V. E. Stepanov. Limit distributions of certain characteristics of random mappings. Theory of Probability and Applications, 14:612--626, 1969.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Then an element of Fn is called a random mapping. Each random mapping f can be represented by a functional graph, i.e. the graph consisting of the nodes 1; 2; n, and of the edges (i; f(i) i = 1; n. Various characteristics of random mappings have been studied. See e.g. [1, 3, 4, 6, 8, 9, 10, 12]. Arney and Bender [1] examined a more general model: They considered mappings such that the number of preimages of every point lies in a given set D of nonnegative integers (with 0 2 D) or, equivalently, the degrees of the nodes of the functional graph have to be in D. Let F D n denote the set ....

V. E. Stepanov, Limit distributions of certain characteristics of random mappings, Th. Prob. Appl. 14 (1969), 612-626.

....and temperature. There is a general consensus that random transformations [8] in which images of points are i.i.d. independent and identically distributed) are the most promising phenomenological models of discretizations of dynamical systems. Further references on random transformations include [5, 15, 21]. This consensus is supported, on the one hand, by convincing heuristic arguments [7, 14] and, on the other hand, by excellent quantitative agreement between the theoretical predictions of these models and the results of extensive computer experiments, especially in situations where the ....

....this with an example. For a completely random mapping Phi on f1; 2; ng (see definition in the next section) the distribution of ff( Phi) has a limit F 1 ff as n 1 which can be calculated theoretically. This is a consequence Theorem 2. 1 below or may be proved by the methods of [21]. Heuristics suggest that the empirical distribution functions F ff ( Delta jH 1 ) and F ff ( Delta jH 2 ) should approximate this limit. Figure 2 presents graphs of these empirical distribution functions along with the prediction F 1 ff . Again, there is good agreement. The construction of ....

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Stepanov, V.E. Limit distributions of certain characteristics of random mappings. Theory Prob. Appl. 14, 612--626. 1969.

....rather than follow every particle of gas in a room, we use the simpler microscopic mathematical models of statistical physics to estimate the macroscopic characteristics of the gas such as the entropy or temperature. Different classes of random mappings [6] including mappings with attracting [26] and absorbing centres, have been used as phenomenological models with excellent success. Although the theory of random mappings is quite rich and well developed, a new role for these mappings has led to new questions. One important question arises because these mappings are, like any ....

....above to be not greater than 0:3. Thus the likely error of the first kind is of magnitude N Gamma1=2 and with overwhelming probability is bounded for a fixed large N by 2N Gamma1=2 : On the other hand, the second error is of magnitude N Gamma1=2 , as it follows from explicit formulas from [26]. This error is systematic and is positive with a coefficient depending on an unknown parameter a. Thus, using the Principle of Correspondence again, we can further sharpen the above hypothesis to Hypothesis 5.4. The moments M i;k b T (f;N) of the mapping f are close for each i; k to the ....

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V.E. Stepanov, Limit distributions of certain characteristics of random mappings, Theory Prob. Appl., 14 (1969), 612--626.

....a random mapping OE i from [ i ] to [ i ] But the cyclic decomposition of N i n N i 1 in the algorithm and the cycles of the random mapping OE i are typically quite different. Indeed, for large i , the number of cycles in OE i is close, with high conditional probability to 1 2 log i (Stepanov [18]) As for the algorithm, the number of cycles deleted in one iteration is close, on average to =2, thus bounded. 3 Proof of Theorem 1 We now introduce the generating function g n (z) E(z Xn ) z 0. Then, by the Markov property, for 1 g (z) z Gamma1 X =0 p ; g (z) 4) and g ....

V.E.Stepanov, Limit distributions of certain characteristics of random mappings, Theory of Probability and its Applications 14 (1969) 612-626.

....the following: Proposition 6 [44, 50] If V n is the length of the nth longest excursion of B away from 0 over the time interval [0; 1] then (V n ) has pd( 1 2 ; 0) distribution if B is Brownian motion; 16) V n ) has pd( 1 2 ; 1 2 ) distribution if B is Brownian bridge. 17) Stepanov [55] encountered asymptotics involving pd( 1 2 ; 1 2 ) in the study of the asymptotic distribution of the sizes of tree components in a random mapping. The connection with Brownian bridge in this setting is explained in Aldous Pitman [1] The pd(ff; 0) distribution also arises as the asymptotic ....

V.E. Stepanov. Limit distributions of certain characteristics of random mappings. Theory Probab. Appl., 14:612--626, 1969.

....for background information on random number generators. Random mappings are the subject of a vast collection of works; Mutafciev s survey [26] cites 113 references For general presentations, we direct the reader to the classic paper of Harris [19] the papers by Arney and Bender [1] and Stepanov [44]. In this area, the contribution of the Russian school which uses essentially probabilistic methods, as shown by Kolchin s book Random Mappings [24] is notable. For completeness, we mention several recent papers not referenced in [24] namely [4, 7, 11, 20, 21, 30, 32] In addition, there is ....

....on random mappings appear to have been found in the 1950 s by a variety of methods including exact enumerations, discrete probability or generating functions. The paper by Harris [19] provides a first extensive approach to problems discussed in this section. Further results are given by Stepanov [44] or Arney and Bender [1] and our presentation follows similar lines. 3.1 Direct Parameters Let [ be a parameter of functional graph (or equivalently, mapping) such as the number of connected components. We introduce the quantities n = X 2Fn [ and 4(z) X n0 n z n n ; ....

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V. E. Stepanov. Limit distributions of certain characteristics of random mappings. Theory of Probability and Applications, 14:612--626, 1969.

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V. E. Stepanov. Limit distributions of certain characteristics of random mappings. Theory of Probability and Applications, 14:612--626, 1969.

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