R'enyi A., Probability Theory, North-Holland, Amsterdam, 1970.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....analytically, the devil staircase behavior. To be more realistic, one has to consider the effect of nonhyperbolicity. Chaotic saddles in nonhyperbolic systems typically possess multifractality. In this case, D 0 D 1 is no longer valid, and it is necessary to use the dimension spectrum D q [35,63 65] to characterize the multifractality. Here we address the following question: does the devil staircase behavior still persist when the dynamics is nonhyperbolic and multifractal We are not able to answer this question analytically and, therefore, our approach will be to perform detailed numerical ....

....the fractal dimension spectrum, one utilizes a grid of boxes of size # and compute the natural measure i contained in each box. It is known that the PIM triple algorithm can typically yield the natural measure of the chaotic saddle [37] The dimension spectrum D q is then defined as follows [35,63 65]: D q lim ##0 i ln # , 21) where K K(#) is the total number of boxes with i 0. The dimension spectrum D q characterizes the fractal structure of the natural measure at different scales. In particular, smaller and smaller scales are characterized as q is increased. Amongst the ....

A. Renyi, Probability Theory, North-Holland, Amsterdam, 1970.

....them, where E[ denotes the expectation value. The basic idea of this approach is to find transformations # 0 (x 0 ) and # 1 (x 1 ) such that the absolute value of the correlation coe#cient between the transformed variables is maximized. More specifically, the so called maximal correlation [14] #(x 0 , x 1 ) sup # # 0 ,# # 1 R(# # 0 (x 0 ) # # 1 (x 1 ) 2) is calculated. The supremum is taken over all Borel measurable functions # # 0 and # # 1 . The functions # 0 (x 0 ) and # 1 (x 1 ) for which the supremum is attained are called optimal transformations. Generalizing the ....

A. Renyi. Probability Theory. Akademiai Kiado, Budapest, 1970.

....is perhaps best introduced using the concepts of generalized probability distributions and generalized random variables, which are extensions of the corresponding ordinary notions to random experiments that cannot always be observed. This presentation follows R enyi s original work [R en61, R en70] Consider a discrete probability space over Omega and let Omega 1 2 2 Omega with 1 ] 0. 1 and P define a generalized discrete probability space that differs from a probability space only by the fact thatP[ Omega 1 ] 1 is possible. A random variable X 1 defined on a generalized ....

Alfred R'enyi, Probability theory, North-Holland, Amsterdam, 1970.

....uniformly for some C 0and# =max( arg z j ) See [5] for details. We now prove the claims concerning (5) Since f j has a power series with nonnegative coe#cients: i) The first part of (c) holds. ########### ###### ## ############# # ####### ### # (ii) By Renyi s number 2 on p. 341 of [23], the first part of (b) holds. iii) f hasapowerserieswithnonnegativecoe#cientsa(k)and #(f ) A =#(f) Since #(f) ZZ , it follows from (iii) that (u) f ( u ) if and only if u = u and so the proof of (c) is complete. The second half of (b) follows from Lemma 6 in ....

A. Renyi, Probability Theory, North-Holland, Amsterdam (1970).

....it simply fills out with more data points from the larger, combined trace. The Shannon entropy can increase in this circumstance, even though the small spikes are at a place that makes them a continuation of existing PDF structure. To overcome this difficulty we propose the use of Renyi entropy [29], a generalization of the Shannon entropy, defined as: Kq (x) 1 log 2 p i (2) The parameter q specifies the order of the Renyi entropy. In the limit as q 1 the Renyi entropy converges to the Shannon entropy (i.e. limq#1 Kq (x) H(x) Renyi entropy shares many properties with ....

A. Renyi and L. Vekerdi. Probability Theory. North-Holland, Amsterdam, 1970.

.... with an n dimensional 4 standard normal distribution (i.e. each coordinate r i is an independent random variable with standard normal distribution) It can be seen that r is spherically symmetric, namely the direction specified by the vector r 2 R n is uniformly distributed (see for instance [Ren70]) In the random hyperplane rounding technique a random vector r of the above distribution is chosen 1 and the vectors v 1 : v n are partitioned into two sets according to the sign of the inner product hv i ; ri. That is, a cut (U; V n U) is defined by the set U = fi j hv i ; ri 0g. 2 ....

A. Renyi. Probability theory. Elsevier, New York, 1970.

....with probability 1, to the true probability value as the length of the sequences increases. More generally, this statement holds if the sites evolve identically and with limits on the degree of pairwise correlation between states at di#erent sites, as allowed by Bernstein s Theorem (see Renyi, [8]) In either situation we have the following result. COROLLARY 2 A computationally e#cient and statistically consistent algorithm to reconstruct unrooted phylogenetic trees from aligned sequence data satisfying the i.i.d. or weaker) assumption described above (as well as (A1) A2) is the ....

A. Renyi, "Probability Theory", North Holland Publishing, Amsterdam, 1970.

....r i are independent random variables, each component having the standard normal distribution. It is easy to verify that this distribution is spherically symmetric, in that the direction specified by the vector r is uniformly distributed. Refer to Feller [15, v. II] Knuth [29, v. 2] and Renyi [36] for further details about the higher dimensional normal distributions. Subsequently, the phrase random d dimensional vector will always denote a vector chosen from the d dimensional standard normal distribution. A crucial property of the normal distribution which motivates its use in our ....

....distributions. Subsequently, the phrase random d dimensional vector will always denote a vector chosen from the d dimensional standard normal distribution. A crucial property of the normal distribution which motivates its use in our algorithm is the following theorem paraphrased from Renyi [36] (see also Section III.4 of Feller [15, v. II] Theorem 7.2 (Theorem IV.16.3 [36] Let r = r 1 ; r n ) be a random n dimensional vector. The projections of r onto two lines 1 and 2 are independent (and normally distributed) if and only if 1 and 2 are orthogonal. Alternatively, ....

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A. Renyi. Probability Theory. Elsevier, New York, 1970.

.... normal distribution N(0; 1) The size of the projection of v on r can be bounded as follows : Pr[j(v; r)j ffikvk] Pr[jr 1 j ffi] 1 Gamma ffi: Pr[ v; r) 2 [0; ffi]kvk] Pr[r 1 2 [0; ffi] 2 [ffi=8; ffi=2] Where ffi 2 [0; 1] Proof : As the distribution of r is spherically symmetric ([Ren70] Theorem IV.16.3) we may assume that v is the vector (kvk; 0; 0) thus (v; r) is exactly r 1 kvk. 2 14 Lemma 4.6 Let (X; Y ) be the partition obtained after the random hyperplane rounding technique. Let i be an all bad vertex of degree d. With probability at least ffi it is the case that ....

A. Renyi. Probability theory. Elsevier, New York, 1970.

....x point of G, the restriction that the initial distribution must assign positive probability to any matching implies the Markov chain M( must converge to both u and . As a Markov chain has a unique limit distribution we get u = Using standard Markov chain techniques (see for example [Ren70]) for every i we get jj i 1 u jj jj i u jj where jj jj denotes the variation distance de ned as the maximum of the di erence of probabilities over all subsets. 6 This technique can be extended to other mating operations for which the mating veri es the symmetry property P (x; y; z) ....

A. Renyi. Probability Theory. North-Holland, Amsterdam, 1970.

....point of G, the restriction that the initial distribution #must assign positive probability to any matching implies the Markov chain M(#) must converge to both # u and #. As a Markov chain has a unique limit distribution we get # u = #. Using standard Markov chain techniques (see for example [Ren70]) for every i we get # i 1 # u # i # u where denotes the variation distance defined as the maximum of the di#erence of probabilities over all subsets. 6 This technique can be extended to other mating operations for which the mating verifies the symmetry ....

A. Renyi. Probability Theory. North-Holland, Amsterdam, 1970.

....of which this may be viewed as a prototype, can also be considered from the point of view of information theory. Let the uncertainty or entropy associated with a probability vector p be defined by the familiar formula associated with Wiener and Shannon (Shannon Weaver 1949, Kullback 1959, Renyi 1970, Barron 1986, Cover Thomas 1991) U = Gamma n X i=1 p i log 2 p i : 3.1) This quantity ranges from 0 if we have complete information about the system (p i = 1 for a single i) to log 2 (n ) if we have no information (p i = 1=n for all i) Conversely, the information associated with p ....

Renyi, A. 1970 Probability theory. Amsterdam: North-Holland.

....in this section and those that follow apply to random heuristic search in general, and speak therefore to both microscopic and macroscopic behavior. 18 4. 2 Fundamental Concepts Standard terminology from probability theory is used in this section (in the context of Markov chains for example, see [4,13] for the definition of a closed set of states, an absorbing state, etc. An instance of random heuristic search is called: ffl Ergodic, if some some power of the transition matrix Q is positive. ffl Absorbing, if, in the Markov chain which represents it, every closed set of states contains an ....

....function. 3 If RHS is ergodic, absorbing, regular, focused, hyperbolic, or normal, then both and G are also called ergodic, absorbing, regular, focused, hyperbolic, or normal (respectively) The following observations are, given the previous definitions, standard results from probability theory [4,13]. When RHS is ergodic, every state must be visited infinitely often. Moreover, in that case T = lim k 1 T Q k exists and is independent of the initial population distribution . The rows of Q 1 are each T , which is a left eigenvector of Q corresponding to the simple and maximal ....

A. Renyi, Probability Theory (North-Holland, Amsterdam, 1970).

....that the initial distribution Delta must assign positive probability to any matching implies the Markov chain M( Delta) must converge to both Pi u and Delta. As a Markov chain has a unique limit distribution we get Pi u = Delta. Using standard Markov chain techniques (see for example [6]) for every i we get jj Pi i 1 Gamma Pi u jj jj Pi i Gamma Pi u jj where jj Delta jj denotes the variation distance defined as the maximum of the difference of probabilities over all subsets. This technique can be extended to other mating operations for which the mating verifies the ....

A. Renyi. Probability Theory. North-Holland, Amsterdam, 1970.

.... is E x 2 2D 2 S [H(X 1 jX 2 = x 2 ) We sometimes refer to the entropy H(X) of random variable X, which is equal to H(D) We sometimes refer to the conditional entropy H(X 1 jX 2 ) of X 1 conditioned on X 2 , which is equal to H(D 1 jD 2 ) The following variant definition of entropy is due to [Renyi : 70] Definition 2.2.2 (Renyi entropy) Let D be a distribution on a set S. The Renyi entropy of D is HRen (D) Gamma log(Pr[X = Y ] where X 2D S and Y 2D S are independent. There are distributions that have arbitrarily large entropy but have only a couple of bits of Renyi entropy. Proposition ....

Renyi, A., Probability Theory, North-Holland, Amsterdam, 1970.

.... We have implemented a computer code that reconstructs the time series in a E dimensional embedding space (see Section 2) computes the distances d j,k,E (n) between the reference point j and its kth nearest neighbour and approximates the fractal dimension with the so called information dimension (Renyi, 1970), i.e D 2 ln(n k) ln 1 m S m j51 d j,k,E (n) 6) Since the estimation of the pointwise dimension turns out to be computationally unstable due to the intrinsic lacunarity of fractal sets, the stability of the results must be checked for a large range of embedding dimensions and ....

Renyi, A. (1970) Probability Theory. Amsterdam: North-Holland.

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R'enyi A., Probability Theory, North-Holland, Amsterdam, 1970.

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A. Renyi. Probability Theory. North-Holland, 1970.

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Renyi, A., Probability Theory, North-Holland, Amsterdam, 1970.

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A. Renyi. Probability Theory. North Holland, Amsterdam, 1971.

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A. Renyi. Probability Theory, Noth-Holland, Amsterdam, 1970

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A. Renyi. Probability theory. Springer Verlag, 1970.

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A. Renyi. Probability Theory. North-Holland,Amsterdam, 1970.

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Alfred Renyi. Probability Theory. Akademiai Kiado, Budapest.

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A. Renyi. Probability theory. Elsevier, New York, 1970.

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