On the strong law of large numbers and additive functions

Let f(n) be a strongly additive complex-valued arithmetic function. Under mild conditions on f, we prove the following weighted strong law of large numbers: if X,X ₁,X ₂, … is any sequence of integrable i.i.d. random variables, then

Game-theoretic versions of strong law of large numbers for unbounded variables

We consider strong law of large numbers (SLLN) in the framework of game-theoretic probability of Shafer and Vovk (Shafer, G. and Vovk, V. 2001, Probability and Finance: It's Only a Game! (New York: Wiley)). We prove several versions of SLLN for the case that Reality's moves are unbounded. Our game-theoretic versions of SLLN largely correspond to standard measure-theoretic results. However game-theoretic proofs are different from measure-theoretic ones in the explicit consideration of various hedges. In measure-theoretic proofs existence of moments is assumed, whereas in our game-theoretic proofs we assume availability of various hedges to Skeptic for finite prices.     

A bivariate strong law of large numbers

Probability Theory and Related Fields -     

A generalized strong law of large numbers

Strong laws of large numbers have been stated in the literature for measurable functions taking on values on different spaces. In this paper, a strong law of large numbers which generalizes some previous ones (like those for real-valued random variables and compact random sets) is established. This law is an example of a strong law of large numbers for Borel measurable nonseparably valued elements of a metric space. Received: 24 February 1998 / Revised version: 3 January 1999     

Some estimates for large deviations and their application to strong law of large numbers

Siberian Mathematical Journal -     

A remark on the strong law of large numbers

The strong law of large numbers for independent and identically distributed random variablesX _i ,i=1, 2, 3,... with finite expectationE|X ₁| can be stated as, for any >0, the number of integersn such that \varepsilon $$ " align="middle" border="0"> ,N is finite a. s. It is known thatEN < iffEX ₁ ² < and that ² EN var X₁ as 0, ifE X ₁ ² <. Here we consider the asymptotic behaviour ofEN (n) asn, whereN (n) is the number of integerskn such that \varepsilon $$ " align="middle" border="0"> andE N ₁ ² =.     

Relative stability and the strong law of large numbers

Summary LetX ₁,X ₂,..., be i.i.d. random variables andS _n=X ₁+X ₂+. +X _n. In this paper we simplify Rogozin's condition forS _n/B _n ±1for someB _n+, which generalises Hinin's condition for relative stability ofS _n. We also consider convergence of subsequences ofS _n/B _n. As an application of our methods, we extend a result of Chow and Robbins to show thatS _n/B _n±1 a.s. for someB _n + if and only if 0<¦EX¦E¦X¦<+ .     

An elementary proof of the strong law of large numbers

Summary In the following note we present a proof for the strong law of large numbers which is not only elementary, in the sense that it does not use Kolmogorov's inequality, but it is also more applicable because we only require the random variables to be pairwise independent. An extension to separable Banach space-valuedr-dimensional arrays of random vectors is also discussed. For the weak law of large numbers concerning pairwise independent random variables, which follows from our result, see Theorem 5.2.2 in Chung [1].     

Fluctuation of averages in the strong law of large numbers

Translated from Matematicheskie Zametki, Vol. 50, No. 5, pp. 151–153, November, 1991.     

A strong law of large numbers for non-additive probabilities

In this paper, with the notion of independence for random variables under upper expectations, we derive a strong law of large numbers for non-additive probabilities. This result is a natural extension of the classical Kolmogorov’s strong law of large numbers to the case where the probability is no longer additive. As an application of our result, we give most frequent interpretation for Bernoulli-type experiments with ambiguity.     

The strong law of large numbers for random processes

Analogs of the Kolmogorov, Zygmund-Martsenkevich, and Brunk-Prokhorov strong law of large numbers are proved for martingales with continuous parameter. A new generalization of the Brunk—Prokhorov strong law of large numbers is given for martingales with discrete times. Along with convergence almost everywhere, we also prove the average convergence.     

Characterization of Hilbert spaces by the strong law of large numbers

Hilbert spaces are characterized through the validity of the strong law of large numbers. Other characterizations of Hilbert spaces are also given at the same time in this note.     

Some remarks on the strong law of large numbers for Banach-space valued weakly integrable random variables

The purpose of this paper is to show the equivalence of almost sure convergence of S_n/n, n ≥ 1 and lim sup_n→∞S_n/n < ∞ a.e., where S_n = X₁ + X₂ + … + X_n and X₁, X₂,… are independent identically distributed random elements in a separable Banach space with EX₁ < ∞. This result disproves a result of Pop-Stojanovic [8].     

Necessary and sufficient conditions for the strong law of large numbers

The well-known characterization indicated in the title involves the moving maximal dyadic averages of the sequence (X _k : k = 1, 2, …) of random variables in Probability Theory. In the present paper, we offer another characterization of the SLLN which does not require to form any maximum. Instead, it involves only a specially selected sequence of moving averages. The results are also extended for random fields (X _kℓ: k, ℓ = 1, 2, …).     

On the strong law of large numbers for dependent random variables

Let {X _n} _n ^=1/∞ be a sequence of random variables with partial sumsS _n, and let {ie241-1} be the σ-algebra generated byX ₁,…,X _n. Letf be a function fromR toR and suppose {ie241-2}. Under conditions off and moment conditions on theX' _ns, we show thatS _n/n converges a.e. (almost everywhere). We give several applications of this result. Research supported by N.S.F. Grant MCS 77-26809