Baum–Katz type theorems with exact threshold

Abstract

Let be either a sequence of arbitrary random variables, or a martingale difference sequence, or a centred sequence with a suitable level of negative dependence. We prove Baum–Katz type theorems by only assuming that the variables satisfy a uniform moment bound condition. We also prove that this condition is best possible even for sequences of centred, independent random variables. This leads to Marcinkiewicz–Zygmund type strong laws of large numbers with estimate for the rate of convergence.

Keywords:

• Complete convergence
• Marcinkiewicz–Zygmund strong law of large numbers
• rate of convergence
• independent random variables
• martingale difference sequences

AMS Subject Classifications:

• Primary: 60F15
• 60G42
• 60G50
• Secondary: 60F10

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author was supported by the National Research, Development and Innovation Office–NKFIH, [grant number 104178]. The second author’s research was supported by the grant EFOP-3.6.1-16-2016-00001 (“Complex improvement of research capacities and services at Eszterhazy Karoly University”).