**Appendix C**

**Proof of Lemma 3.** This proof follows simply by putting *μ*0 = 0 in Equations (A1), (A2), (A3), (A6) and (A10) in the proof for Lemma 1.

**Figure A1.** Histogram of wrapped log-ratio transformed gold price data (in US dollars) with wrapped stable density.

**Figure A2.** Histogram of wrapped log-ratio transformed silver price data(in US dollars)with wrapped stable density.

**Figure A3.** Histogram of wrapped log-ratio transformed gold price data(in INR)with wrapped stable density.

**Figure A4.** Histogram of wrapped log-ratio transformed silver price data (in INR) with wrapped stable density.

**Figure A5.** Histogram of wrapped log-ratio transformed Box and Jenkins data with wrapped stable density.
