A useful central limit theorem for m-dependent variables

Metrika ◽  
1971 ◽  
Vol 17 (1) ◽  
pp. 116-128 ◽  
Author(s):  
P. Schönfeld
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Dependent Variables

The central limit theorem for m-dependent variables

1955 ◽  
Vol 51 (1) ◽  
pp. 92-95 ◽  
Author(s):  
P. H. Diananda
Keyword(s):  
Standard Deviation ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Limit Theorems ◽  
Sufficient Conditions ◽  
The Central Limit Theorem ◽  
Lindeberg Condition ◽  
Dependent Variables ◽  
Basic Ideas

In a previous paper (4) central limit theorems were obtained for sequences of m-dependent random variables (r.v.'s) asymptotically stationary to second order, the sufficient conditions being akin to the Lindeberg condition (3). In this paper similar theorems are obtained for sequences of m-dependent r.v.'s with bounded variances and with the property that for large n, where s′n is the standard deviation of the nth partial sum of the sequence. The same basic ideas as in (4) are used, but the proofs have been simplified. The results of this paper are examined in relation to earlier ones of Hoeffding and Robbins(5) and of the author (4). The cases of identically distributed r.v.'s and of vector r.v.'s are mentioned.

FCLTs for Dependent Variables

2021 ◽  
pp. 699-723
Author(s):  
James Davidson
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Time Domain ◽  
Functional Central Limit Theorem ◽  
Mixing Processes ◽  
Brownian Motions ◽  
Dependent Variables ◽  
The Time Domain ◽  
Functional Central Limit

After some technical preliminaries, this chapter gives two contrasting proofs of the functional central limit theorem for near‐epoch dependent functions of mixing processes. It goes on to consider variants of the result for nonstationary increments in which the limits are transformed Brownian motions, subject to distortions of the time domain. The multivariate case of the result is also given.

The central limit theorem for m-dependent variables asymptotically stationary to second order

1954 ◽  
Vol 50 (2) ◽  
pp. 287-292 ◽  
Author(s):  
P. H. Diananda
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Limit Theorems ◽  
Second Order ◽  
Sufficient Condition ◽  
Independent Variables ◽  
The Central Limit Theorem ◽  
Dependent Variables ◽  
Basic Ideas

In a recent paper (3) the Lindeberg-Lévy theorem (2) was extended for certain types of stationary dependent variables. In the present paper mainly the same basic ideas as were used in (3) are employed to give central limit theorems for m-dependent scalar variables (a) stationary to second order and (b) asymptotically stationary to second order, the sufficient condition in each case being akin to the Linde-berg condition ((1), p. 57) for independent variables. The analogue of the main theorem for vector variables is given. An extension of the Lindeberg-Lévy theorem is derived.

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