Discovering of boundedness and continuity of random fields by means of partition entropic scheme
aa r X i v : . [ m a t h . P R ] O c t DISCOVERING OF BOUNDEDNESS AND CONTINUITYOF RANDOM FIELDS BY MEANS OFPARTITION ENTROPIC SCHEME.Eugene Ostrovsky, Leonid Sirota.
Department of Mathematics, Bar-Ilan University,Ramat Gan, 52900, Israel,e-mails: [email protected];[email protected].
Abstract.
We construct a new sufficient conditions for boundedness or continuity of arbi-trary random fields relying on the so-called partition scheme, alike in the classicalmajorizing measure method.We deduce also the used in the practice (statistics, method Monte-Carlo etc.)exact exponential estimates for tail of distribution of maximum for random fieldsatisfying formulated in this report conditions.
Key words and phrases:
Tails of distributions, separable random process (r.p.)and random fields (r.f.), distance and semi-distance, metric entropy and metric en-tropy conditions, Young-Orlicz function, boundedness and continuity of randomfields almost everywhere, probabilistic module of continuity, partition scheme, sub-gaussian variables, diameter, ordinary and Grand Lebesgue norm and spaces, Orlicznorm and spaces, disjoint sets and functions, majorizing measure method.
Mathematics Subject Classification (2000): primary 60G17; secondary 60E07;60G70.
Let (
T, d ) be arbitrary compact metric space with a set T equipped with a distance(or, more generally, semi-distance) function d = d ( t, s ) , t, s ∈ T. Let also ξ = ξ ( t ) , t ∈ T be numerical valued separable stochastic continuous random process orseparable random field. Denote by the P c ( ξ ) a probability that almost every pathof ξ = ξ ( t ) is continuous: P c ( ξ ) def = P ( ξ ( · ) ∈ C ( T, d )) , (1 . c )1s well as denote by the P b ( ξ ) a probability that almost every path of ξ = ξ ( t ) isbounded: P b ( ξ ) = P (sup t ∈ T | ξ ( t ) | < ∞ ) . (1 . b )It may be interest in many applications: method Monte-Carlo, statistics etc.the concrete estimate of tail for maximum distribution P T ( u ) = P T,ξ ( u ) = P (sup t ∈ T | ξ ( t ) | > u ) , u ≥ C > , as well as the tail estimate for module of continuity of the r.f. ξ ( · ) , see [9], [10], [19],[20] etc. We will talk about the sufficient conditions imposed on the finite-dimensional distributions of the r.f. ξ ( t ) under which P c ( ξ ) = 1 or P b ( ξ ) = 1 . This problem may be named as classical. There are at last two approaches forhis solving: the so-called entropy approach, where the metric entropy is calculatedrelative the distance, more precisely, semi-distance function generated by the r.f. ξ ( t ) itself, see e.g. [13], [14], [31], [1], [3], [4], [5], [20], [23], [28]; the second one isnamed as majorizing measure approach, see [3], [32] - [35], [19] etc. Several notations, definitions and facts. A. A triplet (Ω , B , P ) , where Ω = { ω } or Ω = { x } is arbitrary set, B is non-trivial sigma-algebra of subsets Ω and P is non-zero non-negative completely additivenormed: P (Ω) = 1 measure defined on the B is called as ordinary a probabilisticspace.We denote as usually for the random variable ξ (r.v.) (i.e. measurable functionΩ → R ) its classical Lebesgue-Riesz norm by | ξ | p = [ E | ξ | p ] /p = (cid:20)Z Ω | ξ ( ω ) | p P ( dω ) (cid:21) /p , p ≥
1; (1 . L p = { ξ, | ξ | p < ∞} . B. The so-called Grand Lebesgue Space Gψ equipped with a norm || · || Gψ consists by definition on all the numerical valued random variables defined on theour probability space and having a following finite norm Gψ = { ξ, || ξ || Gψ < ∞} , || ξ || Gψ def = sup p ≥ " | ξ | p ψ ( p ) . (1 . ψ = ψ ( p ) is some continuous strictly positive function such that there existslim p →∞ ψ ( p ) = ∞ . These spaces are Banach rearrangement invariant complete spaces. The detailinvestigation of this spaces (and more general ones) see in [17], [24]. See also [6], [7],[11], [12], [14] etc. 2 n important for us fact about considered here spaces is proved in [14], [20],[22]: they coincide with some exponential Orlicz’s spaces Or (Φ ψ ) . For instance, if ψ ( p ) = ψ / ( p ) := √ p, p ≥ , then the space Gψ / consists on all the subgaussian(non-centered, in general case) r.v. Or (Φ ψ / ) for which by definition Φ ψ / ( u ) =exp( u / − . The Gaussian distributed r.v. η belongs to this space. Another example: letΩ = (0 ,
1) with usually Lebesgue measure and f / ( ω ) = q | log ω | , ω > f / (0) = 0 . It is easy to calculate using Stirling’s formula for the Gamma function: | f / | p ≍ √ p, p ∈ (1 , ∞ ) . The correspondent tail behavior: P ( f / > u ) = exp( − u ) . More detail, let the function ψ ( · ) ∈ G Ψ = G Ψ ∞ be such that the new generatedby ψ function ν ( p ) = ν ψ ( p ) := p ln ψ ( p ) , p ∈ [1 , ∞ )is convex. The r.v. η belongs to the space Gψ if and only if it belongs to the Orlicz’sspace L ( N ψ ) with the correspondent exponential continuous Young-Orlicz function N ψ ( u ) := exp (cid:16) − ν ∗ ψ (ln | u | ) (cid:17) , | u | ≥ e,N ψ ( u ) := C u , | u | < e, C e = exp (cid:16) − ν ∗ ψ (1) (cid:17) , and herewith of course both the Banach space norms: || · || L ( N ψ ) and || · || Gψ areequivalent.One can also complete characterize (under formulated here conditions) the be-longing of the non-zero r.v. ξ to the space Gψ by means of its tail behavior: ξ ∈ Gψ ⇔ ∃ K = const ∈ (0 , ∞ ) , max( P ( ξ > u ) , P ( ξ < − u )) ≤ exp (cid:16) − ν ∗ ψ (ln | u | /K ) (cid:17) , u ≥ Ke, see [14], [20], p. 33-35.The case when in (1.3) the supremum is calculated over finite interval is inves-tigated in [17], [24], [25]: G b ψ = { ξ, || ξ || G b ψ < ∞} , || ξ || G b ψ def = sup ≤ p , (1 . ψ = ψ ( p ) is continuous function in the semi-open interval 1 ≤ p < b such that lim p ↑ b ψ ( p ) = ∞ ; the case when ψ ( b − < ∞ is trivial.3e define formally in the case when b < ∞ ψ ( p ) := + ∞ for all the values p > b. An used further example: ψ ( β,b ) ( p ) def = ( b − p ) − β , ≤ p < b, β = const > G β,b ( p ) := G b ψ ( β,b ) ( p ) . C. Recall that sets A , A , A i ∈ B are named disjoint, if A ∩ A = ∅ . Thesequence of functions { h n } , n = 1 , , . . . is said to be disjoint , or more exactly pairwise disjoint, if ∀ i, j : i = j ⇒ h i · h j a.e. = 0 . (1 . { h n } is pairwise disjoint, then | X n h n | pp = X n | h n | pp , sup n | h n ( x ) | p = X n | h n ( x ) | p , p = const > . (1 . D. We denote as ordinary for any measurable set
A, A ∈ B its indicator functionby I ( A ) = I A ( ω ) . E. Let ξ = ξ ( t ) , t ∈ T be again separable random field (process) such that ∃ b = const ∈ (1 , ∞ ] , ∀ p ∈ [1 , b ) ⇒ sup t ∈ T | ξ ( t ) | p < ∞ . Then the r.f. ξ ( · ) generated the so-called natural Gψ − function by the formula ψ ( p ) = ψ ( ξ ) ( p ) def = sup t ∈ T | ξ ( t ) | p , ≤ p < b. (1 . ∀ t ∈ T ⇒ ξ ( t ) ∈ Gψ ( ξ ) and moreover sup t ∈ T || ξ ( t ) || Gψ ( ξ ) = 1 . (1 . F. Let ψ = ψ ( p ) be some function from the class Gψ b , b = const ∈ (1 , ∞ ] , suchthat all the values ξ ( t ) , t ∈ T belongs uniformly to the space Gψ.
One can supposewithout loss of generality sup t ∈ T || ξ ( t ) || Gψ = 1 . (1 . ψ ( · ) may be picked the naturalfunction for the r.f. ξ : ψ ( p ) := ψ ( ξ ) ( p ) , if of course there exists.Define by means of the function ψ ( · ) the so - called natural (bounded) distance(more precisely, semi-distance) d ψ ( t, s ) , t, s ∈ T on the set T : d ψ ( t, s ) := || ξ ( t ) − ξ ( s ) || Gψ, (1 . d ψ ( t, s ) ≤ . Denote also by D = D ψ the diameter of the set T relative the distance d ψ :4 = D ( T, d ψ ) = diam( T, d ψ ) def = sup t,s ∈ T d ψ ( t, s ) , (1 . H ( T, d ψ , ǫ ) the metric entropy of the set T relative the distance d ψ at thepoint ǫ, < ǫ < D. The so-called entropy integral has by definition a formΘ(
T, d ψ , δ ) def = 9 Z δ exp [ v ∗ (ln 2 + H ( T, d ψ , ǫ ))] dǫ, < δ ≤ D, (1 . T, d ψ , D ) def = 9 Z D exp [ v ∗ (ln 2 + H ( T, d ψ , ǫ ))] dǫ, (1 . a )where v ( y ) = ln ψ (1 /y ) , y ∈ (1 /b, v ∗ ( x ) def = inf y ∈ (1 /b, ( xy + v ( y )) . (1 . v → v ∗ is named as co-transform of Young-Fenchel, or Legendre,in contradiction to the classical Young-Fenchel transform f ∗ ( x ) def = sup y ∈ dom f ( xy − f ( y )) . (1 . T, d ψ , D ) < ∞ , then ξ ( t ) is d ψ ( · , · ) continuous almostsurely and herewith || sup t ∈ T | ξ ( t ) | || Gψ ≤ Θ( T, d ψ , D ) , (1 . P sup t ∈ T | ξ ( t ) | > u ! ≤ exp n − ν ∗ ψ [ ln ( u/ Θ( T, d ψ , D )) ] o , u ≥ . (1 . || sup t,s : d ψ ( t,s ) ≤ δ | ξ ( t ) − ξ ( s ) | || Gψ ≤ Z δ exp [ v ∗ (ln 2 + H ( T, d ψ , ǫ ))] dǫ = Θ( T, d ψ , δ ) . (1 . ω ξ,Gψ,d ψ ( δ ) def = || sup t,s : d ψ ≤ δ | ξ ( t ) − ξ ( s ) | || Gψ is named as ordinary probabilistic module of continuity for the (uniform continuous)r.f. ξ ( · ) relative the distance function d ψ . Obviously,5im δ ↓ ω ξ,Gψ,d ψ ( δ ) = 0 ⇒ P c ( ξ ) = 1 . Analogous estimates holds true is the r.f. ξ = ξ ( t ) satisfies the so-called ma-jorizing measure condition, in particular, if sup t ∈ T || ξ ( t ) || Gψ < ∞ , see [16], [32]-[35]. The following hypothesis has been formulated in the article [27], 2008 year:”Let θ = θ ( t ) , t ∈ T be arbitrary separable random field, centered: E θ ( t ) =0 , bounded with probability one: sup t ∈ T | θ ( t ) | < ∞ (mod P ) , moreover, may becontinuous, if the set T is compact metric space relative some distance.Assume in addition that for some Young (or Young-Orlicz) function Φ( · ) andcorrespondent Orlicz norm || · || Or (Φ)sup t ∈ T || θ ( t ) || Or (Φ) < ∞ . (2 . || ξ || Or (Φ) of a r.v. (measurable function) ξ isdefined as follows: || ξ || Or (Φ) = inf k,k> (cid:26)Z Ω Φ( | ξ ( ω ) | /k ) P ( dω ) ≤ (cid:27) . The Young function Φ( · ) is by definition arbitrary even convex continuousstrictly increasing on the non-negative right-hand semi-axis such thatΦ(0) = 0 , lim u →∞ Φ( u ) = ∞ . Let also Ψ( · ) be arbitrary another Young function such that lim u →∞ Ψ( u ) = ∞ , Ψ << Φ , which denotes by definition ∀ λ > ⇒ lim u →∞ Ψ( λu )Φ( u ) = 0 , (2 . << Φ implies in particular that the (unit) ball in the space Or (Ψ)is precompact set in the space Or (Φ) . Open question: there holds (or not) ” || sup t ∈ T | θ ( t ) | || Or (Ψ) < ∞ . (2 . || sup t ∈ T | θ ( t ) | || Or (Φ) < ∞ . (2 . a )6ne can formulate the analogous question replacing the Orlicz spaces by GrandLebesgue ones, see [27].The detail investigation of the theory of Orlicz’s spaces including the case ofunbounded source measure P may be found in the monographs [29], [30].This conclusions are true for the centered (separable) Gaussian fields, [3], if thefield θ ( · ) satisfies the so-called entropy or generic chaining condition [18], [20], [19],[16], [32], [33], [34]; in the case when θ ( · ) belongs to the domain of attraction of Lawof Iterated Logarithm [21] etc.Notice that if the field θ ( t ) is continuous (mod P ) and satisfies the condition(2.1), then there exists an Young function Ψ( · ) , Ψ( · ) << Φ( · ) for which the in-equality (2.3) there holds, see [18].The negative answers on these questions are obtained: in the article [26], thecase of Orlicz spaces; in [27], more general and more strictly case of Grand LebesgueSpaces. We recall briefly using further the correspondent example from the lastreport [27]. Example 2.1.1.
We choose in the sequel in this pilcrow as the capacity of compact metricspace (
T, d ) the set of positive integer numbers with infinite associated point whichwe denote by ∞ : T = { , , , . . . , ∞} . (2 . d is defined as follows: d ( i, j ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i − j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , i, j < ∞ ; d ( i, ∞ ) = d ( ∞ , i ) = 1 i , i < ∞ ; (2 . d ( ∞ , ∞ ) = 0 . The pair (
T, d ) is compact (closed) metric space and the set T has an uniquelimit point t = ∞ . For instance, lim n →∞ d ( n, ∞ ) = 0 . Let Ω = (0 ,
1) with ordinary Lebesgue measure. Let also f = f ( x ) , x ∈ Ω = (0 ,
1) be non-zero non-negative integrable function belonging to the space L . Define a following ψ − function: ν ( p ) = | f | p = (cid:20)Z | f ( x ) | p dx (cid:21) /p , ≤ p ≤ . (2 . ν ( · ) is a natural function for the function f. Evidently, ν ( · ) ∈ Gψ = Gψ (0 , . Introduce also the following numerical sequences c n := n β , β = const > , n = 2 , , . . . ; (2 . n := C ( β ) · n − β − , C ( β ) : ∞ X n =1 ∆ n = 1; a n = a ( n ) := ∞ X m = n ∆ n ; (2 . θ ( t ) = g n , n = t, t, n ∈ T, Ω = { x } ,g n ( x ) = c ( n ) f x − a ( n )∆( n ) ! I ( a ( n +1) ,a ( n )) ( x ) , x ∈ Ω , g ∞ ( x ) = 0; (2 . g ( x ) = ∞ X n =1 g n ( x ) = ∞ X n =1 c n f x − a ( n )∆( n ) ! I ( a ( n +1) ,a ( n )) ( x ) . (2 . { g n ( x ) } consists on the non-negative and disjointfunctions, thereforesup n g n ( x ) = X n g n ( x ) = g ( x ) , | sup n g n | pp = X n | g n | pp . (2 . g n are disjoint and following sup n | g n ( x ) | < ∞ almost surely.We calculate using the relations (2.7)-(2.11): | g n | pp = c p ( n ) ∆ n ν p ( p ) = C ( β ) n pβ − β − ν p ( p ) , ≤ p ≤ , (2 . p ∈ [1 , sup n | g n | pp ≤ C ( β ) ν (4) < ∞ (2 . a )or equivalently sup n | g n ( · ) | < ∞ . (2 . b )Moreover, g n → ǫ be arbitrary positive number.We get applying the estimate (2.12) at the value p = 1 and Tchebychev-Markovinequality X n P ( | g n | > ǫ ) ≤ C ( β ) X n n − β − ǫ < ∞ . Our conclusion follows immediately from the lemma of Borel-Cantelli.So, the random process θ ( t ) = g n , where n = t satisfies the condition (2.1) rel-ative the Ψ − function ψ (4) ( p ) := 1 , ≤ p ≤ d = d ( t, s ) . Let us now find the exact up to multiplicative constant expression for the naturalfunction of the r.v. sup n | g n ( x ) | as p → − . We have: | sup n | g n | | pp = X n | g n | pp = X n c p ( n ) ∆ n ν p ( p ) == C ( β ) ν p ( p ) X n n pβ − β − ∼ C ( β )4 − p ; (2 . | sup n | g n | | p ∼ C ( β )(4 − p ) − / . (2 . − function ψ ( p ) thefunction ψ (4) ( p ) , which is in turn equivalent to the following Ψ − function ψ (4) ( p ) := 1 , ≤ p < , and correspondingly to take φ ( p ) := (4 − p ) − / def = ψ (1 / , ( p ) , ≤ p < . (2 . φ ( · ) << ψ (0 , ( · ) (2 . || sup n | g n | || Gφ = ∞ , (2 . Remark 2.0.
In order to obtain the centered needed process θ ( t ) with at thesame properties, we consider the sequence ˜ g n ( x ) = ǫ ( n ) · g n ( x ) , where { ǫ ( n ) } is aRademacher sequence independent on the { g n } defined perhaps on some sufficientlyrich probability space: P ( ǫ ( n ) = 1) = P ( ǫ ( n ) = −
1) = 1 /
2; (2 . | ˜ g n ( x ) | = | g n ( x ) | , | ˜ g n | p = | g n | p (2 . { ˜ g n } is also pairwise disjoint (Rademacher’s symmetrization).This completes the grounding of using for us properties of our (counter - )example. Remark 2.1.
The constructed process θ ( t ) give us a new example of centeredcontinuous random process with relatively light tails of finite-dimensional distribu-tion, but for which the so-called entropy and generic chains series divergent. Remark 2.2.
The properties of our example remains true if we use instead thespace of continuous function C ( T, d ) arbitrary separable Banach space.
Definition 3.1.
The representation of the form T = ∪ ∞ m =1 T m , (3 . T ∼ { T m } , on the (measurable) subsets T m , not necessary to be disjoint, issaid to be a partition, or equally covering of the set T. We have for any partition 9 T ( u ) ≤ ∞ X m =1 P T m ( u ) , u > . (3 . ξ ( · ) on the arbitrary subset T m . In detail,denote: ψ m ( p ) := sup t ∈ T m | ξ ( t ) | p , (3 . ∃ b m ∈ (1 , ∞ ] , ∀ p < b m ⇒ ψ m ( p ) < ∞ . (3 . b m < ∞ ∀ p > b m ⇒ ψ ( p ) = + ∞ . Further, put v ( m ) ( y ) := ln ψ m (1 /y ) , y ∈ (0 , D m := diam( T m , d ψ m ) , (3 . Z ( m ) := Θ m ( T m , d ( ψ m ) , D m ) , (3 . Y ( { T m } , u ) def = ∞ X m =1 exp n − ν ∗ ψ m [ ln( u/Z ( m )) ] o . (3 . Theorem 3.1.
Suppose that for some partition T ∼ { T m } lim u →∞ Y ( { T m } , u ) = 0 . (3 . ξ ( t ) is bounded with probability one and moreover P (sup t ∈ T | ξ ( t ) | > u ) ≤ Y ( { T m } , u ) . (3 . Proof.
We have using (1.16) P sup t ∈ T m | ξ ( t ) | > u ! ≤ exp n − ν ∗ ψ m [ ln( u/Z ( m )) ] o , u ≥ . (3 . Corollary 3.1.
We conclude under conditions of theorem (3.1) P (sup t ∈ T | ξ ( t ) | > u ) ≤ inf { T m } Y ( { T m } , u ) def = R ( u ) , u ≥ . (3 . T ∼ { T m } of the set T. Remark 3.1.
The estimates (3.10) and (3.11) are exponential non-improvablestill in the previous entropy approach, i.e. without partition, see [20], chapters 3,4.
An example.
It is easy to see that the our example in the second sectionsatisfies all the conditions of theorem 3.1. The partition for the considered thereinr.f. is trivial. 10
Partition scheme. Main result-continuity.
The condition (3.8) guarantee us only the boundedness of almost all the paths ofthe r.f. ξ ( t ) . We will discuss in this section the sufficient conditions relative theappropriate distance for the continuity of ξ ( · ) based on the offered in this articlepartition scheme.We will follow the article of V.A.Dmitrovsky [2].Some new notations. Introduce a new ψ − function τ = τ ( p ) as follows: τ ( p ) := (cid:20) p Z ∞ u p − R ( u ) du (cid:21) /p , (4 . " E ( sup t ∈ T | ξ ( t ) | ) p /p ≤ τ ( p ) , (4 . p >
1; define b = sup { p, p > , τ ( p ) < ∞ ; then sup t || ξ ( t ) || Gτ ≤ . The correspondent bounded natural distance ρ = ρ ( t, s ) may be defined asfollows: ρ ( t, s ) := || ξ ( t ) − ξ ( s ) || Gτ. (4 . || sup t,s : ρ ( t,s ) ≤ δ | ξ ( t ) − ξ ( s ) | || Gτ ≤ Θ( T, ρ, δ ) , (4 . ω ξ,Gτ,ρ ( δ ) ≤ Θ( T, ρ, δ ) , δ ∈ (0 , diam( T, ρ )) . (4 . Proposition 4.1.
We conclude under formulated above in this section nota-tions and conditions: the r.f. ξ = ξ ( t ) is ρ ( · , · ) is uniform continuous with probabilityone. Example 4.1 - 2.1.
Let us return to the example 2.1 considered in the secondsection. We deduce after some calculations taking into account proposition 4.1 P ( sup n | g n | > u ) ≤ C ( β ) ln uu , u > e. (4 . g n is proved in the second section; we intend to provefurther the probabilistic continuity. It is sufficient to consider the unique limit point n = ∞ . Define the next Young functionΦ( u ) = e u , | u | ≤ e, Φ( u ) = u ln | u | , | u | > e. The distance function ρ ( n, m ) is here equivalent to the Orlicz’s natural distance11 ( n, m ) := || g n − g m || L (Φ)and is equivalent in turn to the source distance d ( m, n ) , m, n = 1 , , . . . , ∞ . Our statement: lim n →∞ || sup k ≥ n | g k | || L (Φ) = 0 , by virtue of proposition 4.1. References [1]
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