We consider weighted sums of independent random variables regulated by an increment sequence. We provide operative conditions that ensure strong law of large numbers for such sums to hold in both the centered and non-centered case. The existing criteria for the strong law are either implicit or assume some sufficient decay for the sequence of coefficients. In our set up we allow for arbitrary sequence of coefficients, possibly random, provided the random variables regulated by such increments satisfy some mild concentration conditions. In the non-centered case, convergence can be translated into the behavior of a deterministic sequence and it becomes a game of mass provided the expectation of the random variables is a function of the increments. We show how different limiting scenarios can emerge by identifying several classes of increments, for which concrete examples will be offered.
In the paper, upper bounds for the rate of convergence in laws of large numbers for mixed Poisson random sums are constructed. As a measure of the distance between the limit and pre-limit laws, the Zolotarev $ζ$-metric is used. The obtained results extend the known convergence rate estimates for geometric random sums (in the famous R{é}nyi theorem) to a considerably wider class of random indices with mixed Poisson distributions including, e. g., those with the (generalized) negative binomial distribution.
This paper develops Rio's method [C. R. Acad. Sci. Paris Sér. I Math., 1995] to prove the weak law of large numbers for maximal partial sums of pairwise independent random variables. The method allows us to avoid using the Kolmogorov maximal inequality. As an application, a weak law of large numbers for maximal partial sums of pairwise independent random variables under a uniform integrability condition is also established. The sharpness of the result is illustrated by an example.
We consider moderately trimmed sums of non-negative i.i.d. random variables. We show that for every distribution function there exists a proper moderate trimming such that for the trimmed sum a non-trivial strong law of large numbers holds. In case that the distribution function has regularly varying tails we give necessary and sufficient conditions on the trimming for a strong law of large numbers to hold.
Let $\{X_n;n\ge 1\}$ be a sequence of independent and identically distributed random variables on a sub-linear expectation space $(Ω,\mathscr{H},\widehat{\mathbb E})$, $S_n=X_1+\ldots+X_n$. We consider the moments of $\max_{n\ge 1}|S_n|/\sqrt{2n\log\log n}$. The sufficient and necessary conditions for the moments to be finite are given. As an application, we obtain the law of the iterated logarithm for moving average processes of independent and identically distributed random variables.
Let $\{V_{i,j}; (i,j)\in\N^2\}$ be a two-dimensional array of i.i.d.\ random variables. The limit laws of the sum of independent random products $$ Z_n=\sum_{i=1}^{N_n} \prod_{j=1}^{n} e^{V_{i,j}} $$ as $n,N_n\to\infty$ have been investigated by a number of authors. Depending on the growth rate of $N_n$, the random variable $Z_n$ obeys a central limit theorem, or has limiting $α$-stable distribution. The latter result is true for non-lattice $V_{i,j}$ only. Our aim is to study the lattice case. We prove that although the (suitably normalized) sequence $Z_n$ fails to converge in distribution, i…
The validity of the strong law of large numbers for multiple sums $S_n$ of independent identically distributed random variables $Z_k$, $k\leq n$, with $r$-dimensional indices is equivalent to the integrability of $|Z|(\log^+|Z|)^{r-1}$, where $Z$ is the typical summand. We consider the strong law of large numbers for more general normalisations, without assuming that the summands $Z_k$ are identically distributed, and prove a multiple sum generalisation of the Brunk--Prohorov strong law of large numbers. In the case of identical finite moments of irder $2q$ with integer $q\geq1$, we show that …
The problem of convergence in law of normed sums of exchangeable random variables is examined. First, the problem is studied w.r.t. arrays of exchangeable random variables, and the special role played by mixtures of products of stable laws - as limits in law of normed sums in different rows of the array - is emphasized. Necessary and sufficient conditions for convergence to a specific form in the above class of measures are then given. Moreover, sufficient conditions for convergence of sums in a single row are proved. Finally, a potentially useful variant of the formulation of the results just…
We present some product representations for random variables with the Linnik, Mittag-Leffler and Weibull distributions and establish the relationship between the mixing distributions in these representations. Based on these representations, we prove some limit theorems for a wide class of rather simple statistics constructed from samples with random sized including, e. g., random sums of independent random variables with finite variances, maximum random sums, extreme order statistics, in which the Linnik and Mittag-Leffler distributions play the role of limit laws. Thus we demonstrate that the…
In high-frequency statistics and econometrics sums of functionals of increments of stochastic processes are commonly used and statistical inference is based on the asymptotic behaviour of these sums as the mesh of the observation times tends to zero. Inspired by the famous Hayashi-Yoshida estimator for the quadratic covariation process based on two asynchronously observed stochastic processes we investigate similar sums based on increments of two asynchronously observed stochastic processes for general functionals. We find that our results differ from corresponding results in the setting of eq…
Laws of the iterated and single logarithm for sums of independent indicators, with applications to the Ginibre point process and Karlin's occupancy scheme
For a given set of random variables $X_1,\ldots,X_d$ we seek as large a family as possible of random variables $Y_1,\ldots,Y_d$ such that the marginal laws and the laws of the sums match: $Y_i\,{\buildrel d \over =}\,X_i$ and $\sum_iY_i\,{\buildrel d \over =}\,\sum_iX_i$. Under the assumption that $X_1,\ldots,X_d$ are independent and belong to any of the Meixner classes, we give a full characterisation of the random variables $Y_1,\ldots,Y_d$ and propose a practical construction by means of a finite mean square expansion. When $X_1,\ldots,X_d$ are identically distributed but not necessarily in…
We consider delayed sums of the type S_{n+an}-Sn where a_n is possibly a positive integer valued random variable satisfying certain conditions and S_n is the sum of independent random variables X_n with distribution functions F_n in {G_1, G_2} . We study the limiting behavior of delayed sums and prove laws of the iterated logarithm of Chover- type. These results extend the results in Vasudeva and Divanji (1992) and Chen (2008).
In this work we investigate the asymptotic behaviour of weighted partial sums of a particular class of random variables related to Oppenheim series expansions. More precisely, we verify convergence in probability as well as almost sure convergence to a strictly positive and finite constant without assuming any dependence structure or the existence of means. Results of this kind are known as exact weak and exact strong laws.
In this paper, based on the initiation of the notion of negatively associated random variables under nonlinear probability, a strong limit theorem for weighted sums of random variables within the same frame is achieved without assumptions of independence and identical distribution, from which the Marcinkiewich-Zygmund type and Kolmogorov type strong laws of large numbers are derived. In addition, as applications of our results, Stranssen type invariance principles of negatively associated random variables and vertically independent random variables are proposed respectively.
We investigate spatial evolutionary games with death-birth updating in large finite populations. Within growing spatial structures subject to appropriate conditions, the density processes of a fixed type are proven to converge to the Wright-Fisher diffusions with drift. In addition, convergence in the Wasserstein distance of the laws of their occupation measures holds. The proofs of these results develop along an equivalence between the laws of the evolutionary games and certain voter models and rely on the analogous results of voter models on large finite sets by convergences of the Radon-Nik…
Continuing the study reported in Satheesh (2001),(arXiv:math.PR/0304499 dated 01May2003) here we study certain aspects of randomization in infinitely divisible (ID) and max-infinitely divisible (MID) laws. They generalize ID and MID laws. In particular we study mixtures of ID & MID laws, its relation to random sums & random maximums, corresponding stationary processes & extremal processes and some of their properties. It is shown that mixtures of ID laws and mixtures of MID laws appear as limits of random sums and random maximums respectively. We identify a class of probability generating func…
We consider a general multidimensional affine recursion with corresponding Markov operator $P$ and a unique $P$-stationary measure. We show spectral gap properties on Hölder spaces for the corresponding Fourier operators and we deduce convergence to stable laws for the Birkhoff sums along the recursion. The parameters of the stable laws are expressed in terms of basic quantities depending essentially on the matricial multiplicative part of $P$. Spectral gap properties of $P$ and homogeneity at infinity of the $P$-stationary measure play an important role in the proofs.
Two approaches are suggested to the definition of asymmetric generalized Weibull distribution. These approaches are based on the representation of the two-sided Weibull distributions as variance-mean normal mixtures or more general scale-location mixtures of the normal laws. Since both of these mixtures can be limit laws in limit theorems for random sums of independent random variables, these approaches can provide additional arguments in favor of asymmetric two-sided Weibull-type models of statistical regularities observed in some problems related to stopped random walks, in particular, in pr…
We study the convergence of centered and normalized sums of i.i.d. random elements of the space $\mathcal{D}$ of c{á}dl{á}g functions endowed with Skorohod's $J\_1$ topology, to stable distributions in $\mathcal D$. Our results are based on the concept of regular variation on metric spaces and on point process convergence. We provide some applications, in particular to the empirical process of the renewal-reward process.
An example due to Erdos and Fortet shows that, for a lacunary sequence of integers (q_n) and a trigonometric polynomial f, the asymptotic distribution of normalized sums of f(q_k x) can be a mixture of gaussian laws. Here we give a generalization of their example interpreted as the limiting behavior of some modified ergodic sums in the framework of dynamical systems.
We prove game-theoretic generalizations of some well known zero-one laws. Our proofs make the martingales behind the laws explicit, and our results illustrate how martingale arguments can have implications going beyond measure-theoretic probability.
In earlier stages in the introduction to asymptotic methods in probability theory, the weak convergence of sequences $(X_n)_{n\geq 1}$ of Binomial of random variables (\textit{rv}'s) to a Poisson law is classical and easy-to prove. A version of such a result concerning sequences $(Y_n)_{n\geq 1}$ of negative binomial \textit{rv}'s also exists. In both cases, $X_n$ and $Y_n-n$ are by-row sums $S_n[X]$ and $S_n[Y]$ of arrays of Bernoulli \textit{rv}'s and corrected geometric \textit{rv}'s respectively. When considered in the general frame of asymptotic theorems of by-row sums of \textit{rv}'s of…
In this paper we obtain the limit distribution for partial sums with a random number of terms following a class of mixed Poisson distributions. The resulting weak limit is a mixing between a normal distribution and an exponential family, which we call by normal exponential family (NEF) laws. A new stability concept is introduced and a relationship between α-stable distributions and NEF laws is established. We propose estimation of the parameters of the NEF models through the method of moments and also by the maximum likelihood method, which is performed via an Expectation-Maximization algorith…
Strong laws of large numbers are established for random fields with weak or strong dependence. These limit theorems are applicable to random fields with heavy-tailed distributions including fractional stable random fields. The conditions for SLLN are described in terms of the $p$-th moments of the partial sums of the random fields, which are convenient to verify. The main technical tool in this paper is a maximal inequality for the moments of partial sums of random fields that extends the technique of Levental, Chobanyan and Salehi \cite{chobanyan-l-s} for a sequence of random variables indexe…
The main purpose of this paper is to obtain strong laws of large numbers for arrays or weighted sums of random variables under a scenario of dependence. Namely, for triangular arrays $\{X_{n,k}, \, 1 \leqslant k \leqslant n, \, n \geqslant 1 \}$ of row-wise extended negatively dependent random variables weakly mean dominated by a random variable $X \in \mathscr{L}_{1}$ and sequences $\{b_{n} \}$ of positive constants, conditions are given to ensure $\sum_{k=1}^{n} \left(X_{n,k} - \mathbb{E} \, X_{n,k} \right)/b_{n} \overset{\textnormal{a.s.}}{\longrightarrow} 0$. Our statements also allow us t…
When the distribution of a random (N) sum of independent copies of a r.v X is of the same type as that of X we say that X is N-sum stable. In this paper we consider a generalization of stability of geometric sums by studying distributions that are stable under summation w.r.t Harris law. We show that the notion of stability of random sums can be extended to include the case when X is discrete. Finally we propose a method to identify the probability law of N for which X is N-sum stable. See also Satheesh and Nair (2002), (Some classes of distributions on ther non-negative lattice, J. Ind. Stati…
A classical result of K. L. Chung and W. Feller deals with the partial sums $S_k$ arising in a fair coin-tossing game. If $N_n$ is the number of "positive" terms among $S_1, S_2,\dots,S_n$ then the quantity $P(N_{2n}=2r)$ takes an elegant form. We lift the restriction on an even number of tosses and give a simple expression for $P(N_{2n+1}=r)$, $r=0,1,2,\dots,2n+1$. We get to this result by adapting the Feynman-Kac methodology.
General Central limit theorem deals with weak limits (in type) of sums of row-elements of array random variables. In some situations as in the invariance principle problem, the sums may include only parts of the row-elements. For strictly stationary arrays (stationary for each row), there is no change to the asymptotic results. But for non-stationary data, especially for dependent data, asymptotic laws of partial sums moving in rows may require extra-conditions to exist. This paper deals with central limit theorems with Gaussian limits for non-stationary data. Our main focus is on dependent da…
For a distribution $F^{*τ}$ of a random sum $S_τ=ξ_1+...+ξ_τ$ of i.i.d. random variables with a common distribution $F$ on the half-line $[0,\infty)$, we study the limits of the ratios of tails $\bar{F^{*τ}}(x)/\bar{F}(x)$ as $x\to\infty$ (here, $τ$ is a counting random variable which does not depend on $\{ξ_n\}_{n\ge1}$). We also consider applications of the results obtained to random walks, compound Poisson distributions, infinitely divisible laws, and subcritical branching processes.
Kolmogorov's exponential inequalities are basic tools for studying the strong limit theorems such as the classical laws of the iterated logarithm for both independent and dependent random variables. This paper establishes the Kolmogorov type exponential inequalities of the partial sums of independent random variables as well as negatively dependent random variables under the sub-linear expectations. As applications of the exponential inequalities, the laws of the iterated logarithm in the sense of non-additive capacities are proved for independent or negatively dependent identically distribute…
We consider a Markov chain $\{X_n\}_{n=0}^\8$ on $\R^d$ defined by the stochastic recursion $X_{n}=M_n X_{n-1}+Q_n$, where $(Q_n,M_n)$ are i.i.d. random variables taking values in the affine group $H=\R^d\rtimes {\rm GL}(\R^d)$. Assume that $M_n$ takes values in the similarity group of $\R^d$, and the Markov chain has a unique stationary measure $ν$, which has unbounded support. We denote by $|M_n|$ the expansion coefficient of $M_n$ and we assume $\E |M|^\a=1$ for some positive $\a$. We show that the partial sums $S_n=\sum_{k=0}^n X_k$, properly normalized, converge to a normal law ($\a\ge 2$…
We establish strong invariance principles for sums of stationary and ergodic processes with nearly optimal bounds. Applications to linear and some nonlinear processes are discussed. Strong laws of large numbers and laws of the iterated logarithm are also obtained under easily verifiable conditions.
Several classical results on boundary crossing probabilities of Brownian motion and random walks are extended to asymptotically Gaussian random fields, which include sums of i.i.d. random variables with multidimensional indices, multivariate empirical processes, and scan statistics in change-point and signal detection as special cases. Some key ingredients in these extensions are moderate deviation approximations to marginal tail probabilities and weak convergence of the conditional distributions of certain ``clumps'' around high-level crossings. We also discuss how these results are related t…
The sub-linear expectation or called G-expectation is a nonlinear expectation having advantage of modeling non-additive probability problems and the volatility uncertainty in finance. Let $\{X_n;n\ge 1\}$ be a sequence of independent random variables in a sub-linear expectation space $(Ω, \mathscr{H}, \widehat{\mathbb E})$. Denote $S_n=\sum_{k=1}^n X_k$ and $V_n^2=\sum_{k=1}^n X_k^2$. In this paper, a moderate deviation for self-normalized sums, that is, the asymptotic capacity of the event $\{S_n/V_n \ge x_n \}$ for $x_n=o(\sqrt{n})$, is found both for identically distributed random variables…
We study the situations when the solution to a weighted stochastic recursion has a power law tail. To this end, we develop two complementary approaches, the first one extends Goldie's (1991) implicit renewal theorem to cover recursions on trees; and the second one is based on a direct sample path large deviations analysis of weighted recursive random sums. We believe that these methods may be of independent interest in the analysis of more general weighted branching processes as well as in the analysis of algorithms.
A general method to prove strong laws of large numbers for random fields is given. It is based on the Hájek - Rényi type method presented in Noszály and Tómács \cite{noszaly} and in Tómács and Líbor \cite{thomas06}. Noszály and Tómács \cite{noszaly} obtained a Hájek-Rényi type maximal inequality for random fields using moments inequalities. Recently, Tómács and Líbor \cite{thomas06} obtained a Hájek-Rényi type maximal inequality for random sequences based on probabilities, but not for random fields. In this paper we present a Hájek-Rényi type maximal inequality for random fields, using probabi…
A quantum walker moves on the integers with four extra degrees of freedom, performing a coin-shift operation to alter its internal state and position at discrete units of time. The time evolution is described by a unitary process. We focus on finding the limit probability law for the position of the walker and study it by means of Fourier analysis. The quantum walker exhibits both localization and a ballistic behavior. Our two results are given as limit theorems for a 2-period time-dependent walk and they describe the location of the walker after it has repeated the unitary process a large num…
We present new mixture representations for the generalized Linnik distribution in terms of normal, Laplace, exponential and stable laws and establish the relationship between the mixing distributions in these representations. Based on these representations, we prove some limit theorems for a wide class of rather simple statistics constructed from samples with random sized including, e. g., random sums of independent random variables with finite variances and maximum random sums, in which the generalized Linnik distribution plays the role of the limit law. Thus we demonstrate that the scheme of…
For nonstationary, strongly mixing sequences of random variables taking their values in a finite-dimensional Euclidean space, with the partial sums being normalized via matrix multiplication, with certain standard conditions being met, the possible limit distributions are precisely the operator-selfdecomposable laws.
In this work we discuss a parameter $σ$ on weighted $k$-element multisets of $[n]= \{1,\dots ,n\}$. The sums of weighted $k$-multisets are related to $k$-subsets, $k$-multisets, as well as special instances of truncated interpolated multiple zeta values. We study properties of this parameter using symbolic combinatorics. We rederive and extend certain identities for $ζ^{t}_n(\{m\}_k)$. Moreover, we introduce random variables on the $k$-element multisets and derive their distributions, as well as limit laws for $k$ or $n$ tending to infinity.
Ecole Polytechnique, IP Paris（巴黎理工学院（IP Paris））；Université Paris-Saclay（巴黎-萨克雷大学）；Université Paris-Dauphine, PSL Research University（巴黎多芬纳大学（PSL研究大学））
In this paper we study a type of games regularized by the relative entropy, where the players' strategies are coupled through a random environment variable. Besides the existence and the uniqueness of equilibria of such games, we prove that the marginal laws of the corresponding mean-field Langevin systems can converge towards the games' equilibria in different settings. As applications, the dynamic games can be treated as games on a random environment when one treats the time horizon as the environment. In practice, our results can be applied to analysing the stochastic gradient descent algor…
