Strong Law of Large Numbers and Functional Central Limit Theorem
نویسنده
چکیده
20.1. Additional technical results on weak convergence Given two metric spaces S1, S2 and a measurable function f : S1 → S2, suppose S1 is equipped with some probability measure P. This induces a probability measure on S2 which is denoted by Pf−1 and is defined by Pf−1(A) = P(f−1(A) for every measurable set A ⊂ S2. Then for any random variable X : S2 → R, its expectation EPf −1 [X] is equal to EP[X(f)]. (Convince yourself that this is the case by looking at the special case when f is a simple function ).
منابع مشابه
A Note on the Strong Law of Large Numbers
Petrov (1996) proved the connection between general moment conditions and the applicability of the strong law of large numbers to a sequence of pairwise independent and identically distributed random variables. This note examines this connection to a sequence of pairwise negative quadrant dependent (NQD) and identically distributed random variables. As a consequence of the main theorem ...
متن کاملOrdinary Clt and Wlln Versions of L =
The familiar queueing principle expressed by the formula L=^\W (Little's law) can be interpreted as a relation among strong laws of large numbers (SLLNs). Here we prove central-limit-theorem (CLT) and weak-law-of-large-numbers (WLLN) versions of L = \W. For example, if the sequence of ordered pairs of interarrival times and waiting times is strictly stationary and satisfies a joint CLT, then th...
متن کاملA PRELUDE TO THE THEORY OF RANDOM WALKS IN RANDOM ENVIRONMENTS
A random walk on a lattice is one of the most fundamental models in probability theory. When the random walk is inhomogenous and its inhomogeniety comes from an ergodic stationary process, the walk is called a random walk in a random environment (RWRE). The basic questions such as the law of large numbers (LLN), the central limit theorem (CLT), and the large deviation principle (LDP) are ...
متن کاملCumulative record times in a Poisson process
We obtain a strong law of large numbers and a functional central limit theorem, as t → ∞, for the number of records up to time t and the Lebesgue measure (length) of the subset of the time interval [0, t] during which the Poisson process is in a record lifetime. strong law of large numbers.
متن کاملec 2 00 7 Cumulative record times in a Poisson process
We obtain a strong law of large numbers and a functional central limit theorem, as t → ∞, for the number of records up to time t and the Lebesgue measure (length) of the subset of the time interval [0, t] during which the Poisson process is in a record lifetime. strong law of large numbers.
متن کاملApplications of Central Limit Theorems for Equity-linked Insurance
In both the past literature and industrial practice, it was often implicitly used without any justification that the classical strong law of large numbers applies to the modeling of equitylinked insurance. However, as all policyholders’ benefits are linked to common equity indices or funds, the classical assumption of independent claims is clearly inappropriate for equity-linked insurance. In o...
متن کامل