We investigate notions of randomness in the space C½2 N of non-empty closed subsets of f0, 1g N . A probability measure is given and a version of the Martin-Lo¨f test for randomness is defined. Å 0 2 random closed sets exist but there are no random Å 0 1 closed sets. It is shown that any random closed set is perfect, has measure 0, and has box dimension log 2 ð4=3Þ. A random closed set has no n-c.e. elements. A closed subset of 2 N may be defined as the set of infinite paths through a tree and so the problem of compressibility of trees is explored. If T n ¼ T \ f0, 1g n , then for any random closed set ½T where T has no dead ends, KðT n Þ ! n À Oð1Þ but for any k, KðT n Þ 2 nÀk þ Oð1Þ, where K() is the prefix-free complexity of 2 f0, 1g à .

We model a selection process arising in certain storage problems. A sequence (X 1, · ··, Xn ) of non-negative, independent and identically distributed random variables is given. F(x) denotes the common distribution of the Xi′s. With F(x) given we seek a decision rule for selecting a maximum number of the Xi′s subject to the following constraints: (1) the sum of the elements selected must not exceed a given constant c > 0, and (2) the Xi′s must be inspected in strict sequence with the decision to accept or reject an element being final at the time it is inspected. We prove first that there exists such a rule of threshold type, i.e. the ith element inspected is accepted if and only if it is no larger than a threshold which depends only on i and the sum of the elements already accepted. Next, we prove that if F(x) ~ Axα as x → 0 for some A, α> 0, then for fixed c the expected number, En (c), selected by an optimal threshold is characterized by Asymptotics as c → ∞and n → ∞with c/n held fixed are derived, and connections with several closely related, well-known problems are brought out and discussed.

We prove that there exists a noncomputable c.e. real which is low for weak 2-randomness, a definition of randomness due to Kurtz, and that all reals which are low for weak 2-randomness are low for Martin-Löf randomness.

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