{P}_{k, \mathcal{B},\theta}\left\{\frac{1}{n} \inf_{\vartheta \in {\Gamma}_{\delta,\theta}} \lambda_{\mathcal{B},\vartheta}(k, k+n) < \frac{1}{n} \log \left( {\frac{A}{{\pi_{k}^{A}}}}\right) \right.\\ &&\qquad\quad\left. +\frac{1}{n} \left[|\log{\Pi}_{k-1+n}^{A}| + \log W({\Gamma}_{\delta,\theta}) +\log p_{\mathcal{B}}\right]\vphantom{\left( {\frac{A}{{\pi_{k}^{A}}}}\right)}\right\}. \end{array} $$

It is easy to see that for n ≥ N_A the last probability does not exceed the probability

$$ \begin{array}{@{}rcl@{}} &&\mathsf{P}_{k, \mathcal{B},\theta}\left\{\frac{1}{n} \inf_{\vartheta \in {\Gamma}_{\delta,\theta}} \lambda_{\mathcal{B},\vartheta}(k, k+n) < I_{\mathcal{B},\theta} +\mu - \varepsilon \quad\qquad +\frac{1}{n} \left( |\log{\Pi}_{k-1+n}^{A}| + \log W({\Gamma}_{\delta,\theta}) +\log p_{\mathcal{B}}\right)\right\}. \end{array} $$

Since, by condition CP1, $N_{A}^{-1} |\log {\Pi }_{k-1+N_{A}}^{A}| \to \mu$ as A →∞, for a sufficiently large value of A there exists a small κ = κ_A (κ_A → 0 as A →∞) such that

$$ \left|\mu - \frac{|\log{\Pi}_{k-1+N_{A}}^{A}|}{N_{A}} \right| < \kappa. $$

(A.6)

Hence, for all sufficiently large A and n such that $\frac {1}{n}[|\log p_{\mathcal {B}}| + |\log W({\Gamma }_{\delta ,\theta })|] \le \varepsilon /2$, we have

$$ \begin{array}{@{}rcl@{}} \mathsf{P}_{k, \mathcal{B},\theta}\left( {{T_{A}^{W}}-k >n}\right) &\le& \mathsf{P}_{k,\mathcal{B}} \left\{\frac{1}{n} \inf_{\vartheta \in {\Gamma}_{\delta,\theta}} \lambda_{\mathcal{B},\vartheta}(k, k+n) < I_{\mathcal{B},\theta} - \varepsilon- \kappa \quad\qquad- \frac{1}{n}\left[\log p_{\mathcal{B}} + \log W({\Gamma}_{\delta,\theta})\right]\right\} \le \mathsf{P}_{k,\mathcal{B},\theta}\left\{\frac{1}{n} \inf_{\vartheta \in {\Gamma}_{\delta,\theta}} \lambda_{\mathcal{B},\vartheta}(k, k+n) < I_{\mathcal{B},\theta} - \varepsilon/2\right\}. \end{array} $$

(A.7)

By Lemma A.1, for any $k\in \mathbb {Z}_{+}$, $\mathcal {B}\in \mathcal {P}$, and 𝜃 ∈Θ we have the following inequality

$$ \begin{array}{@{}rcl@{}} \mathsf{E}_{k,\mathcal{B},\theta}\left[{({T_{A}^{W}}-k)^{+}}\right]^{r} \le {N_{A}^{r}} + r 2^{r-1} \sum\limits_{n=N_{A}}^{\infty} n^{r-1} \mathsf{P}_{k,\mathcal{B},\theta}\left( {{T_{A}^{W}}-k > n}\right), \end{array} $$

(A.8)

which along with (A.7) yields

$$ \begin{array}{@{}rcl@{}} \mathsf{E}_{k,\mathcal{B},\theta}\left[\left( {T_{A}^{W}}-k\right)^{+}\right]^{r} \le \left( {1+ \left\lfloor\frac{ \log (A/{\pi_{k}^{A}})}{I_{\mathcal{B},\theta}+\mu-\varepsilon}\right\rfloor}\right)^{r} + r 2^{r-1} {\Upsilon}_{r}(\varepsilon/2,\mathcal{B}, \theta). \end{array} $$

(A.9)

Now, note that

$$ \mathsf{PFA}_{\pi}({T_{A}^{W}}) \ge \sum\limits_{i=k}^{\infty} {\pi_{i}^{A}} \mathsf{P}_{\infty}({T_{A}^{W}} \le i) \ge \mathsf{P}_{\infty}({T_{A}^{W}} \le k) \sum\limits_{i=k}^{\infty} {\pi_{i}^{A}} = \mathsf{P}_{\infty}({T_{A}^{W}} \le k) {\Pi}_{k-1}^{A} , $$

and hence,

$$ \mathsf{P}_{\infty}({T_{A}^{W}} > k) \ge 1- \mathsf{PFA}_{\pi}({T_{A}^{W}})/{\Pi}_{k-1}^{A} \ge 1- [(A+1) {\Pi}_{k-1}^{A}]^{-1} , \quad k \in \mathbb{Z}_{+}. $$

(A.10)

Recall that we set ${{\Pi }_{k}^{A}}={\Pi }^{\alpha }_{k}$ with α = α_A = 1/(1 + A). It follows from Eqs. A.9 and A.10 that

$$ \begin{array}{@{}rcl@{}} \mathcal{R}_{k, \mathcal{B},\theta}^{r}({T_{A}^{W}}) =\frac{\mathsf{E}_{k, \mathcal{B},\theta}[{({{T_{A}^{W}}-k})^{+}}]^{r}}{\mathsf{P}_{\infty}({T_{A}^{W}} > k)} \le \frac{\left( {1+\left\lfloor \frac{\log (A/{\pi_{k}^{A}})}{I_{\mathcal{B},\theta}+\mu-\varepsilon} \right\rfloor}\right)^{r} + r 2^{r-1} {\Upsilon}_{r}(\varepsilon/2,\mathcal{B}, \theta)}{1- 1/(A {\Pi}^{A}_{k-1})}. \\ \end{array} $$

(A.11)

Since by condition C₂, ${\Upsilon }_{r}(\varepsilon /2,\mathcal {B}, \theta ) < \infty$ for all $\mathcal {B}\in \mathcal {P}$, 𝜃 ∈Θ, and ε > 0 and, by condition CP3, $(A{\Pi }_{k-1}^{A})^{-1} \to 0$, $|\log {\pi _{k}^{A}}|/\log A \to 0$ as A →∞, inequality (A.11) implies the asymptotic inequality

$$ \overline{\mathcal{R}}_{k,\mathcal{B},\theta}^{r}({T_{A}^{W}}) \le \left( {\frac{\log A}{I_{\mathcal{B},\theta}+\mu-\varepsilon}}\right)^{r} (1+o(1)), \quad A \to \infty. $$

Since ε can be arbitrarily small, this implies the asymptotic upper bound (A.4) (for all 0 < m ≤ r, $\mathcal {B}\in \mathcal {P}$, and 𝜃 ∈Θ). This upper bound and the lower bound (A.2) prove the asymptotic relation (5.3). The proof of (i) is complete.

The proof of part (ii). Using the inequalities (A.11) and $1-\mathsf {PFA}_{\pi }({T_{A}^{W}}) \ge A/(1+A)$, we obtain that for any $0<\varepsilon < I_{\mathcal {B},\theta }+\mu$

$$ \begin{array}{ll} \overline{\mathcal{R}}_{\mathcal{B},\theta}^{r}({T_{A}^{W}}) = \frac{{\sum}_{k=0}^{\infty} {\pi_{k}^{A}} \mathsf{E}_{k,\mathcal{B},\theta}[{({T_{A}^{W}}-k)^{+}}]^{r}}{1-\mathsf{PFA}_{\pi}({T_{A}^{W}})} \le\frac{{\sum\limits_{k=0}^{\infty}} {\pi_{k}^{A}} \left( {1+\left\lfloor\frac{\log (A/{\pi_{k}^{A}})}{I_{\mathcal{B},\theta}+\mu-\varepsilon} \right\rfloor}\right)^{r} + r 2^{r-1} {\Upsilon}_{r}(\varepsilon/2,\mathcal{B},\theta)}{A/(1+A)} . \end{array} $$

(A.12)

By condition C₂, ${\Upsilon }_{r}(\varepsilon /2,\mathcal {B},\theta ) < \infty$ for any ε > 0, $\mathcal {B}\in \mathcal {P}$, and 𝜃 ∈Θ and, by condition CP2, ${\sum }_{k=0}^{\infty } {\pi _{k}^{A}} |{\log \pi _{k}^{A}}|^{r} =o(|\log A|^{r})$ as A →∞, which implies that for all $\mathcal {B}\in \mathcal {P}$ and 𝜃 ∈Θ

$$ \overline{\mathcal{R}}_{\mathcal{B},\theta}^{r}({T_{A}^{W}}) \le \left( {\frac{\log A}{I_{\mathcal{B},\theta}+\mu - \varepsilon}}\right)^{r} (1+o(1)), \quad A \to \infty. $$

Since ε can be arbitrarily small, the asymptotic upper bound (A.5) follows and the proof of the asymptotic approximation (5.4) is complete.

1.2 Proof of Proposition 2

As before, ${\pi _{k}^{A}}=\pi _{k}^{\alpha _{A}}$, so $\bar \nu _{A}=\bar \nu _{\alpha _{A}}$ and $\omega _{A}=\omega _{\alpha _{A}}$, where |log α_A|∼ log A.

For ε ∈ (0,1), let

$$ M_{A}=M_{A}(\varepsilon,\mathcal{B},\theta) = (1-\varepsilon) \frac{\log A}{I_{\mathcal{B},\theta}} . $$

Recall that

$$ \mathsf{P}_{k, \mathcal{B},\theta}({\widetilde{T}_{A}^{W}}>k)=\mathsf{P}_{\infty} ({\widetilde{T}_{A}^{W}}>k) \ge 1- \frac{k+ \omega_{A}} {A} , \quad k \in \mathbb{Z}_{+} $$

(see Eq. 5.7), so using Chebyshev’s inequality, we obtain

$$ \begin{array}{@{}rcl@{}} \mathcal{R}_{k,\mathcal{B},\theta}^{r}({\widetilde{T}_{A}^{W}}) \ge {M_{A}^{r}} \mathsf{P}_{k, \mathcal{B},\theta}({\widetilde{T}_{A}^{W}} -k > M_{A}) \\ \ge {M_{A}^{r}}\left[{\mathsf{P}_{k,\mathcal{B},\theta}({\widetilde{T}_{A}^{W}}>k) - \mathsf{P}_{k, \mathcal{B},\theta}(k < {\widetilde{T}_{A}^{W}} < k+M_{A})}\right] \\ \ge {M_{A}^{r}}\left[{1- \frac{\omega_{A} +k } {A} - \mathsf{P}_{k, \mathcal{B},\theta}(k < {\widetilde{T}_{A}^{W}} < k+M_{A})}\right]. \end{array} $$

(A.13)

Analogously to Eq. 4.9,

$$ \mathsf{P}_{k, \mathcal{B},\theta}({k < T < k+ M_{A}}) \le U_{M_{A},k}(T) + \beta_{M_{A},k}(\varepsilon, \mathcal{B},\theta). $$

(A.14)

Since

$$ \mathsf{P}_{\infty}({0 < {\widetilde{T}_{A}^{W}} - k <M_{A}}) \le \mathsf{P}_{\infty}({{\widetilde{T}_{A}^{W}} < k+ M_{A}}) \le (k+\omega_{A} + M_{A})/A, $$

we have

$$ U_{M_{A},k}({\widetilde{T}_{A}^{W}}) \le \frac{k+ \omega_{A}+(1-\varepsilon) I_{\mathcal{B},\theta}^{-1} \log A}{A^{\varepsilon^{2}}}. $$

(A.15)

By condition (5.9),

$$ \lim_{A\to \infty} \frac{{\log (\omega_{A}+\bar\nu_{A})}}{\log A} = 0, $$

(A.16)

which implies that ω_A = o(A^γ) as A →∞ for any γ > 0. Therefore, $U_{M_{A},k}({\widetilde {T}_{A}^{W}})\to 0$ as A →∞ for any fixed k. Also, $\beta _{M_{A},k}(\varepsilon ,\mathcal {B},\theta )\to 0$ by condition C₁, so that $\mathsf {P}_{k, \mathcal {B},\theta }({0 < {\widetilde {T}_{A}^{W}} -k < M_{A}})\to 0$ for any fixed k. It follows from Eq. A.13 that for an arbitrary ε ∈ (0,1) as A →∞

$$ \mathcal{R}_{k,\mathcal{B},\theta}^{r}({\widetilde{T}_{A}^{W}}) \ge \left( {\frac{(1-\varepsilon) \log A}{I_{\mathcal{B},\theta}}}\right)^{r} (1+o(1)), $$

which yields the asymptotic lower bound (for any fixed $k\in \mathbb {Z}_{+}$, $\mathcal {B}\in \mathcal {P}$, and 𝜃 ∈Θ)

$$ \liminf_{A\to\infty}\frac{\mathcal{R}_{k,\mathcal{B},\theta}^{r}({\widetilde{T}_{A}^{W}})}{(\log A)^{r}} \ge \frac{1}{I_{\mathcal{B},\theta}^{r}} . $$

(A.17)

To prove (5.10) it suffices to show that this bound is attained by ${\widetilde {T}_{A}^{W}}$, i.e.,

$$ \limsup_{A\to\infty} \frac{\mathcal{R}_{k, \mathcal{B},\theta}^{r}({\widetilde{T}_{A}^{W}})}{(\log A)^{r}} \le \frac{1}{I_{\mathcal{B},\theta}^{r}} . $$

(A.18)

Define

$$ \widetilde{M}_{A}= \widetilde{M}_{A}(\varepsilon,\mathcal{B},\theta)=1+\left\lfloor \frac{\log A}{I_{\mathcal{B},\theta}-\varepsilon} \right\rfloor. $$

By Lemma A.1, for any $k\in \mathbb {Z}_{+}$, $\mathcal {B}\in \mathcal {P}$, and 𝜃 ∈Θ,

$$ \begin{array}{@{}rcl@{}} & \mathsf{E}_{k, \mathcal{B},\theta}\left[({\widetilde{T}_{A}^{W}}-k)^{+}\right]^{r} \le \widetilde{M}_{A}^{r} + r 2^{r-1} \sum\limits_{n= \widetilde{M}_{A}}^{\infty} n^{r-1} \mathsf{P}_{k, \mathcal{B},\theta}\left( {\widetilde{T}_{A}^{W}} > n\right), \end{array} $$

(A.19)

and since for any n ≥ 1,

$$ \log R_{\textbf{p},W}^{\pi}(k+n) \ge \log {\Lambda}_{\textbf{p},W}(k,k+n) \ge \inf_{\vartheta \in {\Gamma}_{\delta,\theta}} \lambda_{\mathcal{B},\vartheta}(k, k+n) + \log W({\Gamma}_{\delta,\theta})+\log p_{\mathcal{B}} , $$

in just the same way as in the proof of Proposition 1 (setting ${\pi _{k}^{A}}=1$) we obtain that for all $n \ge \widetilde {M}_{A}$

$$ \begin{array}{@{}rcl@{}} \mathsf{P}_{k, \mathcal{B},\theta}\left( {\widetilde{T}_{A}^{W}} > n\right) &\le& \mathsf{P}_{k, \mathcal{B},\theta}\left\{\frac{1}{n} \inf_{\vartheta \in {\Gamma}_{\delta,\theta}} \lambda_{\mathcal{B},\vartheta}(k, k+n) < I_{\mathcal{B},\theta} - \varepsilon \right.\\ &&\qquad\quad \left.+\frac{1}{n}[{|\log W({\Gamma}_{\delta,\theta})| +|\log p_{\mathcal{B}}|}]\right\}. \end{array} $$

Hence, for all sufficiently large n such that $\frac {1}{n}\left [|\log W({\Gamma }_{\delta ,\theta })| +|\log p_{\mathcal {B}}|\right ] \le \varepsilon /2$,

$$ \begin{array}{@{}rcl@{}} \mathsf{P}_{k, \mathcal{B},\theta}\left( {\widetilde{T}_{A}^{W}}-k >n\right) & \le \mathsf{P}_{k,\mathcal{B},\theta}\left( \frac{1}{n}\inf\limits_{\vartheta \in {\Gamma}_{\delta,\theta}} \lambda_{\mathcal{B},\vartheta}(k, k+n) < I_{\mathcal{B},\theta} - \varepsilon/2\right). \end{array} $$

(A.20)

Using Eqs. A.19 and A.20, we obtain

$$ \mathsf{E}_{k, \mathcal{B},\theta}\left[\left( {\widetilde{T}_{A}^{W}}-k\right)^{+}\right]^{r} \le \left( 1+\left\lfloor \frac{\log A}{I_{\mathcal{B},\theta}-\varepsilon} \right\rfloor\right)^{r} + r 2^{r-1} {\Upsilon}_{r}(\varepsilon/2,\mathcal{B}, \theta), $$

(A.21)

which along with the inequality $\mathsf {P}_{\infty }({\widetilde {T}_{A}^{W}} > k) > 1- (\omega _{A} +k)/A$ (see Eq. 5.7) implies the inequality

$$ \begin{array}{@{}rcl@{}} \mathcal{R}_{k, \mathcal{B},\theta}^{r}({\widetilde{T}_{A}^{W}}) =\frac{\mathsf{E}_{k, \mathcal{B},\theta}\left[{({{\widetilde{T}_{A}^{W}}-k})^{+}}\right]^{r}}{\mathsf{P}_{\infty}({\widetilde{T}_{A}^{W}} > k)} \le \frac{\left( 1+\left\lfloor \frac{\log A}{I_{\mathcal{B},\theta}-\varepsilon} \right\rfloor\right)^{r} + r 2^{r-1} {\Upsilon}_{r}(\varepsilon/2,\mathcal{B}, \theta)}{1- (\omega_{A} +k)/A}. \\ \end{array} $$

(A.22)

Since due to Eq. A.16ω_A/A → 0 and, by condition C₂, ${\Upsilon }_{r}(\varepsilon /2,\mathcal {B}, \theta ) < \infty$ for all ε > 0, $\mathcal {B}\in \mathcal {P}$, 𝜃 ∈Θ, inequality (A.22) implies the asymptotic inequality

$$ \mathcal{R}_{k, \mathcal{B},\theta}^{r}({\widetilde{T}_{A}^{W}})\le \left( \frac{\log A}{I_{\mathcal{B},\theta} - \varepsilon}\right)^{r} (1+o(1)), \quad A \to \infty. $$

Since ε can be arbitrarily small the asymptotic upper bound (A.18) follows and the proof of the asymptotic approximation (5.10) is complete.

In order to prove (5.11) note first that, using Eq. A.13, yields the lower bound

$$ \begin{array}{@{}rcl@{}} \overline{\mathcal{R}}_{\mathcal{B},\theta}^{r}(\widetilde{T}_{A}) & \ge {M_{A}^{r}}\left[1-\frac{\bar\nu_{A}+\omega_{A}}{A} - \mathsf{P}^{\pi}_{\mathcal{B},\theta} \left( 0 < \widetilde{T}_{A}-\nu < M_{A}\right)\right] . \end{array} $$

(A.23)

Let K_A be an integer number that approaches infinity as A →∞ with rate O(A^γ), γ > 0. Now, using Eqs. A.14 and A.15, we obtain

$$ \begin{array}{@{}rcl@{}} &&\mathsf{P}^{\pi}_{\mathcal{B},\theta}(0< {\widetilde{T}_{A}^{W}} -\nu < M_{A}) = \sum\limits_{k=0}^{\infty} {\pi_{k}^{A}} \mathsf{P}_{k,\mathcal{B},\theta}\left( 0 < {\widetilde{T}_{A}^{W}} -k < M_{A}\right) \\ &\le& \mathsf{P}(\nu > K_{A}) + \sum\limits_{k=0}^{\infty} {\pi_{k}^{A}} U_{M_{A},k}({\widetilde{T}_{A}^{W}}) + \sum\limits_{k=0}^{K_{A}} {\pi_{k}^{A}} \beta_{M_{A}, k}(\varepsilon,\mathcal{B},\theta) \\ & \le& \mathsf{P}(\nu > K_{A}) + \frac{\bar\nu_{A}+\omega_{A}+ (1-\varepsilon) I_{\mathcal{B},\theta}^{-1} \log A}{A^{\varepsilon^{2}}} + \sum\limits_{k=0}^{K_{A}} {\pi_{k}^{A}} \beta_{M_{A},k}(\varepsilon,\mathcal{B},\theta) . \end{array} $$

(A.24)

Note that due to Eq. A.16$(\omega _{A}+\bar \nu _{A})/A^{\gamma } \to 0$ as A →∞ for any γ > 0. As a result, the first two terms in Eq. A.24 go to zero as A →∞ (by Markov’s inequality $\mathsf {P}(\nu >K_{A}) \le \bar \nu _{A}/K_{A} = \bar \nu _{A}/O(A^{\gamma }) \to 0$) and the last term also goes to zero by condition C₁ and Lebesgue’s dominated convergence theorem. Thus, for all 0 < ε < 1, $\mathsf {P}^{\pi }_{\mathcal {B}}(0< {\widetilde {T}_{A}^{W}} -\nu < M_{A})$ approaches 0 as A →∞. Using inequality (A.23), we obtain that for any 0 < ε < 1 as A →∞

$$ \overline{\mathcal{R}}_{\mathcal{B},\theta}^{r}({\widetilde{T}_{A}^{W}}) \ge (1-\varepsilon)^{r} \left( \frac{\log A}{I_{\mathcal{B},\theta}}\right)^{r} (1+o(1)), $$

which yields the asymptotic lower bound (for any r > 0, $\mathcal {B}\in \mathcal {P}$, and 𝜃 ∈Θ)

$$ \liminf_{A\to\infty}\frac{\overline{\mathcal{R}}_{\pi,\mathcal{B},\theta}^{r}({\widetilde{T}_{A}^{W}})}{(\log A)^{r}} \ge \frac{1}{I_{\mathcal{B},\theta}^{r}}. $$

(A.25)

To obtain the upper bound it suffices to use inequality (A.21), which along with the fact that $\mathsf {PFA}_{\pi }({\widetilde {T}_{A}^{W}}) \le (\bar \nu _{A} +\omega _{A})/A$ yields (for every $0<\varepsilon < I_{\mathcal {B},\theta }$)

$$ \begin{array}{@{}rcl@{}} \overline{\mathcal{R}}_{\mathcal{B},\theta}^{r}({\widetilde{T}_{A}^{W}}) = \frac{{\sum}_{k=0}^{\infty} {\pi_{k}^{A}} \mathsf{E}_{k,\mathcal{B},\theta}\left[({\widetilde{T}_{A}^{W}}-k)^{+}\right]^{r}}{1-\mathsf{PFA}_{\pi}({\widetilde{T}_{A}^{W}} )} \le\frac{\left( 1+\frac{\log A}{I_{\mathcal{B},\theta}-\varepsilon}\right)^{r} + r 2^{r-1} {\Upsilon}_{r}(\varepsilon/2,\mathcal{B},\theta)}{1-(\omega_{A}+\bar\nu_{A})/A}. \end{array} $$

Since $(\omega _{A}+\bar \nu _{A})/A\to 0$ and, by condition C₂, ${\Upsilon }_{r}(\varepsilon /2,\mathcal {B},\theta ) < \infty$ for any ε > 0, $\mathcal {B}\in \mathcal {P}$, and 𝜃 ∈Θ we obtain that, for every $0<\varepsilon < I_{\mathcal {B},\theta }$ as A →∞,

$$ \overline{\mathcal{R}}_{\mathcal{B},\theta}^{r}({\widetilde{T}_{A}^{W}}) \le \left( \frac{\log A}{I_{\mathcal{B},\theta}-\varepsilon}\right)^{r} (1+o(1)) , $$

which implies

$$ \limsup_{A\to\infty} \frac{\overline{\mathcal{R}}_{\mathcal{B},\theta}^{r}({\widetilde{T}_{A}^{W}})}{(\log A)^{r}} \le \frac{1}{I_{\mathcal{B},\theta}^{r}} $$

(A.26)

since ε can be arbitrarily small.

Applying the bounds (A.25) and (A.26) together completes the proof of Eq. 5.11.