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On the Suprema of Bernoulli Processes (Slides) On the boundedness of Bernoulli processes Rafał Latała (based on the joint work with Witold Bednorz) University of Warsaw Berkeley, October 1 2013 Introduction One of the fundamental issues of probability theory is the study of suprema of stochastic processes. Besides various practical motivations it is closely related to such theoretical problems as regularity of sample paths of stochastic processes, convergence of othogonal and random series, estimates of norms of random vectors and random matrices, limit theorems for random vectors and empirical processes, combinatorial matching theorems. In particular in many situations one needs to estimate the quantity E supt∈T Xt , where (Xt )t∈T is a stochastic process. The modern approach to this problem is based on chaining techniques, present already in the works of Kolmogorov and successfully developed over the last 40 years. To avoid measurability problems one may either assume that T is countable or define E sup Xt := sup E sup Xt , t∈T F t∈F where the supremum is taken over all finite sets F ⊂ T . Gaussian Processes Let (Gt )t∈T be a centered Gaussian process and g(T ) := E sup Gt . t∈T In this case the boundedness of the process (which by the concentration properties of Gaussian processes is equivalent to the condition g(T ) < ∞) is related to the geometry of the metric 2 1/2 space (T , d), where d(t, s) := (E(Gt − Gs ) ) . R.Dudley’67 derived an upper bound for g(T ) in terms of entropy numbers: Z ∞ g(T ) ≤ L log1/2 N(T , d, r)dr, 0 where L is a universal constant and N(T , d, r) is the minimal number of balls with radius r that cover T . Dudley’s bound may be reversed for stationary processes, but not in general. Majorizing Measure Theorem Two-sided bound for g(T ) was found by X.Fernique’74 (upper bound) and M.Talagrand’87 (lower bound) in terms of majorizing measures. Let Z ∞ 1/2 1 γ2(T ) := inf sup log dx, t∈T 0 µ(B(t, x)) where the infimum is taken over all probability measures on T and B(t, ε) denotes the closed ball in (T , d) with radius x, centered at t. Theorem (Fernique-Talagrand) There exists a universal constant L < ∞ such that for all centered Gaussian processes, 1 γ (T ) ≤ g(T ) ≤ Lγ (T ). L 2 2 In particular g(T ) < ∞ if and only if γ2(T ) < ∞. Majorizing measures without measures In general finding a majorizing measure in a concrete situation is a highly nontrivial task. Talagrand proposed a more combinatorial approach to this problem. An increasing sequence (An)n≥0 of partitions of the set T is called 2n admissible if A0 = {T } and |An| ≤ Nn := 2 . Talagrand showed that the quantity γ2(T ) is equivalent to ∞ X n/2 inf sup 2 ∆(An(t)), t∈T n=0 where the infimum runs over all admissible sequences of partitions. Here An(t) is the unique set in An which contains t and ∆(A) denotes the diameter of the set A. So to bound a supremum of some Gaussian process one may either construct a measure or build a partition. Bernoulli Processes Any separable Gaussian process has a canonical Karhunen-Loève P type representation ( i≥1 ti gi )t∈T , where g1, g2,... are i.i.d. standard normal Gaussian N (0, 1) r.v’s and T is a subset of `2. Another fundamental class of processes is obtained when in such a sum one replaces the Gaussian r.v’s (gi ) by independent random signs. Let (εi )i≥1 be a Bernoulli sequence, i.e. a sequence of i.i.d. symmetric r.v.’s taking values ±1. For any t ∈ `2 the series P i≥1 ti εi converges a.s. and we may define a Bernoulli process X 2 ti εi , T ⊂ ` . t∈T i≥1 It is natural to ask when the Bernoulli process is a.s. bounded or, equivalently, when X b(T ) := E sup ti εi < ∞? t∈T i≥1 Two easy upper bounds The first bound follows by the uniform bound P P | i ti εi | ≤ i |ti | = ktk1, so that b(T ) ≤ sup ktk1. t∈T Another bound is based on the domination by the canonical P Gaussian process Gt := i ti gi . Assuming independence of (gi ) and (εi ), Jensen’s inequality implies X X g(T ) = E sup ti gi = E sup ti εi |gi | t∈T i t∈T i s 2 ≥ sup X t ε |g | = b(T ). E i i E i π t∈T i Bernoulli Conjecture 1 2 l Obviously also if T ⊂ T1 + T2 = {t + t : t ∈ Tl } then b(T ) ≤ b(T1) + b(T2), hence n rπ o b(T ) ≤ inf sup ktk1 + g(T2): T ⊂ T1 + T2 t∈T1 2 n o ≤ inf sup ktk1 + Lγ2(T2): T ⊂ T1 + T2 . t∈T1 It was open for about 25 years (under the name of Bernoulli conjecture) whether the above estimate may be reversed. Next theorem provides a positive solution. Theorem (Bednorz, L.’13+) For any set T ⊂ `2 with b(T ) < ∞ we may find a decomposition T ⊂ T + T with sup P |t | ≤ Lb(T ) and g(T ) ≤ Lb(T ). 1 2 t∈T1 i≥1 i 2 Kwapień’s Conjecture Problem Let (F , k k) be a normed space and (ui ) be a sequence of vectors P in F such that the series i≥1 ui εi converges a.s. Does there exist a universal constant L and a decomposition ui = vi + wi such that X X X X sup vi ηi ≤ LE ui εi and E wi gi ≤ LE ui εi ? ηi =±1 i≥1 i≥1 i≥1 i≥1 The positive solution of the Bernoulli Conjecture shows that the answer is positive for F = `∞, in general one may only assume that F is a subspace of `∞. The difficulty here is that our proof gives very little additional information about the decomposition, in particular there is no reason for sets T1 and T2 to be contained in the linear space spanned by the index set T . Another bound for Bernoulli processes p 1/p For a random variable X and p > 0 we set kXkp := (E|X| ) . Corollary Suppose that (Xt )t∈T is a Bernoulli process with b(T ) < ∞. Then there exist t1, t2,... ∈ `2 such that T − T ⊂ conv{tn : n ≥ 1} and kXtn klog(n+2) ≤ Lb(T ) for all n ≥ 1. The converse statement easily follows from the union bound and Chebyshev’s inequality. Indeed, suppose that n T − T ⊂ conv{t : n ≥ 1} and kXtn klog(n+2) ≤ M. Then X P sup Xs ≥ uM ≤ P sup Xtn ≥ uM ≤ P(Xtn ≥ ukXtn klog(n+2)) s∈T −T n≥1 n≥1 ≤ X u− log(n+2) n≥1 and integration by parts easily yields for any t0 ∈ T , b(T ) = E sup(Xt − Xt0 ) = E sup(Xt−t0 ) ≤ E sup Xs ≤ LM. t∈T t∈T s∈T −T Proof of the Gaussian case Proof of the lower bound for g(T ) is based on two facts - Gaussian concentration and Sudakov minoration. Theorem (Gaussian concentration) Let (Gt )t∈T be a centered Gaussian process, then for any u > 0, u2 P sup Gt − g(T ) ≥ u) ≤ 2 exp − 2 , t∈T 2σ 2 2 where σ := supt∈T E|Gt | . Theorem (Sudakov minoration) Suppose that t1,..., tm are such that kGtl − Gtl0 k2 ≥ a for all l 6= l0. Then 1 p E sup Gtl ≥ a log m. l≤m L Concentration and minoration for Bernoulli processes Theorem (Talagrand’88) Let T ⊂ `2 be such that b(T ) < ∞, then for any u > 0, u2 sup X t ε − b(T ) ≥ u) ≤ L exp − , P i i Lσ2 t∈T i≥1 2 2 where σ := supt∈T ktk2. Theorem (Talagrand’93) 2 Suppose that vectors t1,..., tm ∈ ` (I) and numbers a, b > 0 satisfy ∀l6=l0 ktl − tl0 k2 ≥ a and ∀l ktl k∞ ≤ b. Then 1 n a2 o sup X t ε ≥ min aplog m, . E l,i i L b l≤m i∈I About the proof of BC It is however not an easy task to combine in a right way concentration and minoration properties of Bernoulli processes. Our proof builds on many ideas developed over the years by Michel Talagrand. One of difficulties lies in the fact that there is no direct way of producing a decomposition t = t1 + t2 for t ∈ T such that 1 2 supt∈T kt k1 ≤ Lb(T ) and γ2({t : t ∈ T }) ≤ Lb(T ). Following Talagrand we connect decompositions of the set T with suitable sequences of its admissible partitions (An)n≥0. To each set A ∈ An we will assign an integer jn(A) and a point πn(A) ∈ T . The main novelty in the next statement is the introduction of sets In(A). Partitions and Bernoulli decomposition Theorem Suppose that (√An)n≥0 is an admissible sequence of partitions s.t. −j (T ) i) kt − sk2 ≤ Mr 0 for t, s ∈ T, 0 ii) if n ≥ 1, An 3 A ⊂ A ∈ An−1 then either 0 0 a) jn(A) = jn−1(A ) and πn(A) = πn−1(A ) or 0 0 b) jn(A) > jn−1(A ), πn(A) ∈ A and X 2 −2jn(A) n −2jn(A) min{(ti − πn(A)i ) , r } ≤ M2 r for all t ∈ A, i∈In(A) where for any t ∈ A, −jk (t) In(A) = In(t) := i : |πk+1(t)i −πk (t)i | ≤ r for 0 ≤ k ≤ n−1 .
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