Precise rates in the law of logarithm for the moment convergence of i.i.d. random variables

Let be a sequence of i.i.d. random variables with EX=0 and EX²=σ²<∞. Set , Mn=maxk?n|Sk|, n?1. Let r>1, then we obtain

     

Moderate deviations and Erdös-Rényi-Shepp laws for weighted sums

Let X₀,X₁,… be i.i.d. random variables with E(X₀)=0, E(X²₀)=1 and E(exp{tX₀})<∞ for any |t|<t₀. We prove that the weighted sums V(n)=∑_j=0^∞a_j(n)X_j, n?1 obey a moderately large deviation principle if the weights satisfy certain regularity conditions. Then we prove a new version of the Erdös-Rényi-Shepp laws for the weighted sums.     

The inverse M-matrix problem

One aspect of the inverse M-matrix problem can be posed as follows. Given a positive n × n matrix A=(a_ij) which has been scaled to have unit diagonal elements and off-diagonal elements which satisfy 0 < y ? a_ij ? x < 1, what additional element conditions will guarantee that the inverse of A exists and is an M-matrix? That is, if A^?1=B=(b_ij), then b_ii> 0 and b_ij ? 0 for i≠_j. If n=2 or x=y no further conditions are needed, but if n ? 3 and y < x, then the following is a tight sufficient condition. Define an interpolation parameter s via x²=sy+(1?s)y²; then B is an M-matrix if s^?1 ? n?2. Moreover, if all off-diagonal elements of A have the value y except for a_ij=a_jj=x when i=n?1, n and 1 ? _j ? n?2, then the condition on both necessary and sufficient for B to be an M-matrix.     

A class of Fibonacci-type sequences

Let {L_n:n ? 1} be a sequence of the form

where {a_j} and {b_j} are positive integers, and e=max_i,j{a_i, b_j}. A necessary and sufficient condition on the integers {a_j} and {b_j} is given so that, for all choices of positive initial values L₁, L₂,…,L_e,L_n=Σ^p_j=1L_n?aj for all large enough n.     

A generalization of the Strassen converse

This note is devoted to a generalization of the Strassen converse. Let gn:R^∞→[0,∞], n?1 be a sequence of measurable functions such that, for every n?1, and for all x,y∈R^∞, where 0<C<∞ is a constant which is independent of n. Let be a sequence of i.i.d. random variables. Assume that there exist r?1 and a function ?:[0,∞)→[0,∞) with limt→∞?(t)=∞, depending only on the sequence such that lim supn→∞gn(X₁,X₂,…)=?(Er|X|) a.s. whenever Er|X|<∞ and EX=0. We prove the converse result, namely that lim supn→∞gn(X₁,X₂,…)<∞ a.s. implies Er|X|<∞ (and EX=0 if, in addition, lim supn→∞gn(c,c,…)=∞ for all c≠0). Some applications are provided to illustrate this result.     

Precise Rates in the Law of Iterated Logarithm for the Moment of I.I.D. Random Variables

Let{X,Xn;n≥1} be a sequence of i,i.d, random variables, E X = 0, E X^2 = σ^2 〈 ∞.Set Sn=X1＋X2＋…＋Xn,Mn=max k≤n│Sk│,n≥1.Let an=O（1/loglogn）.In this paper,we prove that,for b〉-1,lim ε→0 →^2（b＋1）∑n=1^∞ （loglogn）^b/nlogn n^1/2 E{Mn-σ（ε＋an）√2nloglogn}＋σ2^-b/（b＋1）（2b＋3）E│N│^2b＋3∑k=0^∞ （-1）k/（2k＋1）^2b＋3 holds if and only if EX=0 and EX^2=σ^2〈∞.     

Essential self-adjointness of Schrödinger-type operators

The essential self-adjointness of the strongly elliptic operator L = ∑_j,k=1ⁿ (?_j ? ib_j(x)) a_jk(x)(?_k ? ib_k(x)) + q(x) acting on C₀^∞(Rⁿ) is considered, where the matrix (a_jk) is real and symmetric, b_j and q are real, a_jk and b_j are sufficiently smooth, and q?L_loc². It has been shown by Ural'ceva and also Laptev that if q is bounded below and n ? 3 the minimal operator may not be self-adjoint if the principal coefficients rise too rapidly. Thus a theorem of Weyl for ordinary differential operators does not extend to partial differential operators. In this paper it is shown that if q is bounded below and if the principal coefficients are “well behaved” within a sequence of closed shells which go to infinity, then the minimal operator is self-adjoint. It is also shown that a number of results which were known to be true when q is sufficiently smooth may be extended to include the case where q?L_loc². The principal tools used are a distribution inequality due to Tosio Kato and a general maximum principle due to Walter Littman.     

Utilizing Shelve Slots: Sufficiency Conditions for Some Easy Instances of Hard Problems

In this paper we present a minimal set of conditions sufficient to assure the existence of a solution to a system of nonnegative linear diophantine equations. More specifically, suppose we are given a finite item set U = {u₁, u₂, . . . , u_k} together with a "size" v_i ≡ v(u_i) ∈ Z⁺, such that v_i ≠ v_j for i ≠ j, a "frequency" a_i ≡ a(u_i) ∈ Z⁺, and a positive integer (shelf length) L ∈ Z⁺ with the following conditions: (i) L = ∏ⁿ_j=1p_j(p_j ∈ Z⁺ ∀j, p_j ≠ p_l for j ≠ l) and v_i = ∏ _j∈Aip_j, A_i ⊆ {l, 2, . . . , n} for i = 1, . . . , n; (ii) (A_i\{⋂^k_j=1A_j}) ∩ (A_l\{⋂^k_j=1A_j}) = ⊘∀i ≠ l. Note that v_i|L (divides L) for each i. If for a given m ∈ Z⁺, ∑ⁿ_i=1a_iv_i = mL (i.e., the total size of all the items equals the total length of the shelf space), we prove that conditions (i) and (ii) are sufficient conditions for the existence of a set of integers {b₁₁, b₁₂, . . . , b_1m, b₂₁, . . . , b_n1, . . . , b_nm}⊆ N such that ∑^m_j=1b_ij = a_i, i = 1, . . . , k, and ∑^k_i=1b_ijv_i = L, j =1, . . . , m (i.e., m shelves of length L can be fully utilized). We indicate a number of special cases of well known NP-complete problems which are subsequently decided in polynomial time.     

Relative probabilities of majority winners under partial information

Suppose the only observable characteristic of each of n random variables that is uniformly distributed on the six rankings of objects in a three-element set is its first-ranked object. Let ?(n₁,n₂,n₃) be the probability that one of the three objects has majorities over the other two within the rankings when n_j of the n rankings have the jth object in first place. It is assumed that n is odd, so that ?(n₁,n₂,n₃)=1 only if n_j≥(n+1)/2 for some j.It is shown that $\text{?(a+1,b,c)<?(a,b+1,c) if a <b,a ≤ c ≤ b+1}$ and max {b,c}≤(n?1)/2. It follows from this that ? is minimized for fixed n if and only if n_j?n_k≤1 for all j,k? {1,2,3}. However, ? does not necessarily increase when two of its arguments get farther apart. For example, ?(b,b,3)>?(b?1,b+1,3) for b≥28, and ?(b,b,2b?1)>?(b?1,b+1,2b?1) for b≥12.     

Precise asymptotics in Chung’s law of the iterated logarithm

Let X, X1, X2,... be i.i.d, random variables with mean zero and positive, finite variance σ^2, and set Sn = X1 ＋... ＋ Xn, n≥1. The author proves that, if EX^2I{｜X｜≥t} = 0（（log log t）^-1） as t→∞, then for any a〉-1 and b〉 -1,lim ε↑1/√1＋a（1/√1＋a-ε）b＋1 ∑n=1^∞（logn）^a（loglogn）^b/nP{max κ≤n｜Sκ｜≤√σ^2π^2n/8loglogn（ε＋an）}=4/π（1/2（1＋a）^3/2）^b＋1 Г（b＋1）,whenever an = o（1/log log n）. The author obtains the sufficient and necessary conditions for this kind of results to hold.     

Prophet Regions for Independent [0, 1]-Valued Random Variables with Random Discounting

Abstract

Let X ₁, X ₂… and B ₁, B ₂… be mutually independent [0, 1]-valued random variables, with EB _j  = β > 0 for all j. Let Y _j  = B ₁ … sB _j?1 X _j for j ≥ 1. A complete comparison is made between the optimal stopping value V(Y ₁,…,Y _n ):=sup{EY _τ:τ is a stopping rule for Y ₁,…,Y _n } and E(max _1≤j≤n Y _j ). It is shown that the set of ordered pairs {(x, y):x = V(Y ₁,…,Y _n ), y = E(max _1≤j≤n Y _j ) for some sequence Y ₁,…,Y _n obtained as described} is precisely the set {(x, y):0 ≤ x ≤ 1, x ≤ y ≤ Ψ _{n, β}(x)}, where Ψ _{n, β}(x) = [(1 ? β)n + 2β]x ? β^?(n?2) x ² if x ≤ β ^n?1, and Ψ _{n, β}(x) = min _j≥1{(1 ? β)jx + β ^j } otherwise. Sharp difference and ratio prophet inequalities are derived from this result, and an analogous comparison for infinite sequences is obtained.     

Asymptotic Behavior of the Maximum of Sums of I.I.D. Random Variables along Monotone Blocks

Let {X_i, Y_i}_i=1,2,... be an i.i.d. sequence of bivariate random vectors with P(Y₁ = y) = 0 for all y. Put M_n(j) = max_0≤k≤n-j (X_k+1 + ... X_k+j)I_k,j, where I_k,k+j = I{Y_k+1 < ⋯ < Y_k+j} denotes the indicator function for the event in brackets, 1 ≤ j ≤ n. Let L_n be the largest index l ≤ n for which I_k,k+l = 1 for some k = 0, 1, ..., n - l. The strong law of large numbers for “the maximal gain over the longest increasing runs,” i.e., for M_n(L_n) has been recently derived for the case where X₁ has a finite moment of order 3 + ε, ε > 0. Assuming that X₁ has a finite mean, we prove for any a = 0, 1, ..., that the s.l.l.n. for M(L_n - a) is equivalent to EX ₁ ^3+a I{X₁ > 0} < ∞. We derive also some new results for the a.s. asymptotics of L_n. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 311, 2004, pp. 179–189.     

On convergence rates and the expectation of sums of random variables

Let X_i be iidrv's and S_n=X₁+X₂+…+X_n. When EX²₁<+∞, by the law of the iterated logarithm $\text{(S}{}_{\mathrm{n}}\text{?α}{}_{\mathrm{n}}\text{)}\text{(n log n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{→0 a.s.}$ for some constants α_n. Thus the r.v. $\text{Y=sup}{}_{\mathrm{n?1}}\text{[|S}{}_{\mathrm{n}}\text{?α}{}_{\mathrm{n}}\text{|?(δn log n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{]}{}^{\mathrm{+}}$ is a.s.finite when δ>0. We prove a rate of convergence theorem related to the classical results of Baum and Katz, and apply it to show, without the prior assumption EX²₁<+∞ that EY^h<+∞ if and only if $\text{E}\text{|X}{}_1\text{|}{}^{\mathrm{2+h}}\text{[log|X}{}_1\text{|]}{}^{-1}\text{<+∞}$ for 0<h<1 and δ> $\text{h}\text{E}\text{(X}{}_1\text{?}\text{E}\text{X}{}_1\text{)}{}^2$, whereas $\text{E}\text{Y}{}^{\mathrm{h}}\text{=+∞}$ whenever h>0 and $\text{0<δ<h}\text{E}\text{(X}{}_1\text{?}\text{E}\text{X}{}_1\text{)}{}^2$.     

A NONCLASSICAL LAW OF THE ITERATED LOGARITHM FOR I.I.D. SQUARE INTEGRABLE RANDOM VARIABLES. II

Let {X _n , n≥1} be a sequence of i.i.d. random variable with EX ₁=0 and E X ₁ ²=1 and let {b _n , n≥1} be a sequence of positive constants monotonically approaching infinity such that lim inf _n→∞ b _n /log log n=1. It is proved that lim sup _n→∞ ∑ _i=1 ⁿ X _i /√2nb _n =1 almost certainly thereby extending the work of Klesov and Rosalsky[4] Klesov, O. and Rosalsky, A. 2001. A Nonclassical Law of the Iterated Logarithm for I.I.D. Square Integrable Random Variables. Stoch. Anal. Appl., 19: 627–641. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar] to a much larger class of sequences {b _n , n≥1}. The tools employed in the argument are results of Bulinskii[1] Bulinskii, A.V. 1977. On Normalization in the Law of the Iterated Logarithm. Teor. Veroyatnost. i Primen., 22: 407–409. In Russian, English translation in Theory Probab. Appl., 22 (1977), 398–399 [Google Scholar] and Feller[2] Feller, W. 1943. The General Form of the So-Called Law of the Iterated Logarithm. Trans. Am. Math. Soc., 54: 373–402.  [Google Scholar] and the Strassen[5] Strassen, V. 1964. An Invariance Principle for the Law of the Iterated Logarithm. Z. Wahrsch. Verw. Gebiete, 3: 211–226. [Crossref] , [Google Scholar] strong invariance principle.     

A note on the functional law of iterated logarithm for maxima of Gaussian sequences

Let {X_n, n ≥ 1} be a real-valued stationary Gaussian sequence with mean zero and variance one. Let M_n = max{X_t, i ≤ n} and H_n(t) = (M_[nt] ? b_n)a_n^?1 be the maximum resp. the properly normalised maximum process, where $\text{c}{}_{\mathrm{n}}\text{= (2}\text{log}\text{n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, $\text{a}{}_{\mathrm{n}}\text{=}\text{(}\text{log log}\text{n)}\text{c}{}_{\mathrm{n}}$ and $\text{b}{}_{\mathrm{n}}\text{= c}{}_{\mathrm{n}}\text{?}\text{1}\text{2}\text{(}\text{log}\text{(4π}\text{log}\text{n))}\text{c}{}_{\mathrm{n}}$. We characterize the almost sure limit functions of (H_n)_n≥3 in the set of non-negative, non-decreasing, right-continuous, real-valued functions on (0, ∞), if r(n) (log n)^3?Δ = O(1) for all Δ > 0 or if r(n) (log n)^2?Δ = O(1) for all Δ > 0 and r(n) convex and fulfills another regularity condition, where r(n) is the correlation function of the Gaussian sequence.     

Limit laws for record values

{X_n,n?1} are i.i.d. random variables with continuous d.f. F(x). X_j is a record value of this sequence if X_j>max{X₁,…,X_j?1}. Consider the sequence of such record values {X_Ln,n?1}. Set R(x)=-log(1?F(x)). There exist B_n > 0 such that . in probability (i.p.) iff i.p. iff $\text{{R(kx)?R(x)}}\text{R}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(kx)}$ → ∞ as x→∞ for all k>1. Similar criteria hold for the existence of constants A_n such that X_Ln?A_n → 0 i.p. Limiting record value distributions are of the form N(-log(-logG(x))) where G(·) is an extreme value distribution and N(·) is the standard normal distribution. Domain of attraction criteria for each of the three types of limit laws can be derived by appealing to a duality theorem relating the limiting record value distributions to the extreme value distributions. Repeated use is made of the following lemma: If $\text{P}\text{{X}{}_{\mathrm{n}}\text{?x}=1?e}{}^{\mathrm{-x}}\text{,x?0}$, then X_Ln=Y₀+…+Y_n where the Y_j's are i.i.d. and $\text{P}\text{{Y}{}_{\mathrm{j}}\text{?x}=1?e}{}^{\mathrm{-x}}$.     

The Dirichlet problem for the Kohn Laplacian on the Heisenberg group,I

For (x,y,t)∈$\text{R}$ⁿ × $\text{R}$ⁿ × $\text{R}$, denote $\text{X}{}_{\mathrm{j}}\text{=}\text{?}\text{?x}{}_{\mathrm{j}}\text{+ 2y}{}_{\mathrm{j}}\text{?}\text{?t}\text{, y}{}_{\mathrm{j}}\text{=}\text{?}\text{?y}{}_{\mathrm{j}}\text{? 2x}{}_{\mathrm{j}}\text{?}\text{?t}$ and $\text{L}{}_{\mathrm{\alpha }}\text{=?}\text{1}\text{4}\text{∑}\text{j=1}\text{n}\text{X}{}_{\mathrm{j}}{}^2\text{+ Y}{}_{\mathrm{j}}{}^2\text{+}\text{?}\text{?t}$. When α = n ? 2q, $\text{L}$_a represents the action of the Kohn Laplacian □_b on q-forms on the Heisenberg group. For ?n < α < n, we construct a parametrix for the Dirichlet problem in smooth domains D near non-characteristic points of ?D. A point w of ?D is non-characteristic if one of X₁,…, X_n, Y₁,…, Y_n is transverse to ?D at w. This yields sharp local estimates in the Dirichlet problem in the appropriate non-isotropic Lipschitz classes. The main new tool is a “convolution calculus” of pseudo-differential operators that can be applied to the relevant layer potentials, for which the usual asymptotic composition formula is false. Characteristic points are treated in Part II.     

On constant discrete programming problems

Let H be a subset of the set S_n of all permutations

$\text{1}\text{2}\text{???}\text{n}\text{s(1)}\text{s(2)}\text{???}\text{s(n)}$

C=6c_ij6 a real n?n matrix L_c(s)=c_1s(1)+c_2s(2)+???+c_ns(n) for s ? H. A pair (H, C) is the existencee of reals a₁,b₁,a₂,b₂,…a_n,b_n, for which c_ij=a₁+b_j if (i,j)?D(H), where D(H)={(i,j):(?h?H)(j=h(i))}.For a pair (H,C) the specifity of it is proved in the case, when H is either a special cyclic class of permutations or a special union of cyclic classes. Specific pairs with minimal sets H are in some sense described.