There is quite old precedent: running amok. "Amok" (or "amuk") is a Malay word, and means exactly the kind of random violence you describe, except that typically a bladed weapon is used, rather than a gun, as seen on occasion in the areas that are now Malaysia and Indonesia. ("Running amok" can refer to mob violence, and to berserkly charging an enemy and fighting them until killed, but the meaning of to targeting people at random (which is the relevant one for you) is old, and perhaps the original meaning.)

The earliest mention in the Western literature is by Nicolo Conti, from the 15th century:

But he who, rather than be a slave, prefers death, seizing a naked sword issues into the street and kills all he meets, until he is slain by some one more powerful than himself

(Nicolo Conti (author), J Winter Jones (translator), "The travels of Nicolo Conti in the East in the early part of the fifteenth century", in R. H. Major (ed), India in the Fifteenth Century, The Hakluyt Society, 1857. https://archive.org/details/travelsofnicoloc00wint )

For more on amok, see John C. Spores, Running amok: an historical inquiry, Ohio University Center for International Studies, Center for Southeast Asian Studies, 1988. Spores notes the similarity of traditional amok to the modern random shooting. It can also be noted that 20th centur cases can involve the use of guns instead of, or in addition to, the traditional bladed weapons: Karl Schmidt, Lee Hill, George Guthrie, "Running Amok", International Journal of Social Psychiatry 23(4), 264-274 (1977).

Further references:

Hempel AG, Levine RE, Meloy JR, Westermeyer J., "A cross-cultural review of sudden mass assault by a single individual in the oriental and occidental cultures", J Forensic Sci. 2000 May;45(3):582-8. Compares late 20th century Laotian amok cases with North American "sudden mass assault by a single individual", or "SMASI", cases. The authors note that "Similarities among samples far outweigh differences".