Abstract Generated abstract
This paper studies the law of the iterated logarithm for sums of independent random variables represented as measurable elements of a complete topological semifield equipped with a Boolean algebra and measure. After formulating integration, probability, independence, and Laplace transforms in this setting, the authors prove a main sufficient condition based on truncation idempotents and convergence of a series controlling the large parts of the variables. The result extends Kolmogorov’s classical theorem and yields further criteria under variance growth and moment-type assumptions, including cases related to Petrov’s theorem. The paper also notes that weakening the smallness condition in the truncation can lead to failure of the law.
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MATHEMATICS
Academician of the Academy of Sciences of the Uzbek SSR T. A. SARYMSAKOV, Ya. Kh. KUCHAROV
ON THE LAW OF THE ITERATED LOGARITHM
The present article is devoted to the law of the iterated logarithm for a sequence of independent random variables. In this exposition the latter are represented by measurable elements of a topological semifield. The results obtained are a generalization and refinement of certain theorems available in the literature on the iterated logarithm.
In what follows we shall adhere to the definitions and notation of works \((^{1,2})\). At the same time, for continuity of exposition, we give here a number of definitions and facts concerning topological semifields and their Boolean algebras.
- Let \(E\) be a complete topological semifield, \(\overline K\) the cone of nonnegative elements in \(E\), and \(\nabla\) the topological Boolean algebra of all idempotents from \(E\) \((^1)\).
Definition 1. A subset \(I \subset \nabla\) will be called a confinal if: 1) from \(e \in I\) it follows that \(Ce \in I\); 2) from \(e, g \in I\) it follows that \(e \vee g \in I\), \(e \wedge g \in I\).
Definition 2. A nonnegative continuous function \(m\), defined on the confinal \(I\), taking finite or infinite values, finitely additive for disjoint elements of the confinal \(I\) and equal to zero only on the idempotent \(\theta\) (the zero of the algebra \(\nabla\)), will be called a measure.
Further, let \(e \in \nabla\). Define the number
\[ m^*(e)=\inf_{\substack{a_i \in I\\ \bigvee_{i=1}^{\infty} a_i \ge e}} \sum_{i=1}^{\infty} m(a_i). \]
An element \(e(\in \nabla)\) will be called measurable if for any \(g\) from \(\nabla\)
\[ m^*(g)=m^*(g \wedge e)+m^*(g \wedge Ce). \]
The set of all measurable elements of \(\nabla\) will be denoted by \(I_\nabla\) and \(I_\nabla\) will be called the maximal confinal. The usual properties of a confinal and a measure are given in \((^2)\).
Definition 3. The triple \((E, I_\nabla, m)\) will be called a space with measure.
Throughout this paper it is assumed that the space with measure is fixed.
Definition 4. An element \(x \in E\) is called measurable if its support (see \((^1)\)) \(a(x) \in I_\nabla\) and, for any real number \(\lambda\), the relation \(g_\lambda=\bigvee\{e: xe < \hat{\lambda}e\}\in I\) holds. Without dwelling on the properties of measurable elements (they coincide with the properties of measurable functions), we introduce the concept of the integral of a simple element. An element \(x \in E\) is called simple if it has the form:
\[ x=\sum_{k=1}^{n}\lambda_k e_k, \]
where \(e_1,e_2,\ldots,e_n\) are pairwise disjoint measurable elements from \(I_\nabla\), and \(\lambda_1,\lambda_2,\ldots,\lambda_n\) are real numbers.
The number
\[ \mu(x)=\sum_{k=1}^{\infty}\lambda_k m(e_k) \]
will be called the integral of \(x\), if \(m(e_k)<\infty\) for \(k=1,2,\ldots,n\); in this case \(x\) is called an integrable simple element. Just as in the classical theory of integration, this integral can be extended to measurable elements of the semiring \(E\), and its properties are analogous to the classical ones. Elements \(E\) for which the integral exists are called summable, and their set, as usual, is denoted by \(L\). Obviously, \(L\) is a linear topological space in the topology induced from \(E\). An element \(x\in L\cap \bar K\) will be called a distribution if \(\mu(x)=1\).
Introduce the notation \(P=\{x:x\in L\cap \bar K,\ \mu(x)=1\}\). Each element \(x\in P\) generates a continuous probability measure \(\mu_x\), defined on \(I_\nabla\), namely \(\mu_x(e)=\mu(xe)\), where \(e\in I_\nabla\). The triple \((I_\nabla,m,x)\) will be called a probability space; here \(x(\in P)\) is a fixed element (for more details see (2)).
The pair \((\xi,x)\), where \(\xi\in E,\ x\in P\), will be called a random variable; since \(x\) is fixed, we shall simply write \(\xi\). Random variables \(\xi\) and \(\eta\) are called independent if
\[ \mu_x\bigl[a((\xi-\alpha 1)_-)a((\eta-\beta 1)_-)\bigr] = \mu_x\bigl[a((\xi-\alpha 1)_-)\bigr]\mu_x\bigl[(\eta-\beta 1)_-\bigr], \]
where \(\alpha\) and \(\beta\) are real numbers, \(z_- =(-z\vee \theta)\) is the negative part of the element \(z\), and \(1\) is the unit (or maximal element) of the Boolean algebra \(\nabla\).
Along with the semiring \(E\), introduce for consideration the complex semiring \(\mathcal E\), defined by the equality \(\mathcal E=E+iE\). The rules of operation for elements of \(\mathcal E\) are the same as for complex numbers. Below the Laplace transform is introduced, written in our notation. Its introduction is based on the following assumption.
Let \(E\) be a complete semiring \((^{1})\), and let \(A\) be a family of power series absolutely convergent on the entire real line. By \(\varphi(A)\) we denote the family of series obtained from a series belonging to the family \(A\) by replacing the real variable by an element of the semiring \(E\). Then the series from \(\varphi(A)\) converge absolutely on \(E\).
- Consider a sequence \(X_1,X_2,\ldots,X_n,\ldots\) of independent random variables. Everywhere we shall assume that the random variables \(X_k\) have zero mathematical expectations and finite variances.
We shall say that the sequence of sums \(\{S_n\}\) \((S_n=X_1+\cdots+X_n\in E)\) obeys the law of the iterated logarithm (l.i.l.) if
\[ \mu_x\left( \bigvee_{e\in I_\nabla} \left\{e: \bigwedge_{m=1}^{\infty} \bigvee_{n=m}^{\infty} \frac{S_n e}{\chi(\mu(xS_n^2))}=1 \right\} \right)=1, \]
\[ \mu_x\left( \bigvee_{e\in I_\nabla} \left\{e: \bigvee_{m=1}^{\infty} \bigwedge_{n=m}^{\infty} \frac{S_n e}{\chi(\mu(xS_n^2))}=-1 \right\} \right)=1, \]
where \(\chi(t)=\sqrt{2t\ln\ln t}\). Introduce for consideration the idempotent
\[ g_k=\bigvee_{g\in I_\nabla}\{g:|X_k|g\leq \tfrac12 m_k g\}, \tag{1} \]
where
\[ m_k=o\left[\left(\frac{\mu(xS_n^2)}{\ln\ln \mu(xS_n^2)}\right)^{1/2}\right]. \]
Put
\[ \xi_k g_k=(X_k-\mu(xX_k g_k))g_k,\qquad \xi_k\bar g_k=-\mu(xX_k g_k)\bar g_k; \]
\[ \eta_k g_k=\mu(xX_k g_k)g_k;\qquad \eta_k\bar g_k=(X_k+\mu(xX_k g_k))\bar g_k, \]
where \(\bar g_k=1-g_k\).
It is clear from the construction that \(\{\xi_k\}\) and \(\{\eta_k\}\) form sequences of independent random variables.
Main theorem. If, as \(n\to\infty\), \(\mu(xS_n^2)\to\infty\) and the series
\[
\sum_k \frac{\mu(xX_k\bar g_k^{\delta})}{\mu(xS_k^2)},
\]
where \(g_k\) is defined by relation (1), converges, then the law of the iterated logarithm is applicable to the sequence of sums \(\{S_n\}\).
Let us note that for \(\bar g_k=\theta\) \((k=1,2,\ldots)\), the well-known theorem of A. N. Kolmogorov \((^3)\) on the LIL follows from this theorem.
The proof of the first part of the theorem (which is analogous to the well-known proof of A. N. Kolmogorov \((^3)\)) is based on the use of exponential estimates for the sums
\[
\xi_n=\sum_{k=1}^n \xi_k.
\]
The derivation of these estimates, in turn, rests on the equality
\[
\mu\left(xe^{z\xi_n}\right)
=
\exp\left\{\frac{z^2\mu(x\xi_n^2)}{2}(1+\gamma(z,n))\right\}.
\]
Here
\[
\gamma(z,n)=
\frac{(e^\delta-1-\delta)(3-e^\delta+\delta)m_n^*|z|}
{\delta^3(2-e^\delta+\delta)(1-m_n^*|z|/\delta)}\,\theta(z),
\]
where \(|\theta(z)|<1\), \(m_n^*=\max_{k\le n}m_k\), and \(0<\delta<1\).
Theorem 1. If
\[
\lim_{n\to\infty}\frac{1}{n}\mu(xS_n^2)>0,
\qquad
\mu(xX_k^2\bar g_k)=O\left(\frac{1}{f(k)}\right),
\]
where \(f(k)\) is some monotone sequence of positive numbers satisfying the condition
\[
\sum_{k=1}^{\infty}\frac{1}{kf(k)}<\infty,
\]
then the LIL is applicable to the sequence \(\{S_n\}\).
Theorem 2. If
\[
\lim_{n\to\infty}\frac{1}{n}\mu(xS_n^2)>0,
\qquad
\mu\left(x|X_k|^2\varphi(X_k)\right)<\infty,
\]
for all \(k\), and
\[
\overline{\lim}_{n\to\infty}\frac{1}{n}\sum_{k=1}^n
\mu\left(x|X_k|^2\varphi(X_k)\right)<\infty,
\]
then the LIL is applicable to the sequence \(\{S_n\}\), where \(\varphi(X_k)>0\) and
\[
\sum_{k=1}^{\infty}\frac{1}{k\varphi(k)}<\infty.
\]
The proof of these theorems reduces to verifying the conditions of the main theorem.
From Theorem 2, for \(\varphi(X_k)=|X_k|^\delta\) \((\delta>0)\), one obtains the theorem of V. V. Petrov \((^4)\). Finally, as in Marcinkiewicz–Zygmund \((^5)\) and M. Weiss \((^6)\), it is not difficult to construct examples such that if, in our condition (1), \(o\) is replaced by \(O\), then the LIL will not hold.
Tashkent State University
named after V. I. Lenin
Received
3 VI 1968
REFERENCES
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- A. N. Kolmogoroff, Math. Ann., 101, 126 (1929).
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- M. Weiss, J. Math. and Mech., 8, no. 1, 121 (1959).