Limit Theorem For Probability Measures Research Articles

Central limit theorem for probability measures defined by sum-of-digits function in base 2

In this paper we prove a central limit theorem for some probability measures defined as asymtotic densities of integer sets defined via sum-of-digit-function. To any integer a we can associate a measure on Z called $\mu$a such that, for any d, $\mu$a(d) is the asymptotic density of the set of integers n such that s\_2(n + a) -- s\_2(n) = d where s\_2(n) is the number of digits "1" in the binary expansion of n. We express this probability measure as a product of matrices. Then we take a sequence of integers (a\_X(n)) n$\in$N via a balanced Bernoulli process. We prove that, for almost every sequence, and after renormalization by the typical variance, we have a central limit theorem by computing all the moments and proving that they converge towards the moments of the normal law N (0, 1).

Limit Theorems for Non-Commutative Random Variables

We study the weak law of large numbers and the central limit theorem for non-commutative random variables. We first define the concepts of variance and expectation for probability measures on homogeneous spaces, and formulate the weak law of large numbers and the central limit theorem for probability measures on locally compact groups. Then, we consider the non-commutative case, where the homogeneous space is replaced by a C*-algebra \({\cal A}\) that is equipped with a locally compact group G of automorphisms. We define the concepts of variance and expectation in the non-commutative situation. Furthermore, we prove that the weak law of large numbers and the central limit theorem hold for non-commutative random variables on \({\cal A}\) if they hold on the group G of automorphisms.

Multivariate R-O Varying Measures Part I: Uniform Bounds

A finite Borel measure $\mu$ on $\mathbb R^d$ is called R-O varying with index $F$ if there exist a $\operatorname{GL}(\mathbb R^d)$-valued function $f$ varying regularly with index $(-F)$, an increasing function $k : (0,\infty)\to (0,\infty)$ with $k(t)\to\infty$ and $k(t+1)/k(t)\to c\geq 1$ as $t\to\infty$, and a $\sigma$-finite measure $\phi$ on $\mathbb R^d\setminus\{0\}$ such that \[k(t)\cdot(f(k(t))\mu)\to\phi\quad \text{as $t\to\infty$.}\] R-O varying measures generalize regularly varying measures introduced by Meerschaert (see M.~M.~Meerschaert, `Regular variation in $\mathbb R^k$', {\it Proc. Amer. Math. Soc.} 102 (1988) 341--348) and have numerous applications in limit theorems for probability measures. For an R-O varying measure $\mu$ and $-\infty<\infty$ let \begin{equation*} \begin{split} V_a(t,\theta) &= \int_{|\langle x,\theta\rangle|>t} |\langle x,\theta\rangle|^a\,d\mu(x),\\ U_b(t,\theta) &= \int_{|\langle x,\theta\rangle|\leq t} |\langle x,\theta\rangle|^b\,d\mu(x) \end{split} \end{equation*} denote the tail- and truncated moment functions of $\mu$ in the direction $\|\theta\|=1$. The purpose of this paper is to show that R-O variation of a measure implies sharp bounds on the growth rate of the tail- and truncated moment functions depending on the real parts of the eigenvalues of the index $F$ along a compact set of directions. Furthermore, bounds on the ratio of these functions for certain values of $a$ and $b$ are obtained. 1991 Mathematics Subject Classification: 60B10, 28C15.

Limit theorems for probability measures on totally disconnected groups

Limit theorems for probability measures on totally disconnected groups

Limit theorems for probability measures on simply connected nilpotent lie groups

LetX 1,X 2,... be i.i.d. random variables with values in a simply connected nilpotent Lie groupG. Assume the laws of $$(\tau _n (X_1 X_2 \cdots X_{k_n } ))_{n \in \mathbb{N}}$$ to be weakly convergent to a probability measure μ onG, τ n ∈ Aut(G), and (k n)n∈ℕ strictly increasing inℕ. In this paper we want to characterize the possible limit laws. We obtain that every limit law is continuously embeddable and we prove a kind of functional limit theorem. Further, we study the connections between two different concepts of stability (resp. semistability) and limit laws. Finally, we describe the various domains of attraction of measures (resp. of convolution semigroups).

On the central limit theorem in 𝐹-spaces

In this note we improve the known result on the integrability of F-norms with respect to Gaussian measures and obtain a Central Limit Theorem for probability measures on arbitrary separable F-spaces.

Limit theorems for probability measures on non-compact groups and semi-groups

Limit theorems for probability measures on non-compact groups and semi-groups

On the Extension of the Class of Stable Distributions

On the Extension of the Class of Stable Distributions