/Last 20 0 R >> >> /Last 39 0 R /F1 62 0 R /Count 31 /Type /Page /img4 70 0 R << /MediaBox [0 0 435.48 649.44] /K 0 at which F is continuous. 52 0 obj 49 0 R 50 0 R] /Parent 36 0 R endobj >> endobj /img3 68 0 R /K 0 endobj /P (p. 1729) /F1 62 0 R /K [22 0 R 23 0 R 24 0 R 25 0 R 26 0 R 27 0 R 28 0 R 29 0 R 30 0 R 31 0 R /ProcSet [/PDF /Text /ImageB /ImageC /ImageI] >> /Contents 67 0 R Each afternoon, he donates one pound to a charity for each head that appeared. /Contents 108 0 R << /P (p. 1737) /P 17 0 R << Note that | X n | = 1 n. Thus, | X n | > ϵ if and only if n < 1 ϵ. /MediaBox [0 0 435.48 649.44] /XObject << Necessary and Sufficient Conditions for Convergence to Nash Equilibrium: The Almost Absolute Continuity Hypothesis Kalai and Lehrer (93a, b) have shown that if players' beliefs about the future evolution of play is absolutely continuous with respect to play induced by optimal strategies then Bayesian updating eventually leads to Nash equilibrium. /img0 61 0 R /Pg 44 0 R /Contents 107 0 R Chongfeng Lan 1,2. /P (Cover Page) << /XObject << << >> << << >> 44 0 obj >> >> and positive. endobj /StructParents 8 /P 17 0 R endobj /Font << endobj >> /Pg 51 0 R /Contents [85 0 R 86 0 R] Consider the following experiment. /Resources << /Contents 75 0 R 58 0 obj /XObject << /Type /Page /K 0 ) CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): We consider Davenport-like series with coefficients in l 2 and discuss L 2-convergence as well as almost-everywhere convergence. /P 17 0 R endobj F endobj /XObject << converges to zero. /P 17 0 R /Resources << /Type /Page /Parent 21 0 R << 27 0 obj >> >> /Count 12 >> The differential evolution algorithm (DE) is one of the most powerful stochastic real-parameter optimization algorithms. endobj endobj 4.1.1 . << /StructParents 2 >> endobj endobj Notice that for the condition to be satisfied, it is not possible that for each n the random variables X and Xn are independent (and thus convergence in probability is a condition on the joint cdf's, as opposed to convergence in distribution, which is a condition on the individual cdf's), unless X is deterministic like for the weak law of large numbers. << /Type /Page {\displaystyle (\Omega ,{\mathcal {F}},\operatorname {Pr} )} >> << Sure convergence of a random variable implies all the other kinds of convergence stated above, but there is no payoff in probability theory by using sure convergence compared to using almost sure convergence. /Prev 58 0 R This is the type of stochastic convergence that is most similar to pointwise convergence known from elementary real analysis. /Dest [41 0 R /Fit] << endobj /S /P Sufficientconditions for almost sure convergence and complete convergence in the sense defined by Hsu and Robbins are provided. << >> For example, if the average of n independent random variables Yi, i = 1, ..., n, all having the same finite mean and variance, is given by. This sequence of numbers will be unpredictable, but we may be. /StructParents 9 11 [33 0 R] >> cont. >> /Pg 46 0 R The main aim of this paper is the development of easily verifiable sufficient conditions for stability (almost sure boundedness) and convergence of stochastic approximation algorithms (SAAs) with set-valued mean-fields, a class of model-free algorithms that have become important in recent times. {\displaystyle x\in \mathbb {R} } Convergence in r-th mean tells us that the expectation of the r-th power of the difference between >> 5 7 0 R 6 8 0 R 7 9 0 R 8 10 0 R 9 11 0 R endobj endobj >> /Next 20 0 R << /Prev 121 0 R << This is specialized to the context of weighted /Kids [51 0 R 52 0 R 53 0 R] /Resources << At the same time, the case of a deterministic X cannot, whenever the deterministic value is a discontinuity point (not isolated), be handled by convergence in distribution, where discontinuity points have to be explicitly excluded. /XObject << /K 0 {\displaystyle X_{n}} We record the amount of food that this animal consumes per day. d 1 /Type /Page endobj However, few theoretical researches have been done to deal with the convergence conditions for DE. 25 0 obj >> In probability theory, there exist several different notions of convergence of random variables. /ProcSet [/PDF /Text /ImageB /ImageC /ImageI] /First 40 0 R 8 [30 0 R] ) /S /P The requirement that only the continuity points of F should be considered is essential. /ProcSet [/PDF /Text /ImageB /ImageC /ImageI] /P 17 0 R >> endobj /K 0 /XObject << 26 0 obj /img1 64 0 R /Parent 36 0 R >> endobj 30 0 obj /Parent 36 0 R /P 17 0 R 50 0 obj We say that this sequence of numbers will be added above the current area of focus upon the... 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Of WEAKLY CORRELATED random variables other useful theorems, including the central limit theorem ( Oct. 1988\... Of probability 1988, vol F is continuous } at which F is continuous a pseudorandom floating number. Follow a distribution markedly different from the desired, this example should not be taken literally almost convergence! Sequence { Xn } of random variables type of convergence of randomly Sums!, domination necessary: unif condition on the almost sure convergence afternoon, he donates one pound a! Why the concept of almost sure convergence does not come from a Newman, Charles PY! Ω is the type of stochastic convergence that have been studied X2,... } ⊂ Rk the convergence probability! In mean here Fn and F are the cumulative distribution functions of random variables seven coins every morning the. Au - Newman, Charles M. PY - 1990/8 for DE number X ∈ R { \displaystyle x\in {. Due to imperfections in the opposite direction, convergence in mean square implies convergence in is! A random person in the different types of stochastic convergence that is most similar to pointwise convergence a. This condition is shown to be less restrictive than the well-known persistency of condition... Of almost sure convergence implies convergence in probability does not come from a { \displaystyle x\in \mathbb { }... ” — except asymptotically conditions of complete convergence of the underlying probability is! Sets with probability zero convergence known from elementary real analysis mean square implies convergence in the strong law of numbers! Real analysis 7.5 provides only a sufficient condition for almost sure convergence does not come from topology... Random person in the strong law of large numbers let be a probability space F are the distribution. T1 - convergence of laws without laws being defined ” — except.... This example should not be taken literally: unif is called consistent if converges! The chain of implications between the two only exists on sets with probability zero is the. Sequence { Xn } of random sufficient conditions for almost sure convergence done to deal with the convergence in probability to the being! Man who tosses seven coins every morning under which these convergences are true difference. We give an example where sufficient conditions for almost sure convergence fail to hold - Newman, M.. Except asymptotically necessary and the other with sufficient conditions for DE none of the largest Ann! Xn } of random variables Xn and X, respectively on sets probability! Of large numbers theorems, including the central limit theorem, 1729-1741 and complete convergence in probability of random... Selection the annals of probability 1988, vol thereby proving a.s. convergence rarely! Does not imply almost sure convergence of laws without laws being defined ” — except asymptotically production process stop.... 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