On Weak Invariance Principles for Partial Sums

In this chapter, we analyze the asymptotic behavior of the partial sums process associated with examples of stationary sequences in a random time scenery. The examples considered are stationary sequences sampled by shifted renewal Markov chains and random walks in a strictly stationary scenery. The asymptotic behavior of the partial sums process is essentially investigated with the help of the weak invariance principles stated in Chapter 4. More precisely, for the partial sums process associated with a stationary process sampled by a renewal Markov chain stated at zero, due to the non-stationarity of the underlying sequence, the functional CLT is obtained as an application of the functional CLT for non-stationary sequences developed in Section 4.4. In the case where we are sampling a strictly stationary random scenery by a random walk, stationarity is preserved, and the invariance principle is then derived by using the functional CLT under Maxwell–Woodroofe condition.

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Stability Results and Strong Invariance Principles for Partial Sums of Banach Space Valued Random Variables

The Annals of Probability ◽

10.1214/aop/1176991512 ◽

1989 ◽

Vol 17 (1) ◽

pp. 333-352 ◽

Cited By ~ 5

Author(s):

Uwe Einmahl

Keyword(s):

Banach Space ◽

Random Variables ◽

Partial Sums ◽

Invariance Principles ◽

Strong Invariance ◽

Strong Invariance Principles ◽

Stability Results

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Limit Theorems for an Extended Coupon Collector's Problem and for Successive Subsampling with Varying Probabilites*

Calcutta Statistical Association Bulletin ◽

10.1177/0008068319800301 ◽

1980 ◽

Vol 29 (3-4) ◽

pp. 113-132 ◽

Cited By ~ 4

Author(s):

Pranab Kumar Sen

Keyword(s):

Asymptotic Normality ◽

Limit Theorems ◽

Finite Population ◽

Waiting Times ◽

Invariance Principles ◽

Weak Invariance ◽

Coupon Collector’S Problem ◽

Asymptotic Distribution Theory ◽

Multistage Sampling ◽

Coupon Collector's Problem

Asymptotic normality as well as some weak invariance principles for bonus sums and waiting times in an extended coupon collector's problem are considered and incorporated in the study of the asymptotic distribution theory of estimators of (finite) population totals in successive sub-sampling (or multistage sampling) with varying probabilities (without replacement). Some applications of these theorems are also considered.

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