On the Rate of Convergence in the Bernstein–von Mises Theorem for M/M/1 Queue

Abstract

The paper discusses the rate of convergence in the Bernstein–von Mises theorem for M/M/1 queueing system which is observed over a continuous time interval (0, T] where T is a suitable stopping time.

Appendix

Proof of Lemma 3.1

Note that

$$\begin{aligned} f< d(1-r) \Rightarrow f< d g \quad \text {or} \quad g < 1-r. \end{aligned}$$

Then

$$\begin{aligned} P \{ w: f(w)< d (1-r) \} \le 1 - P\{ w: \frac{f(w)}{g(w)} < d \} + P\{ w: | g(w) -1 | >r \}. \end{aligned}$$

Or,

$$\begin{aligned} 1 - P \{ w: f(w) \ge d (1-r)\} \le 1 - P\{ w: \frac{f(w)}{g(w)} < d\} + P\{ w: | g(w) -1 | >r\}, \end{aligned}$$

(without loss of generality, we have assumed here that $g > 0$ a.s.). $\square$

Proof of Lemma 3.2

$$\begin{aligned}&P \bigg \{ \int \limits _{|t| \le r_1(T) \sqrt{I(\theta _0)}} Z(t) \lambda (\theta _0) |\upsilon _T(t)- e^{-\frac{t^2}{2}}| dt> h(1) \lambda (\theta _0) \left( e^{r_1^2(T)} - 1\right) \bigg \} \\&\quad = P \bigg \{ \int \limits _{|t| \le r_1(T) \sqrt{I(\theta _0)}} Z(t) \lambda (\theta _0) e^{-\frac{t^2}{2}} | e^{-\frac{t^2}{2}(b_T -1)}- 1 | dt \\&\quad> h(1) \lambda (\theta _0) \left( e^{r_1^2(T)} - 1\right) \bigg \} \\&\quad \le P \bigg \{ \int \limits _{|t| \le r_1(T) \sqrt{I(\theta _0)}} Z(t) \lambda (\theta _0) e^{-\frac{t^2}{2}} [ e^{\frac{t^2}{2} | b_T -1| }- 1 ] dt \\&\qquad> h(1) \lambda (\theta _0) \left( e^{r_1^2(T)} - 1\right) \bigg \} \quad \text {(since } |e^{-x}-1| \le e^{|x|}-1 \forall x) \\&\quad \le P \bigg \{ h(1)\lambda (\theta _0) [ e^{\frac{{r_1^2(T) I(\theta _)}}{2}| b_T -1| }- 1 ] dt> h(1) \lambda (\theta _0) \left( e^{r_1^2(T)} - 1\right) \bigg \} \\&\quad (\text {by assumption } (A_3)), \\&\quad = P\{| b_T -1|> 2 / I(\theta _0)\} \\&\quad = P\{| b_T -1| > 2 r_1^2(T)~ r_2^2(T)\}. \end{aligned}$$

$\square$

Proof of Lemma 3.3

$$\begin{aligned}&P \bigg \{ \int \limits _{|t| \le r_1(T) \sqrt{I(\theta _0)}} Z(t) \upsilon _T(t) \bigg | \lambda (\theta _0) - \lambda \bigg ( {\widehat{\theta }}_T + \frac{t}{\sqrt{I(\theta _0)}} \bigg ) \bigg | dt \\&\quad> C_1 h(1-r_1(T))(r_2(T) + r_1(T))\bigg \} \\&\quad = P\bigg \{ \int \limits _{|t| \le r_1(T) \sqrt{I(\theta _0)}} Z(t) e^{-\frac{t^2}{2} b_T} \bigg | \lambda (\theta _0) - \lambda \bigg ( {\widehat{\theta }}_T + \frac{t}{\sqrt{I(\theta _0)}} \bigg ) \bigg | dt \\&\quad> C_1 h(1-r_1(T))(r_2(T) + r_1(T)) \bigg \} \\&\quad \le P \bigg \{ \int \limits _{|t| \le r_1(T) \sqrt{I(\theta _0)}} Z(t) e^{-\frac{t^2}{2} (1- |b_T-1|)} \bigg | \lambda (\theta _0) - \lambda \bigg ( {\widehat{\theta }}_T + \frac{t}{\sqrt{I(\theta _0)}} \bigg ) \bigg | dt \\&\quad> C_1 h(1-r_1(T))(r_2(T) + r_1(T))\bigg \} \\&\quad \le P\{|b_T -1|> r_1(T)\} \\&\qquad + P \bigg \{ \int \limits _{|t| \le r_1(T) \sqrt{I(\theta _0)}} e^{-\frac{t^2}{2} (1- r_1(T))} Z(t) dt \bigg | \lambda (\theta _0) - \lambda \bigg ( {\widehat{\theta }}_T + \frac{t}{\sqrt{I(\theta _0)}} \bigg )\bigg | \\&\quad> C_1 h(1-r_1(T)) (r_2(T) + r_1(T))\bigg \} \\&\quad \le P \{ |b_T -1|> r_1(T)\} \\&\qquad + P \bigg \{ \bigg ( \int \limits _{|t| \le r_1(T) \sqrt{I(\theta _0)}} e^{-\frac{t^2}{2} (1- r_1(T))} Z(t) dt \bigg ) \\&\quad \times C_1 \left[ |\theta _0-{\widehat{\theta }}_T|+r_1(T)\right]> C_1 h(1-r_1(T))(r_2(T) + r_1(T))\bigg \} \\&\quad (\text {by assumption }(A_1)), \nonumber \\&\quad \le P\{ |b_T -1|> r_1(T)\} \\&\qquad + P \bigg \{ C_1 h(1-r_1(T))\left[ |\theta _0-{\widehat{\theta }}_T|+r_1(T)\right] \\&\quad> C_1 h(1-r_1(T)) (r_2(T) + r_1(T))\bigg \} \\&\quad (\text {by assumption }(A_3))\\&\quad =P\left\{ |b_T -1|> r_1(T)\right\} + P \left\{ |{\widehat{\theta }}_T -\theta _0 |>r_2(T) \right\} . \end{aligned}$$

$\square$

Proof of Lemma 3.4

$$\begin{aligned}&P \bigg \{ \int \limits _{|t|> r_1(T) \sqrt{I(\theta _0)}} Z(t) \upsilon _T(t) \bigg | \lambda \left( {\widehat{\theta }}_T + \frac{t}{\sqrt{I(\theta _0)}} \right) \bigg | dt \\&\quad> C_3 h\left( 1-r_1(T)- p\right) \left( 2r_2^{\delta }(T) + r_2^{2\delta }(T) \right) \bigg \} \\&\quad \le P \bigg \{ \int \limits _{|t|> r_1(T) \sqrt{I(\theta _0)}} Z(t) e^{-\frac{t^2}{2} \left( 1-|b_T -1|\right) } C_2 \left( 1+\bigg | {\widehat{\theta }}_T + \frac{t}{\sqrt{I(\theta _0)}} \bigg |^{\delta } \right) dt\\&\quad> C_3 h\left( 1-r_1(T)- p\right) \left( 2r_2^{\delta }(T)+r_2^{2\delta }(T) \right) \bigg \} \quad \text {(by assumption }(A_4)) \\&\quad \le P \left\{ |b_T -1|> r_1(T)\right\} \\&\quad + P\bigg \{ \int \limits _{|t|> r_1(T)\sqrt{I(\theta _0)}} Z(t) e^{-\frac{t^2}{2}(1- r_1(T))} \\&\quad \times C_2 \left( 1 + k_{\delta }^{\star } |{\widehat{\theta }}_T -\theta _0|^{\delta } + k_{\delta }^{\star } |\theta _0 |^{\delta } + k_{\delta }^{\star } \bigg |\frac{t}{\sqrt{I(\theta _0)}} \bigg |^{\delta } \right) dt \\&\quad > C_3 h\left( 1-r_1(T)-p\right) \left( 2r_2^{\delta }(T) + r_2^{2\delta }(T) \right) \bigg \} \\ \end{aligned}$$

$$\begin{aligned}&\bigg [\text {since} |a+l|^{\delta } \le k_{\delta } \left( |a|^{\delta } + |l|^{\delta }\right) , \text {where} k_{\delta }=1 \text {or} 2^{\delta -1} \text {according as} \delta \le 1\\&\text {or} \delta \ge 1 \text {and} k_{\delta } = \text {max}\left( k_{\delta }, k_{\delta }^2\right) \bigg ] \\&\quad \le P \left\{ |b_T -1|> r_1(T)\right\} \\&\quad +P \bigg \{ \int \limits _{|t|> r_1(T) \sqrt{I(\theta _0)}} Z(t) e^{-\frac{t^2}{2} \left( 1- r_1(T) - p\right) } e^{-p\frac{t^2}{2}} \\&\quad ~~\times C_2 \bigg (1+k_{\delta }^{\star }|{\widehat{\theta }}_T -\theta _0 |^{\delta } + k_{\delta }^{\star } |\theta _0 |^{\delta } + k_{\delta }^{\star } \bigg | \frac{t}{\sqrt{I(\theta _0)}} \bigg |^{\delta }\bigg ) dt \\&\quad> C_3 h\left( 1-r_1(T)- p\right) \left( 2r_2^{\delta }(T) + r_2^{2\delta }(T) \right) \bigg \} \\&\quad \le P \left\{ |b_T -1|> r_1(T)\right\} \\&\qquad + P \bigg \{ \int \limits _{|t|> r_1(T) \sqrt{I(\theta _0)}} Z(t) e^{-\frac{t^2}{2}\left( 1- r_1(T)-p\right) } \times C_3 \left( \frac{2}{|t|^{\delta }}+\frac{| {\widehat{\theta }}_T-\theta _0|^{\delta }}{|t|^{\delta }} \right) dt \\&\quad> C_3 h\left( 1-r_1(T)- p\right) \left( 2r_2^{\delta }(T) + r_2^{2\delta }(T) \right) \bigg \} \\&\quad \le P \{|b_T -1|> r_1(T)\} \\&\qquad + P \bigg \{ C_3 h\left( 1- r_1(T) - p\right) \left( \frac{2}{(r_1(T\sqrt{I(\theta _0)})^{\delta }} \right) + \left( \frac{|{\widehat{\theta }}_T -\theta _0 |^{\delta }}{(r_1(T) \sqrt{I(\theta _0)})^{\delta }} \right) \\&\quad> C_3 h\left( 1-r_1(T)- p\right) \left( 2r_2^{\delta }(T) + r_2^{2\delta }(T) \right) \bigg \} \quad \text {(by assumption} (A_3)) \\&\quad \le P \{|b_T -1|> r_1(T)\} + P \left\{ \left( 2 r_2^{\delta }(T) + |{\widehat{\theta }}_T -\theta _0 |^{\delta }r_2^{\delta }(T) \right)> 2r_2^{\delta }(T) + r_2^{2\delta }(T)\right\} \\&\quad = P \{|b_T -1|> r_1(T)\} +P\left\{ |{\widehat{\theta }}_T-\theta _0|>r_2(T)\right\} . \end{aligned}$$

$\square$

Proof of Lemma 3.5

Observe that

$$\begin{aligned}&\lambda (\theta _0) \int \limits _{|t|> r_1(T) \sqrt{I(\theta _0)}} Z(t) e^{-\frac{t^2}{2}} dt \nonumber \\&\quad = \lambda (\theta _0) \int \limits _{|t| > r_1(T) \sqrt{I(\theta _0)}} Z(t) e^{-\frac{t^2}{2} (1-p)} e^{-p\frac{t^2}{2}} dt \quad (\text {where } 0< p< 1) \nonumber \\&\quad \le \lambda (\theta _0) e^{-\frac{p}{2} (r_1^2(T) I(\theta _0))}h(1-p) \quad (\text {by assumption} (A_3)) \nonumber \\&\quad < (2/p) \lambda (\theta _0)h(1-p)r_2^2(T) = C_4 h(1-p) r_2^2(T). \end{aligned}$$

(A.1)

Now using Lemmas 3.2 and 3.3 we get,

$$\begin{aligned}&P \bigg \{ \int _{|t| \le r_1(T) \sqrt{I(\theta _0)}} Z(t) \bigg | \upsilon _T(t) \lambda \bigg ( {\widehat{\theta }}_T + \frac{t}{\sqrt{I(\theta _0)}} \bigg ) - \lambda (\theta _0) e^{-\frac{t^2}{2}} \bigg | dt \nonumber \\&\quad> h(1) \lambda (\theta _0) \left( e^{r_1^2(T)} -1\right) + C_1 h(1-r_1(T))(r_2(T) + r_1(T))\bigg \} \nonumber \\&\quad \le P \left\{ |b_T-1|> 2r_1^2(T) r_2^2(T)\right\} + P\left\{ |b_T-1|>r_1(T)\right\} \nonumber \\&\quad + P\left\{ |{\widehat{\theta }}_T-\theta _0|>r_2(T)\right\} . \end{aligned}$$

(A.2)

Again using Lemma 3.4 and (A.1) we get,

$$\begin{aligned}&P \bigg \{ \int _{|t|> r_1(T) \sqrt{I(\theta _0)}}Z(t) \bigg | \upsilon _T(t) \lambda \bigg ({\widehat{\theta }}_T + \frac{t}{\sqrt{I(\theta _0)}} \bigg ) - \lambda (\theta _0) e^{-\frac{t^2}{2}} \bigg | dt \nonumber \\&\quad> C_3 h\left( 1-r_1(T)-p\right) \left( 2 r_2^{\delta }(T) + r_2^{2 \delta }(T) \right) + C_4 h(1-p) r_2^2(T) \bigg \}\nonumber \\&\quad \le P \bigg \{ \int _{|t|> r_1(T) \sqrt{I(\theta _0)}}Z(t) \bigg | \upsilon _T(t) \lambda \left( {\widehat{\theta }}_T + \frac{t}{\sqrt{I(\theta _0)}} \right) \bigg | dt\nonumber \\&\qquad + \bigg | \lambda (\theta _0) \int \limits _{|t|> r_1(T) \sqrt{I(\theta _0)}} Z(t) e^{-\frac{t^2}{2}} dt \bigg | \nonumber \\&\quad> C_3 h\left( 1-r_1(T)-p\right) \left( 2 r_2^{\delta }(T) + r_2^{2 \delta }(T) \right) + C_4 h(1-p) r_2^2(T) \bigg \} \nonumber \\&\quad \le P \bigg \{ \int _{|t|> r_1(T) \sqrt{I(\theta _0)}}Z(t) \upsilon _T(t) \lambda \left( {\widehat{\theta }}_T + \frac{t}{\sqrt{I(\theta _0)}}\right) dt \nonumber \\&\quad> C_3 h\left( 1-r_1(T)-p\right) \left( 2r_2^{\delta }(T)+r_2^{2\delta }(T)\right) \bigg \} \nonumber \\&\quad \le P \left\{ |b_T-1|>r_1(T)\right\} +P\left\{ |{\widehat{\theta }}_T-\theta _0|>r_2(T)\right\} . \end{aligned}$$

(A.3)

Now combining (A.2) and (A.3) we prove the Lemma. $\square$