Expected difference of order statistics in terms of hazard rate
If the hazard rate $ \frac{ F'(x) }{ 1-F(x) } $ is increasing (in $x$), then $ \mathbb E\, ( X_{n:n} - X_{n-1:n} ) $ is decreasing (in $n$), and moreover, completely monotone.
math.PR↗
arXiv subjects
Boris Tsirelson
Publications and source records attributed to Boris Tsirelson.
Can each number be specified by a finite text?
Contrary to popular misconception, the question in the title is far from simple. It involves sets of numbers on the first level, sets of sets of numbers on the second level, and so on, endlessly. The infinite hierarchy of the levels involved distinguishes the concept of "definable number" from such notions as "natural number", "rational number", "algebraic number", "computable number" etc. (Explanatory essay for non-experts.)
math.LO↗
Linear response and moderate deviations: hierarchical approach. V
The Moderate Deviations Principle (MDP) is well-understood for sums of independent random variables, worse understood for stationary random sequences, and scantily understood for random fields. Here it is established for some planary random fields of the form $ X_t = ψ(G_t) $ obtained from a Gaussian random field $ G_t $ via a function $ ψ$, and consequently, for zeroes of the Gaussian Entire Function. Version 2: Appendix "Reader's guide to parts I-V" added. Minor changes, as follows. Formulations corrected: (2.1), 3.12(b), 4.5. Proofs corrected: 2.5, 2.6, 3.12, 3.16, 3.19, 4.13, 4.14, 4.15, 5.14. Formulations clarified: 2.10, 2.11 (former 2.9, 2.10), 3.17, 5.14, 5.15, 5.23. Clarifications/copyedit: remarks 2.11 (former 2.10), 5.17; pages 5, 11, 15, 21, 22, 36, 39, 40, 41, 42; refs [5], [6].
math.PR↗
Linear response and moderate deviations: hierarchical approach. IV
The Moderate Deviations Principle (MDP) is well-understood for sums of independent random variables, worse understood for stationary random sequences, and scantily understood for random fields. Here it is established for splittable random fields integrated against test functions. Version 2: minor changes. Interval [0,1] replaced with [0,1) for better compatibility with Part V. Copyedit pages 3, 4, 6, 16, 22, 23.
math.PR↗
Linear response and moderate deviations: hierarchical approach. II
The Moderate Deviations Principle (MDP) is well-understood for sums of independent random variables, worse understood for stationary random sequences, and scantily understood for random fields. An upper bound for a new class of random fields is obtained here by induction in dimension. Version 3. Sect 1. Stationarity, being not essential in the proofs, is removed from the definitions and the main result formulation. Sect. 2. $[C,2C]$ instead of $[C_1,2C_1]$ before Prop. 2.6; $ a\ge1$ instead of $a\ge C/C_1$ in the last proof; Remark 2.5 added; supremum over shifts in (2.2) (formerly (2.3)); "centered" instead of "CMS". Cosmetic changes: indexing of leaks; Remark 2.11. Sect. 3. Cosmetic change: semicolon after the second display of the proof of Lemma 3.9. References: "response", not "responce".
math.PR↗
Linear response and moderate deviations: hierarchical approach. III
The Moderate Deviations Principle (MDP) is well-understood for sums of independent random variables, worse understood for stationary random sequences, and scantily understood for random fields. Here it is established for splittable random fields. Version 2. Adapted to new Version 3 of [2] arXiv:1706.00991. Terminology: "CMS random fields". Numbers of items in [2] updated. References: "response", not "responce". Otherwise unchanged.
math.PR↗
Linear response and moderate deviations: hierarchical approach. I
This is basically a polished presentation for Sections 1,2 of arXiv:0801.1050. The Moderate Deviations Principle (MDP) is well-understood for sums of independent random variables, worse understood for stationary random sequences, and scantily understood for random fields. Here it is established for a new class of random processes. The approach is promising also for random fields.
math.PR↗
Noise as a Boolean algebra of $σ$-fields
A noise is a kind of homomorphism from a Boolean algebra of domains to the lattice of $σ$-fields. Leaving aside the homomorphism we examine its image, a Boolean algebra of $σ$-fields. The largest extension of such Boolean algebra of $σ$-fields, being well-defined always, is a complete Boolean algebra if and only if the noise is classical, which answers an old question of J. Feldman.
math.PR↗
Random compact set meets the graph of nonrandom continuous function
On the plane, every random compact set with almost surely uncountable first projection intersects with a high probability the graph of some continuous function. Implication: every black noise over the plane fails to factorize when the plane is split by such graph.
math.PR↗
Random Stopping Times in Stopping Problems and Stopping Games
Three notions of random stopping times exist in the literature. We introduce two concepts of equivalence of random stopping times, motivated by optimal stopping problems and stopping games respectively. We prove that these two concepts coincide and that the three notions of random stopping times are equivalent.
math.PR↗
Moderate deviations on different scales: no relations
An example of a discrete-time stationary random process whose sums follow the normal approximation within a given part of the region of moderate deviations, but violate it outside this part.
math.PR↗
From uniform renewal theorem to uniform large and moderate deviations for renewal-reward processes
A uniform key renewal theorem is deduced from the uniform Blackwell's renewal theorem. A uniform LDP (large deviations principle) for renewal-reward processes is obtained, and MDP (moderate deviations principle) is deduced under conditions much weaker than existence of exponential moments.
math.PR↗
Noise as a Boolean algebra of sigma-fields. III. An old question of Jacob Feldman
The noise-type completion C of a noise-type Boolean algebra B is generally not the same as the closure of B. As shown in Part I (Introduction, Theorem 2), C consists of all complemented elements of the closure. It appears that C is the whole closure if and only if B is classical (as defined in Part II, Sect. 1a), which is the main result of this Part III.
math.PR↗
Noise as a Boolean algebra of sigma-fields. II. Classicality, blackness, spectrum
Similarly to noises, Boolean algebras of sigma-fields can be black. A noise may be treated as a homomorphism from a Boolean algebra of regular open sets to a Boolean algebra of sigma-fields. Spectral sets are useful also in this framework.
math.PR↗
Noise as a Boolean algebra of sigma-fields. I. Completion
Nonclassical noises over the plane (such as the black noise of percolation) consist of sigma-fields corresponding to some planar domains. One can treat less regular domains as limits of more regular domains, thus extending the noise and its set of sigma-fields. The greatest extension is investigated in a new general framework.
math.PR↗
Subproduct systems of Hilbert spaces: dimension two
A subproduct system of two-dimensional Hilbert spaces can generate an Arveson system of type I1 only. All possible cases are classified up to isomorphism. This work is triggered by a question of Bhat: can a subproduct system of n-dimensional Hilbert spaces generate an Arveson system of type II or III? The question is still open for n=3,4,...
math.OA↗
Graded algebras and subproduct systems: dimension two
Objects dual to graded algebras are subproduct systems of linear spaces, a purely algebraic counterpart of a notion introduced recently in the context of noncommutative dynamics (Shalit and Solel, Bhat and Mukherjee). A complete classification of these objects in the lowest nontrivial dimension is given in this work, triggered by a question of Bhat.
math.RA↗
On automorphisms of type II Arveson systems (probabilistic approach)
A counterexample to the conjecture that the automorphisms of an arbitrary Arveson system act transitively on its normalized units.
math.OA↗