Convergence in the pth-mean and some Weak Laws of Large Numbers for weighted sums of random elements in separable normed linear spaces

Theorem. Let X_n, n ≥ 1, be a sequence of tight random elements taking values in a separable Banach space B such that |X_n|, n ≥ 1, is uniformly integrable. Let a_nk, n ≥ 1, k ≥ 1, be a double array of real numbers satisfying Σ_{k ≥ 1} |a_nk| ≤ Γ for every n ≥ 1 for some positive constant Γ. Then Σ_{k ≥ 1}a_nkX_k, n ≥ 1, converges to 0 in probability if and only if Σ_{k ≥ 1}a_nkf(X_k), n ≥ 1, converges to 0 in probability for every f in the dual space B¹.     

Smallest singular value of a random rectangular matrix

We prove an optimal estimate of the smallest singular value of a random sub‐Gaussian matrix, valid for all dimensions. For an N × n matrix A with independent and identically distributed sub‐Gaussian entries, the smallest singular value of A is at least of the order √N ? √n ? 1 with high probability. A sharp estimate on the probability is also obtained. © 2009 Wiley Periodicals, Inc.     

Degree distribution in random planar graphs

We prove that for each k?0, the probability that a root vertex in a random planar graph has degree k tends to a computable constant dk, so that the expected number of vertices of degree k is asymptotically dkn, and moreover that k_∑dk=1. The proof uses the tools developed by Giménez and Noy in their solution to the problem of the asymptotic enumeration of planar graphs, and is based on a detailed analysis of the generating functions involved in counting planar graphs. However, in order to keep track of the degree of the root, new technical difficulties arise. We obtain explicit, although quite involved expressions, for the coefficients in the singular expansions of the generating functions of interest, which allow us to use transfer theorems in order to get an explicit expression for the probability generating function p(w)=k_∑dkwk. From this we can compute the dk to any degree of accuracy, and derive the asymptotic estimate dk∼c⋅k−1/2qk for large values of k, where q≈0.67 is a constant defined analytically.     

Only the first term of some series counts

Let X,X₁,X₂,… be i.i.d. random variables, and set S_n=X₁+?+X_n. We prove that for three important distributions of X, namely normal, exponential and geometric, series of the type ∑_n≥1a_nP(|S_n|≥xb_n) or ∑_n≥1a_nP(S_n≥xb_n) behave like their first term as x→∞.     

Randomly weighted sums of subexponential random variables with application to capital allocation

We are interested in the tail behavior of the randomly weighted sum \( \sum _{i=1}^{n}\theta _{i}X_{i}\) , in which the primary random variables X ₁, …, X _n are real valued, independent and subexponentially distributed, while the random weights ?? ₁, …, ?? _n are nonnegative and arbitrarily dependent, but independent of X ₁, …, X _n . For various important cases, we prove that the tail probability of \(\sum _{i=1}^{n}\theta _{i}X_{i}\) is asymptotically equivalent to the sum of the tail probabilities of ?? ₁ X ₁, …, ?? _n X _n , which complies with the principle of a single big jump. An application to capital allocation is proposed.     

On the modular sumset partition problem

A sequence m₁≥m₂≥?≥m_k of k positive integers isn-realizable if there is a partition X₁,X₂,…,X_k of the integer interval [1,n] such that the sum of the elements in X_i is m_i for each i=1,2,…,k. We consider the modular version of the problem and, by using the polynomial method by Alon (1999) [2], we prove that all sequences in Z/pZ of length k≤(p−1)/2 are realizable for any prime p≥3. The bound on k is best possible. An extension of this result is applied to give two results of p-realizable sequences in the integers. The first one is an extension, for n a prime, of the best known sufficient condition for n-realizability. The second one shows that, for n≥(4k)³, an n-feasible sequence of length k isn-realizable if and only if it does not contain forbidden subsequences of elements smaller than n, a natural obstruction forn-realizability.     

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 structure of the distribution of a couple of observable random variables in credibility theory

We consider Bühlmann's classical model in credibility theory and we assume that the set of possible values of the observable random variables X₁, X₂,… is finite, say with n elements. Then the distribution of a couple (X_r, X_s) (r ≠ s) amounts to a square real matrix of order n, that we call a credibility matrix. In order to estimate credibility matrices or to adjust roughly estimated credibility matrices, we study the set of all credibility matrices and some particular subsets of it.     

Self-normalized moderate deviations for independent random variables

Let X1,X2,...be a sequence of independent random variables(r.v.s) belonging to the domain of attraction of a normal or stable law.In this paper,we study moderate deviations for the self-normalized sum ∑ni=1 Xi/Vn,p,where Vn,p =(∑ni=1 |Xi|p)1/p(p>1).Applications to the self-normalized law of the iterated logarithm,Studentized increments of partial sums,t-statistic,and weighted sum of independent and identically distributed(i.i.d.) r.v.s are considered.     

On a characterization of the exponential distribution by spacings

Summary LetX be a non-negative random variable with probability distribution functionF. SupposeX _i,n (i=1,…,n) is theith smallest order statistics in a random sample of sizen fromF. A necessary and sufficient condition forF to be exponential is given which involves the identical distribution of the random variables (n−i)(X _i+1,n−X_i,n) and (n−j)(X _j+1,n−X_j,n) for somei, j andn, (1≦i<j<n). The work was partly completed when the author was at the Dept. of Statistics, University of Brasilia, Brazil.     

Restricted Isometry Property of Matrices with?Independent Columns and Neighborly Polytopes by?Random Sampling

This paper considers compressed sensing matrices and neighborliness of a centrally symmetric convex polytope generated by vectors ±X ₁,…,±X _N ∈ℝ ⁿ , (N≥n). We introduce a class of random sampling matrices and show that they satisfy a restricted isometry property with overwhelming probability. In particular, we prove that matrices with i.i.d. centered and variance 1 entries that satisfy uniformly a subexponential tail inequality possess the restricted isometry property with overwhelming probability. We show that such “sensing” matrices are valid for the exact reconstruction process of m-sparse vectors via ℓ ₁ minimization with m≤Cn/log ²(cN/n). The class of sampling matrices we study includes the case of matrices with columns that are independent isotropic vectors with log-concave densities. We deduce that if K⊂ℝ ⁿ is a convex body and X ₁,…,X _N ∈K are i.i.d. random vectors uniformly distributed on K, then, with overwhelming probability, the symmetric convex hull of these points is an m-centrally-neighborly polytope with m∼n/log ²(cN/n).     

Upper-Lower Class Tests for Weighted i.i.d. Sequences and Martingales

We study the upper-lower class behavior of weighted sums ∑ _k=1 ⁿ a _k X _k , where X _k are i.i.d. random variables with mean 0 and variance 1. In contrast to Feller’s classical results in the case of bounded X _j , we show that the refined LIL behavior of such sums depends not on the growth properties of (a _n ) but on its arithmetical distribution, permitting pathological behavior even for bounded (a _n ). We prove analogous results for weighted sums of stationary martingale difference sequences. These are new even in the unweighted case and complement the sharp results of Einmahl and Mason obtained in the bounded case. Finally, we prove a general upper-lower class test for unbounded martingales, improving several earlier results in the literature.     

On the singularity probability of discrete random matrices

Let n be a large integer and Mn be an n by n complex matrix whose entries are independent (but not necessarily identically distributed) discrete random variables. The main goal of this paper is to prove a general upper bound for the probability that Mn is singular. For a constant 0<p<1 and a constant positive integer r, we will define a property p-bounded of exponent r. Our main result shows that if the entries of Mn satisfy this property, then the probability that Mn is singular is at most (p1/r+on(1)). All of the results in this paper hold for any characteristic zero integral domain replacing the complex numbers. In the special case where the entries of Mn are “fair coin flips” (taking the values +1,−1 each with probability 1/2), our general bound implies that the probability that Mn is singular is at most , improving on the previous best upper bound of , proved by Tao and Vu [Terence Tao, Van Vu, On the singularity probability of random Bernoulli matrices, J. Amer. Math. Soc. 20 (2007) 603-628]. In the special case where the entries of Mn are “lazy coin flips” (taking values +1,−1 each with probability 1/4 and value 0 with probability 1/2), our general bound implies that the probability that Mn is singular is at most , which is asymptotically sharp. Our method is a refinement of those from [Jeff Kahn, János Komlós, Endre Szemerédi, On the probability that a random ±1-matrix is singular, J. Amer. Math. Soc. 8 (1) (1995) 223-240; Terence Tao, Van Vu, On the singularity probability of random Bernoulli matrices, J. Amer. Math. Soc. 20 (2007) 603-628]. In particular, we make a critical use of the structure theorem from [Terence Tao, Van Vu, On the singularity probability of random Bernoulli matrices, J. Amer. Math. Soc. 20 (2007) 603-628], which was obtained using tools from additive combinatorics.     

Comma-free codes: An extension of certain enumerative techniques to recursively defined sequences

The number X_n of increasing subsequences of the n-long random permutation is studied. Asymptotics of the moments of X_n are found; In X_n is shown to grow, in probability, as $\text{a}\text{n}$, 2 ln 2 ? a ? 2.     

On the fourier-stieltjes coefficients of cantor-type distributions

For a givenn-tuple of non-negative numbers (p ₀,p ₁,...,p _n?1) whose sum is equal to unity let μ(t) denote the probability that Σ _j ^{= 1/∞} X _j /n ^j ≦t, where the independent random variablesX _j assume the values 0,1,...,n?1 with probabilitiesp ₀,p ₁,...,p _n?1 respectively. For mostn-tuples we obtain upper and lower bounds on |û(m)|; these estimates involve then-ary representation ofm, or in some cases of 2m, so that a very simple and explicit characterization of the sequences on whichû(m) approaches zero can be given. In particular, for the Cantor middle-third measure, corresponding to the triple (1/2, 0, 1/2), the following criterion is obtained.û(m) approaches zero on a sequenceT of integers if and only if Ω(2m) approaches infinity onT, where Ω(k) is the sum of the following three quantities associated with the ternary representation ofk: the number of runs of zeros, the number of runs of twos and the number of ones. The results obtained are easily extended to the case when then-tuple varies withj (subject to certain mild restrictions).     

A function with prescribed initial derivatives in different Banach spaces

Let X₀ ? X₁ ? ··· ? X_p be Banach spaces with continuous injection of X_k into X_{k + 1} for 0 ? k ? p ? 1, and with X₀ dense in X_p. We seek a function u: [0, 1] → X₀ such that its kth derivative u^(k), k = 0, 1,…, p, is continuous from [0, 1] into x_k, and satisfies the initial condition u^(k)(0) = a_k?X_k. It is shown that such a function exists if and only if the initial values a₀, a₁, …, a_p satisfy a certain condition reminiscent of interpolation theory. This condition always holds when p = 1; when p ? 2, the spaces X_k (k = 0, 1, …, p) may or may not be such that the desired function exists for any given initial values a_k?X_k.     

A characterization of the multivariate normal distribution

Let X_j (j = 1,…,n) be i.i.d. random variables, and let Y′ = (Y₁,…,Y_m) and X′ = (X₁,…,X_n) be independently distributed, and A = (a_jk) be an n × n random coefficient matrix with a_jk = a_jk(Y) for j, k = 1,…,n. Consider the equation U = AX, Kingman and Graybill [Ann. Math. Statist.41 (1970)] have shown U ～ N(O,I) if and only if X ～ N(O,I). provided that certain conditions defined in terms of the a_jk are satisfied. The task of this paper is to delete the identical assumption on X₁,…,X_n and then generalize the results to the vector case. Furthermore, the condition of independence on the random components within each vector is relaxed, and also the question raised by the above authors is answered.     

A note on random walk in random scenery

We consider a random walk in random scenery {Xn=η(S₀)+?+η(Sn),n∈N}, where a centered walk {Sn,n∈N} is independent of the scenery {η(x),x∈Zd}, consisting of symmetric i.i.d. with tail distribution P(η(x)>t)∼exp(−cαtα), with 1?α<d/2. We study the probability, when averaged over both randomness, that {Xn>ny} for y>0, and n large. In this note, we show that the large deviation estimate is of order exp(−ca(ny)), with a=α/(α+1).     

On general summability factor theorem of infinite series

A lower triangular matrix with nonzero principal diagonal entries is called a triangle. In this paper we obtain the sufficient conditions for ∑a_nλ_n to be summable ∣A∣_k whenever ∑a_n is summable ∣T∣_k for a triangle T.     

Spectrum of Markov Generators on Sparse Random Graphs

We investigate the spectrum of the infinitesimal generator of the continuoustime random walk on a randomly weighted oriented graph. This is the non‐Hermitian random n × n matrix L defined by L_jk = X_jk if k ≠ j and L_jj = – Σ_{k ≠ j} L_jk, where (X_jk)_{j ≠ k} are i.i.d. random weights. Under mild assumptions on the law of the weights, we establish convergence as n → ∞ of the empirical spectral distribution of L after centering and rescaling. In particular, our assumptions include sparse random graphs such as the oriented Erd?s‐Rényi graph where each edge is present independently with probability p(n) → 0 as long as np(n) ? (log(n))⁶. The limiting distribution is characterized as an additive Gaussian deformation of the standard circular law. In free probability terms, this coincides with the Brown measure of the free sum of the circular element and a normal operator with Gaussian spectral measure. The density of the limiting distribution is analyzed using a subordination formula. Furthermore, we study the convergence of the invariant measure of L to the uniform distribution and establish estimates on the extremal eigenvalues of L.© 2014 Wiley Periodicals, Inc.