D. W. Loveland. A new interpretation of von Mises' concept of a random sequence. 1966.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... respectively [4] The proof of Theorem 3 is stated in terms of betting strategies and is in fact less involved than the proofs for the more specific results on stochastic sequences given by Daley, which rely on an combinatorial algorithm for constructing stochastic sets, the LMS algorithm [9]. It is known that assertion (i) in Theorem 3 cannot be strengthened to a constant bound in place of f(n) in fact, any sequence that satisfies such a constant bound is an infinite branch of a recursively enumerable tree of constant width and is thus computable. The question of whether in ....

D. W. Loveland. A new interpretation of the von Mises' concept of random sequence. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik, 12:279--294, 1966.

....that if , then every element of is Kolmogorov Loveland stochastic. Again taking 2 slowly enough that = 1, this allowed Shen to conclude that not every Kolmogorov Loveland stochastic sequence is random, thereby solving a twenty year old problem of Kolmogorov [18, 20] and Loveland [25, 26]. Theorems 7.7 and 7.2 have the following immediate consequence concerning such sequences . is a computable sequence of biases that converge to slowly enough that 2 ) 1, then DIM 1 RAND: That is, every sequence that is random with respect to such a bias sequence is an example ....

D. W. Loveland. A new interpretation of von Mises' concept of a random sequence. Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik, 12:279-294, 1966.

....are tight. 1 Introduction The relationship between prediction and gambling has been investigated for decades. In the 1950s, Shannon [21] and Kelly [10] studied prediction and gambling, respectively, as alternative means of characterizing information. In the 1960s, Kolmogorov [11] and Loveland [12] introduced a strong notion of unpredictability of in nite binary sequences, now known as Kolmogorov Loveland stocasticity. In the early 1970s, Schnorr [19, 20] proved that an in nite binary sequence is random (in the sense of Martin L of[15] if and only if no constructive gambling strategy ....

D. W. Loveland. A new interpretation of von Mises' concept of a random sequence. Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik, 12:279-294, 1966.

....stochastic. Again taking to converge to 1 2 slowly enough that P 1 i=0 ( i 1 2 ) 2 = 1, this allowed Shen 0 to conclude that not every Kolmogorov Loveland stochastic sequence is random, thereby solving a twenty year old problem of Kolmogorov [14, 16] and Loveland [21, 22]. Theorems 7.6 and 7.2 have the following immediate consequence concerning such sequences . Corollary 7.8. If is a computable sequence of biases that converge to 1 2 slowly enough that P 1 i=0 ( i 1 2 ) 2 = 1, then RAND DIM 1 RAND: That is, every sequence that is ....

D. W. Loveland. A new interpretation of von Mises' concept of a random sequence. Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik, 12:279-294, 1966.

....and betting games have been introduced. The concepts considered so far are monotone, 7 i.e. the selection of a place or the bet on the value of the sequence at this place only depends on the values of the sequence at the previous places. In the 1960s Kolmogorov [10] and, independently, Loveland [18] gave a more general interpretation of admissible selection rules than that by computable selection functions. A Kolmogorov Loveland (KL) selection rule consists of a pair of partial computable functions that, given a sequence A, depending on the information on A already obtained, first will ....

D. W. Loveland. A new interpretation of von Mises' concept of a random sequence. Z. Math. Logik Grundlagen Math. 12 (1966) 279-294.

.... A of a random sequence R is chosen according to an admissible selection rule , then the limiting frequency of 1 s in the subsequence A is exactly 1 2 : The broadest class of admissible selection rules that has been studied in this context is the class of Kolmogorov Loveland selection rules [13, 14, 19, 20]. These algorithmic rules (which are described in section 2) are more general than earlier selection rules proposed by von Mises [31] Wald [32] and Church [9] in two respects. First, given a sequence S; a Kolmogorov Loveland selection rule may choose bits from S in whatever order arises from ....

....is a constant c 2 N such that, for all w v R; d(w) 2 c : Now let n 0 2 J: Fix a prefix w v R such that d n 0 (w) 1: Then we have 2 c d n 0 (w) 2 c d(w) 2 c 1 X n=m(2c) d n (w) so n 0 m(2c) Thus J is finite. 2 The notion of Kolmogorov Loveland stochasticity was defined in [13, 14, 19, 20]; detailed discussions may be found in [29, 15, 18] A sequence is Kolmogorov Loveland stochastic if any subsequence chosen by a Kolmogorov Loveland selection rule possesses frequency stability, that is, if the proportion of 1 s in initial segments tends toward a limit of 1 2 : A ....

D. W. Loveland. A new interpretation of von Mises' concept of a random sequence. Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik, 12:279--294, 1966.

....introduced by Martin L of and Schnorr, respectively, and can be de ned by betting strategies that can be e ectively approximated in a certain sense and by computable betting strategies. In an attempt to overcome the shortcomings of Mises Church Wald random sequences, Kolmogorov [4] and Loveland [7] independently proposed to drop the monotonicity condition in the de nition of Mises Church Wald random sequences. For the moment, let a Kolmogorov Loveland selection rule be a selection rule that is partially computable but is not necessarily monotonic and call a sequence Kolmogorov Loveland ....

....S is Kolmogorov Loveland stochastic, there are in nitely many such z 0 k , hence T can not be KolmogorovLoveland stochastic. ut Shen [12] observed that the proof of Theorem 6 yields an easy proof of the separation result stated in Corollary 7. The corollary itself was proven before by Loveland [7], however his proof is considerably more involved. Corollary 7. The Kolmogorov Loveland stochastic sequences form a proper subclass of the class of Mises Wald Church stochastic sequences. Proof. The sequence T from the proof of Theorem 6 is not Kolmogorov Loveland stochastic, hence it suces to ....

D. Loveland. A new interpretation of the von Mises' concept of random sequence. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik, 12:279-294, 1966.

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D. W. Loveland. A new interpretation of von Mises' concept of a random sequence. 1966.

No context found.

D. W. Loveland. A new interpretation of von Mises' concept of a random sequence. Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik, 12:279--294, 1966.

No context found.

D. W. Loveland. A new interpretation of the von Mises' concept of random sequence. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik, 12:279-294, 1966.

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