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Limiting distributions of randomly accelerated motions. (English) Zbl 0927.60033
Lith. Math. J. 37, No. 3, 219-229 (1997) and Liet. Mat. Rink. 37, No. 3, 295-308 (1997).
Summary: The process \(\{X(t);t>0\}\), representing the position of a uniformly accelerated particle (with Poisson-paced) changes of its acceleration, is studied. It is shown that the distribution of \(X(t)\) (suitably normalized), conditionally on the number \(n\) of changes of acceleration, tends in distribution to a normal variate as \(n\) goes to infinity. The asymptotic normality of the unconditional distribution of \(X(t)\) for large values of \(t\) is also shown. The study of these limiting distributions is motivated by the difficulty of evaluating exactly the conditional and unconditional probability laws of \(X(t)\). In fact, the results obtained permit us to give useful approximations of the probability distributions of the position of the particle.
MSC:
60F05 Central limit and other weak theorems
60J20 Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
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References:
[1] P. Billingsley,Convergence of Probability Measures, Wiley, New York (1968). · Zbl 0172.21201
[2] Y. S. Chow and H. Teicher,Probability Theory, 2nd edn, Springer-Verlag, Berlin (1978).
[3] P. L. Conti and E. Orsingher, On the distribution of the position of a randomly accelerated particle, submitted for publication (1996). · Zbl 0923.60059
[4] S. K. Foong, First passage time, maximum displacement and Kac’s solution of the telegraph equation,Phys. Rev. A,46, 707–710 (1992). · doi:10.1103/PhysRevA.46.5296
[5] A. V. Glušak and E. Orsingher, General solution of a hyperbolic equation, emerging in the analysis of a motion with random acceleration, to appear inProbability Theory and Mathematical Statistics (1997).
[6] S. Goldstein, On diffusion by discontinuous movements and the telegraph equation,Q. J. Mech. Appl. Math.,4, 129–156 (1951). · Zbl 0045.08102 · doi:10.1093/qjmam/4.2.129
[7] L. Holst and I. S. Rao, Asymptotic spacings theory with applications to the two-sample problem,Can. J. Statistics,9, 79–89 (1981). · Zbl 0477.62028 · doi:10.2307/3315298
[8] M. Kac, A stochastic model related to telegraph’s equation,Rocky Mountain J. Math.,4, 497–509 (1974). · Zbl 0314.60052 · doi:10.1216/RMJ-1974-4-3-497
[9] M. Kelbert and E. Orsingher, On the equation of telegraph type with varying coefficients describing a randomly accelerated motion,Probl. Inf. Trans.,30, 99–103 (1994). · Zbl 0917.35068
[10] R. Z. Khasminskii, On stochastic processes defined by differential equations with a small parameter,Theory Probab. Appl.,9, 211–228 (1966). · Zbl 0168.16002 · doi:10.1137/1111018
[11] F. Proshan and R. Pyke, Tests for monotone failure rate, in:Proceedings of the 5th Berkeley Symposium on Mathematical Statistics and Probability, Vol. 3 (1965), pp. 293–312.
[12] R. Pyke, Spacings,J. Roy. Statist. Soc. Ser. B,27, 395–449 (1965).
[13] A. Rényi, On the theory of order statistics,Acta Math. Hungar.,IV, 191–227 (1953).
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