The arms race increased rather than reduced rivalry and insecurity.
Stalin was convinced that the USA used the atomic bombs on Japan as a warning to the Soviet Union. The USSR felt especially vulnerable in the years 1945-9 until they had developed their own atom bomb.
The result was a massive build up of US missiles
The danger of initiating nuclear war acted as a restraint on both sides. Limited war, such as in Korea and Vietnam, was used to avoid direct confrontation
In the 1950s, the USA developed the tactic of brinkmanship, of being prepared to go to the brink of a nuclear war in order to stop enemy aggression. This was shown most significantly in the Cuban Missile Crisis.
In the 1970s, both sides had enough weapons to totally destroy each other. This led to MAD (Mutually Aided Destruction). A more flexible range of responses was necessary.
The arms race also provided one of the reasons for ending the Cold War because it partially bankrupted the USSR and prompted some of the political events of the late 1980s.