This paper presents a critique of expected utility theory as a descriptive model of decision making under risk, and develops an alternative model, called prospect theory. Choices among risky prospects exhibit several pervasive effects that are inconsistent with the basic tenets of utility theory. In particular, people underweight outcomes that are merely probable in comparison with outcomes that are obtained with certainty. This tendency, called the certainty effect, contributes to risk aversion in choices involving sure gains and to risk seeking in choices involving sure losses. In addition, people generally discard components that are shared by all prospects under consideration. This tendency, called the isolation effect, leads to inconsistent preferences when the same choice is presented in different forms. An alternative theory of choice is developed, in which value is assigned to gains and losses rather than to final assets and in which probabilities are replaced by decision weights. The value function is normally concave for gains, commonly convex for losses, and is generally steeper for losses than for gains. Decision weights are generally lower than the corresponding probabilities, except in the range of low probabilities. Overweighting of low probabilities may contribute to the attractiveness of both insurance and gambling.

We study random towers that are suitable to analyse the statistics of slowly mixing random systems. We obtain upper bounds on the rate of quenched correlation decay in a general setting. We apply our results to the random family of Liverani-Saussol-Vaienti maps with parameters in [α 0 , α 1 ] ⊂ (0, 1) chosen independently with respect to a distribution ν on [α 0 , α 1 ] and show that the quenched decay of correlation is governed by the fastest mixing map in the family. In particular, we prove that for every δ > 0, for almost every ω ∈ [α 0 , α 1 ] Z , the upper bound n 1− 1 α 0 +δ holds on Date: January 30, 2018. 1991 Mathematics Subject Classification. Primary 37A05, 37E05. Key words and phrases. Random dynamical systems, slowly mixing systems, quenched decay of correlations.

We study a class of random transformations built over finitely many intermittent maps sharing a common indifferent fixed point. Using a Young-tower technique, we show that the map with the fastest relaxation rate dominates the asymptotics. In particular, we prove that the rate of correlation decay for the annealed dynamics of the random map is the same as the sharp rate of correlation decay for the map with the fastest relaxation rate.Date: October 1, 2018. 1991 Mathematics Subject Classification. Primary 37A05, 37E05. Key words and phrases. Interval maps with a neutral fixed point, intermittency, random dynamical systems, decay of correlations.1 Annealed dynamics refers to the randomized dynamics, averaged over the randomizing space, see Subsection 2.2 and Theorem 2.3. This should be contrasted with the notion of quenched dynamics, the behaviour of the system with one random choice of the randomizing sequence. The term almost sure dynamics is also used to refer to quenched dynamics. arXiv:1305.6588v3 [math.DS]

We study random transformations built from intermittent maps on the unit interval that share a common neutral fixed point. We focus mainly on random selections of Pomeu-Manneville-type maps Tα using the full parameter range 0 < α < ∞, in general. We derive a number of results around a common theme that illustrates in detail how the constituent map that is fastest mixing (i.e. smallest α) combined with details of the randomizing process, determines the asymptotic properties of the random transformation. Our key result (Theorem 1.1) establishes sharp estimates on the position of return time intervals for the quenched dynamics. The main applications of this estimate are to limit laws (in particular, CLT and stable laws, depending on the parameters chosen in the range 0 < α < 1) for the associated skew product; these are detailed in Theorem 3.2. Since our estimates in Theorem 1.1 also hold for 1 ≤ α < ∞ we study a piecewise affine version of our random transformations, prove existence of an infinite (σ−finite) invariant measure and study the corresponding correlation asymptotics. To the best of our knowledge, this latter kind of result is completely new in the setting of random transformations.Date: August 11, 2016. 1991 Mathematics Subject Classification. Primary 37A05, 37E05. Key words and phrases. Interval maps with a neutral fixed point, intermittency, random dynamical systems, decay of correlations, Central Limit Theorem, stable laws.

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