Abstract
Suppose 0≤t 1<t 2<... are fixed points in time. At timet k , a unit with magnitudeX k and lifetimeL k enters a population or is placed into a system. Suppose that theX k 's are i.i.d. withEX 1=μ, theL k 's are i.i.d., and that theX k 's andL k 's are independent. In this paper we find conditions under which the continuous time process Avr{X k :t k ≤t<t k +L k } is almost surely convergent to μ. We also demonstrate the sharpness of these conditions.
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Rothmann, M.D., Russo, R.P. Strong consistency of the mean on randomly deleted sets. J Theor Probab 7, 119–128 (1994). https://doi.org/10.1007/BF02213362
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DOI: https://doi.org/10.1007/BF02213362