{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,9]],"date-time":"2026-10-09T15:53:18Z","timestamp":1791561198463,"version":"4.3.4"},"reference-count":12,"publisher":"Institute of Mathematical Statistics","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Ann. Appl. Probab."],"published-print":{"date-parts":[[2016,8,1]]},"abstract":"<jats:p>From its beginning, there have been attempts by physicists to formulate quantum mechanics without requiring the use of wave functions. An interesting recent approach takes the point of view that quantum effects arise solely from the interaction of finitely many classical \u201cworlds.\u201d The wave function is then recovered (as a secondary object) from observations of particles in these worlds, without knowing the world from which any particular observation originates. Hall, Deckert and Wiseman [Phys. Rev. X 4 (2014) 041013] have introduced an explicit many-interacting-worlds harmonic oscillator model to provide support for this approach. In this note, we provide a proof of their claim that the particle configuration is asymptotically Gaussian, thus matching the stationary ground-state solution of Schr\u00f6dinger\u2019s equation when the number of worlds goes to infinity. We also construct a Markov chain based on resampling from the particle configuration and show that it converges to an Ornstein\u2013Uhlenbeck process, matching the time-dependent solution as well.<\/jats:p>","DOI":"10.1214\/15-aap1154","type":"journal-article","created":{"date-parts":[[2016,9,1]],"date-time":"2016-09-01T11:57:42Z","timestamp":1472731062000},"page":"2540-2555","source":"Crossref","is-referenced-by-count":6,"title":["Convergence of empirical distributions in an interpretation of quantum mechanics"],"prefix":"10.1214","volume":"26","author":[{"given":"Ian W.","family":"McKeague","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Bruce","family":"Levin","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"108","reference":[{"key":"12","unstructured":"van der Vaart, A. W. (1998). <i>Asymptotic Statistics. 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Springer, Berlin.","DOI":"10.1007\/978-3-319-00327-6_3"},{"key":"3","doi-asserted-by":"publisher","unstructured":"Goldstein, L. and Reinert, G. (1997). Stein\u2019s method and the zero bias transformation with application to simple random sampling. <i>Ann. Appl. Probab.<\/i> <b>7<\/b> 935\u2013952.","DOI":"10.1214\/aoap\/1043862419"},{"key":"2","doi-asserted-by":"publisher","unstructured":"Goldstein, L. (2004). Normal approximation for hierarchical structures. <i>Ann. Appl. Probab.<\/i> <b>14<\/b> 1950\u20131969.","DOI":"10.1214\/105051604000000440"},{"key":"5","doi-asserted-by":"crossref","unstructured":"Hall, M. J. W., Deckert, D.-A. and Wiseman, H. M. (2014). Quantum phenomena modeled by interactions between many classical worlds. <i>Phys. Rev. X<\/i> <b>4<\/b> 041013.","DOI":"10.1103\/PhysRevX.4.041013"},{"key":"6","doi-asserted-by":"publisher","unstructured":"Holland, P. (2005). Computing the wavefunction from trajectories: Particle and wave pictures in quantum mechanics and their relation. <i>Ann. Physics<\/i> <b>315<\/b> 505\u2013531.","DOI":"10.1016\/j.aop.2004.09.008"},{"key":"9","doi-asserted-by":"crossref","unstructured":"Poirier, B. (2010). Bohmian mechanics without pilot waves. <i>Chem. Phys.<\/i> <b>370<\/b> 4\u201314.","DOI":"10.1016\/j.chemphys.2009.12.024"},{"key":"10","doi-asserted-by":"crossref","unstructured":"Schiff, J. and Poirier, B. (2012). Communication: Quantum mechanics without wavefunctions. <i>J. Chem. Phys.<\/i> <b>136<\/b> 031102.","DOI":"10.1063\/1.3680558"},{"key":"11","doi-asserted-by":"publisher","unstructured":"Sebens, C. T. (2015). Quantum mechanics as classical physics. <i>Philos. 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