The incidents occurred in the town centre.
A 36-year-old man from Wallingford has been arrested on suspicion of harrassment without violence.
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He has since been released on police bail.
Police are now appealing for witnesses to the inicidents.
If you have witnessed anything or can help the police with thier investigation, please call 101 or make a report on the Thames Valley Police website, quoting reference number 43230227459.