} are monotonically non-decreasing in k in hazard rate ordering.

Probability II (MATH 2647) M15 The next result is very important for applications. Its proof uses the intrinsic monotonicity structure of the de nition (1.1). Then: a) If P k P (A k) < 1 , … If A Bthen P(A) P(B). Moreover, the methods used yield a stronger result, which does not follow from the optimum property

Let A = \ n 1 [1 k = n A k be the event that in nitely many of the A n occur.

Such monotonicity, when true, is usually easy to prove. Prove the following properties of every probability measure. Contradicting the lemma of monotonicity of Expected Shortfall. Similarly, the critical temperature of an Ising model cannot decrease if the intensity of any pair-interaction is increased. Strict monotonicity, on the other hand, presents new difficulties.

Showing some problems with monotonicity of risk measures. (a) Monotonicity. probability ratio tests within their own class-the uniqueness and the restricted optimum property-are immediate consequences of the monotonicity property, it seems worth-while to prove the latter independently. Monotonicity of throughput in non-Markovian networks - Volume 26 Issue 1 - Pantelis Tsoucas, Jean Walrand b Lemma 1.6 (Borel-Cantelli lemma) . (c) Continuity from below: If Ai" A, that is, A1 A2 :::and [iAi = A, then P(Ai) " P(A). (b) Sub-additivity. O.H. Remark. For example, the critical probability of a percolation process cannot increase if new edges are added to the lattice. The paper shows also that the lemma used in the literature to prove monotonicity of Expected Shortfall is not truth and we will prove the lemma with the op- posite relation. In this paper we prove for a number of distributions that the probability for the value of the sum of the first k (but not before) of i.i.d.r.v. (d) Continuity from above: If Ai # A, that is, A1 A2 :::and \iAi = A, then P(Ai) # P(A). Can one please let me know about the following question: Using monotonicity or calculus prove that a number c can be found between a and b such that 3c^{2}=a^{2}+b^{2} Thanks. If A [iAi then P(A) P i P(Ai). Value of the paper – Mathematical proofs in the field of risk measurement. The main contributions of the present paper are to prove monotonicity and bounds for the exponent \(d_\gamma \) of Theorem ... Alternatively, it can be defined in terms of the return probability for random walk on random planar maps, in which case it was proven to be equal to 2 for all of the planar maps considered in the present paper in . to exceed a given value A is monotonically increasing in the range k < k * (or k < k * + 1 ) where k * = max k such that kμ ≤A.