# The conduct of the sample average when the first moment is infinite

J.L. Mijnheer

### Abstract

Many books about probability and statistics only mention the weak and the strong law of large numbers for samples from distributions with finite expectation. However, these laws also hold for distributions with infinite expectation and then the sample average has to go to infinity with increasing sample size. Being curious about the way in which this would happen, we simulated increasing samples (up to n= 40000) from three distributions with infinite expectation. The results were somewhat surprising at first sight, but understandable after some thought. Most statisticians, when asked, seem to expect a gradual increase of the average with the size of the sample. So did we. In general, however, this proves to be wrong and for different parent distributions different types of conduct appear from this experiment. The samples from the "absolute Cauchy"-distribution are most interesting from a practical point of view: the average takes a high jump from time to time and decreases in between. In practice it might well happen, that the observations causing the jumps would be discarded as outlying observations.
Original language English 37-41 Statistica Neerlandica 22 1 https://doi.org/10.1111/j.1467-9574.1960.tb00616.x Published - 1968

### Fingerprint

Moment
Jump
Cauchy Distribution
Strong law of large numbers
Sample Size
Infinity
Statistics
Decrease
Experiment
Observation

• IR-70659

### Cite this

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title = "The conduct of the sample average when the first moment is infinite",
abstract = "Many books about probability and statistics only mention the weak and the strong law of large numbers for samples from distributions with finite expectation. However, these laws also hold for distributions with infinite expectation and then the sample average has to go to infinity with increasing sample size. Being curious about the way in which this would happen, we simulated increasing samples (up to n= 40000) from three distributions with infinite expectation. The results were somewhat surprising at first sight, but understandable after some thought. Most statisticians, when asked, seem to expect a gradual increase of the average with the size of the sample. So did we. In general, however, this proves to be wrong and for different parent distributions different types of conduct appear from this experiment. The samples from the {"}absolute Cauchy{"}-distribution are most interesting from a practical point of view: the average takes a high jump from time to time and decreases in between. In practice it might well happen, that the observations causing the jumps would be discarded as outlying observations.",
keywords = "IR-70659",
author = "J.L. Mijnheer",
year = "1968",
doi = "10.1111/j.1467-9574.1960.tb00616.x",
language = "English",
volume = "22",
pages = "37--41",
journal = "Statistica Neerlandica",
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}

In: Statistica Neerlandica, Vol. 22, No. 1, 1968, p. 37-41.

TY - JOUR

T1 - The conduct of the sample average when the first moment is infinite

AU - Mijnheer, J.L.

PY - 1968

Y1 - 1968

N2 - Many books about probability and statistics only mention the weak and the strong law of large numbers for samples from distributions with finite expectation. However, these laws also hold for distributions with infinite expectation and then the sample average has to go to infinity with increasing sample size. Being curious about the way in which this would happen, we simulated increasing samples (up to n= 40000) from three distributions with infinite expectation. The results were somewhat surprising at first sight, but understandable after some thought. Most statisticians, when asked, seem to expect a gradual increase of the average with the size of the sample. So did we. In general, however, this proves to be wrong and for different parent distributions different types of conduct appear from this experiment. The samples from the "absolute Cauchy"-distribution are most interesting from a practical point of view: the average takes a high jump from time to time and decreases in between. In practice it might well happen, that the observations causing the jumps would be discarded as outlying observations.

AB - Many books about probability and statistics only mention the weak and the strong law of large numbers for samples from distributions with finite expectation. However, these laws also hold for distributions with infinite expectation and then the sample average has to go to infinity with increasing sample size. Being curious about the way in which this would happen, we simulated increasing samples (up to n= 40000) from three distributions with infinite expectation. The results were somewhat surprising at first sight, but understandable after some thought. Most statisticians, when asked, seem to expect a gradual increase of the average with the size of the sample. So did we. In general, however, this proves to be wrong and for different parent distributions different types of conduct appear from this experiment. The samples from the "absolute Cauchy"-distribution are most interesting from a practical point of view: the average takes a high jump from time to time and decreases in between. In practice it might well happen, that the observations causing the jumps would be discarded as outlying observations.

KW - IR-70659

U2 - 10.1111/j.1467-9574.1960.tb00616.x

DO - 10.1111/j.1467-9574.1960.tb00616.x

M3 - Article

VL - 22

SP - 37

EP - 41

JO - Statistica Neerlandica

JF - Statistica Neerlandica

SN - 0039-0402

IS - 1

ER -