**Theorem 1.**

$$
\sqrt{m}(\hbar - a) \xrightarrow{L} N(0, \gamma' \Sigma \gamma)
$$

*where*

$$\gamma = \frac{1}{\ln 2} \left( \frac{-\cos \mu\_0}{\rho \ln \rho}, \frac{\cos 2\mu\_0}{\rho^{2^a} \ln \rho^{2^a}}, \frac{-\sin \mu\_0}{\rho \ln \rho}, \frac{\sin 2\mu\_0}{\rho^{2^a} \ln \rho^{2^a}} \right)'$$

*and*

$$\underline{\gamma}\prime\Sigma\underline{\gamma} = \frac{1}{(\ln 2)^2} \left[ \frac{1 + \rho^{2^a} - 2\rho^2}{2(\rho \ln \rho)^2} + \frac{1 + \rho^{4^a} - 2(\rho^{2^a})^2}{2(\rho^{2^a} \ln \rho^{2^a})^2} + \frac{2\rho^{2^a + 1} - \rho - \rho^{3^a}}{\rho \ln \rho \rho^{2^a} \ln \rho^{2^a}} \right]$$

**Proof.** We know from Lemma 1 that √*m*(*Tm* − *μ*) *L*−→ *<sup>N</sup>*4(0, Σ)

Therefore, by delta method (given in Casella and Berger (2002)), we ge<sup>t</sup> *γ*

√*m*(*α*<sup>ˆ</sup> − *α*) *L*−→ *N*(0, *γ* <sup>Σ</sup>*γ*) where *g*(*Tm*) = *α*ˆ = 1ln 2 ln ln *R* ¯ 2 ln *R* ¯ 1 = 1 ln 2 ln ln -*C*¯2<sup>2</sup> + *S*¯2<sup>2</sup> ln -*C*¯1<sup>2</sup> + *S*¯1<sup>2</sup> = ⎛⎜⎜⎜⎜⎜⎝ *∂g ∂C*¯1 *∂g ∂C*¯2 *∂g ∂S*¯1 *∂g ∂S*¯2 ⎞⎟⎟⎟⎟⎟⎠ at *μ* = 1 ln 2 ⎛⎜⎜⎜⎜⎜⎜⎜⎝ *C* ¯ 1 −(*C*¯1<sup>2</sup>+*<sup>S</sup>*¯1<sup>2</sup>)ln√*C*¯1<sup>2</sup>+*<sup>S</sup>*¯1<sup>2</sup> *C* ¯ 2 (*C*¯2<sup>2</sup>+*S*¯2<sup>2</sup>)ln√*C*¯2<sup>2</sup>+*S*¯2<sup>2</sup> *S* ¯ 1 −(*C*¯1<sup>2</sup>+*<sup>S</sup>*¯1<sup>2</sup>)ln√*C*¯1<sup>2</sup>+*<sup>S</sup>*¯1<sup>2</sup> *S* ¯ 2 (*C*¯2<sup>2</sup>+*S*¯2<sup>2</sup>)ln√*C*¯2<sup>2</sup>+*S*¯2<sup>2</sup> ⎞⎟⎟⎟⎟⎟⎟⎟⎠ at *μ* = 1 ln 2 ⎛⎜⎜⎜⎜⎜⎝ − cos *μ*0 *ρ* ln *ρ* cos 2*μ*0 *ρ*2*<sup>α</sup>* ln *ρ*2*<sup>α</sup>* − sin *μ*0 *ρ* ln *ρ* sin 2*μ*0 *ρ*2*<sup>α</sup>* ln *ρ*2*<sup>α</sup>* ⎞⎟⎟⎟⎟⎟⎠ *γ* Σ*γ* = 1 (ln 2)<sup>2</sup> 1 + *ρ*2*α* cos<sup>2</sup> 2*μ*0 − <sup>2</sup>*ρ*<sup>2</sup>(cos<sup>4</sup> *μ*0 + sin<sup>4</sup> *μ*0) <sup>2</sup>(*ρ* ln *ρ*)<sup>2</sup> + −*ρ* cos<sup>2</sup> *μ*0 cos 2*μ*0 − *ρ*3*α* cos 3*μ*0 cos *μ*0 cos 2*μ*0 + <sup>2</sup>*ρ*<sup>2</sup>*<sup>α</sup>*+<sup>1</sup> cos<sup>2</sup> *μ*0 cos<sup>2</sup> 2*μ*0 *ρ* ln *ρρ*<sup>2</sup>*<sup>α</sup>* ln *ρ*<sup>2</sup>*<sup>α</sup>* + *ρ*2*α* sin 2*μ*0 sin *μ*0 cos *μ*0 − <sup>2</sup>*ρ*<sup>2</sup> cos<sup>2</sup> *μ*0 sin<sup>2</sup> *μ*0 (*ρ* ln *ρ*)<sup>2</sup> + −*ρ*3*<sup>α</sup>* cos *μ*0 sin 2*μ*0 sin 3*μ*0 − *ρ* sin *μ*0 cos *μ*0 sin 2*μ*0 + <sup>2</sup>*ρ*<sup>2</sup>*<sup>α</sup>*+<sup>1</sup> cos<sup>2</sup> *μ*0 sin<sup>2</sup> 2*μ*0*ρ* ln *ρρ*<sup>2</sup>*<sup>α</sup>* ln *ρ*<sup>2</sup>*<sup>α</sup>* + *ρ*4*α* cos<sup>2</sup> 4*μ*0 + 1 − <sup>2</sup>(*ρ*<sup>2</sup>*<sup>α</sup>* )<sup>2</sup>(cos<sup>4</sup> 2*μ*0 + sin<sup>4</sup> <sup>2</sup>*μ*0) <sup>2</sup>(*ρ*<sup>2</sup>*<sup>α</sup>* ln *ρ*<sup>2</sup>*<sup>α</sup>* )2 + *ρ*4*α* sin 4*μ*0 sin 2*μ*0 cos 2*μ*0 − <sup>2</sup>(*ρ*<sup>2</sup>*<sup>α</sup>* )2 cos<sup>2</sup> 2*μ*0 sin<sup>2</sup> 2*μ*0 (*ρ*<sup>2</sup>*<sup>α</sup>* ln *ρ*<sup>2</sup>*<sup>α</sup>* )2 + −*ρ*3*<sup>α</sup>* sin 3*μ*0 cos 2*μ*0 sin *μ*0 + *ρ* sin<sup>2</sup> *μ*0 cos 2*μ*0 + <sup>2</sup>*ρ*<sup>2</sup>*<sup>α</sup>*+<sup>1</sup> cos<sup>2</sup> 2*μ*0 sin<sup>2</sup> *μ*0 *ρ* ln *ρρ*<sup>2</sup>*<sup>α</sup>* ln *ρ*<sup>2</sup>*<sup>α</sup>* + −*ρ* cos *μ*0 sin *μ*0 sin 2*μ*0 + *ρ*3*α* cos 3*μ*0 sin *μ*0 sin 2*μ*0 + <sup>2</sup>*ρ*<sup>2</sup>*<sup>α</sup>*+<sup>1</sup> sin<sup>2</sup> *μ*0 sin<sup>2</sup> 2*μ*0*ρ* ln *ρρ*<sup>2</sup>*<sup>α</sup>* ln *ρ*<sup>2</sup>*<sup>α</sup>* = 1 (ln 2)<sup>2</sup> 1 + *ρ*2*α* − <sup>2</sup>*ρ*<sup>2</sup> <sup>2</sup>(*ρ* ln *ρ*)<sup>2</sup> + 1 + *ρ*4*α* − <sup>2</sup>(*ρ*<sup>2</sup>*<sup>α</sup>* )2 <sup>2</sup>(*ρ*<sup>2</sup>*<sup>α</sup>* ln *ρ*<sup>2</sup>*<sup>α</sup>* )2 + <sup>2</sup>*ρ*<sup>2</sup>*<sup>α</sup>*+<sup>1</sup> − *ρ* − *ρ*3*α ρ* ln *ρρ*<sup>2</sup>*<sup>α</sup>* ln *ρ*<sup>2</sup>*<sup>α</sup>* (using usual trigonometric identities and formulae)

**Lemma 2.**

*where*

$$\sqrt{m}(T'\_{\mathfrak{m}} - \mu') \xrightarrow{L} \mathcal{N}(0, \sigma^2)$$

$$T\_m' = \mathbb{C}\_{2\nu}$$

*μ is the mean given by*

$$
\mu' = \rho^{2^n}
$$

*and σ*<sup>2</sup> *is the dispersion given by*

$$
\sigma^2 = \frac{\rho^{4^a} + 1 - 2(\rho^{2^a})^2}{2}
$$

**Proof.** The derivations for the proof are given in Appendix B.

Hence, assuming large sample size, central limit theorem Feller (1971) gives *C*¯ 2 *L* −→ *<sup>N</sup>*(*μ* , *σ*<sup>2</sup> *m* ) where *μ* is the mean given by *μ* = *ρ*2*α* and *σ*<sup>2</sup> is the dispersion given by *σ*<sup>2</sup> = *mV*(*C*¯ <sup>2</sup>), that is *σ*<sup>2</sup> = *ρ*4*<sup>α</sup>* <sup>+</sup>1−<sup>2</sup>(*ρ*2*<sup>α</sup>* )2 2.

**Theorem 2.**

$$\sqrt{m}(\hat{\mathfrak{a}}\_1 - \mathfrak{a}) \xrightarrow{L} N\left(0, \sigma^2 \left(\frac{\partial \mathcal{G}}{\partial \mu'}\right)^2\right)$$

*where*

$$\frac{\partial \mathcal{g}}{\partial \mu'} = -\frac{1}{(\ln 2)\rho^{2^x}} \quad \text{and}$$

$$\sigma^2 \left(\frac{\partial \mathcal{g}}{\partial \mu'}\right)^2 = \frac{1}{(\ln 2)^2 \rho^{2^x}}$$

**Proof.** We know from Lemma 2 that √ *m*(*<sup>T</sup> m* − *μ* ) → *N*(0, *σ*<sup>2</sup>) in distribution

Therefore, by delta method (given in Casella and Berger (2002)), we ge<sup>t</sup> √ *m*(*α*ˆ1 − *α*) *L* −→ 

$$\mathcal{N}\left(0, \sigma^2 \left(\frac{\partial \mathcal{g}(T'\_m)}{\partial \mu'}\right)^2\right) \text{ where}$$

$$\mathcal{g}(T'\_{\mathrm{m}}) = -\frac{\ln \mathcal{C}\_2}{\ln 2}$$

$$\frac{\partial \mathcal{g}(T'\_{\mathrm{m}})}{\partial \mu'} = -\frac{1}{\ln 2} \frac{1}{\mathcal{C}\_2} \bigg|\_{\mu'} = -\frac{1}{(\ln 2)\rho^{2^\kappa}}$$

$$\sigma^2 \left(\frac{\partial \mathcal{G}(T'\_{\mathrm{m}})}{\partial \mu'}\right)^2 = \frac{1}{(\ln 2)^2 \rho^{2^\kappa}}$$

**Lemma 3.**

$$
\sqrt{m}(T\_{\text{nr}}^{\prime\prime} - \mu^{\prime\prime}) \xrightarrow{L} N\_2(0, \Sigma^{\prime}),
$$

*where*

$$T\_{m}^{\prime\prime} = (\vec{C}\_1, \vec{C}\_2)^{\prime}$$

*μ is the mean vector given by*

$$
\mu'' = (\rho, \rho^{\widetilde{\mathbb{Z}}^\*})'
$$

*and* Σ *is the dispersion matrix given by:-*

$$
\Sigma' = \begin{pmatrix} A & B \\ -B & C \end{pmatrix},
$$

*where*

$$\stackrel{\cdots}{A} = \frac{\varrho^{2^{\mathbf{x}}} + 1 - 2\rho^2}{2}, \quad B = \frac{\varrho + \rho^{3^{\mathbf{x}}} - 2\rho^{2^{\mathbf{x}} + 1}}{2}, \quad C = \frac{\varrho^{4^{\mathbf{x}}} + 1 - 2(\rho^{2^{\mathbf{x}}})^2}{2}$$

**Proof.** The derivations for the proof are given in Appendix C.

Hence, assuming large sample size, central limit theorem Feller (1971) gives

$$(\mathcal{C}\_1, \mathcal{C}\_2)' \xrightarrow{L} N\_2(\mu'', \frac{\Sigma'}{m})$$

where *μ* is the mean vector given by *μ* = (*ρ*, *ρ*2*α* ) and Σ is the dispersion matrix given by Σ = *A B B C* where *A* = *<sup>ρ</sup>*2*<sup>α</sup>*+1−2*ρ*<sup>2</sup> 2 , *B* = *<sup>ρ</sup>*+*ρ*3*<sup>α</sup>*−2*ρ*2*α*+<sup>1</sup> 2 and *C* = *<sup>ρ</sup>*4*<sup>α</sup>*+1−<sup>2</sup>(*ρ*2*<sup>α</sup>* )2 2 .
