The Swedish music star, real name Tim Bergling, was on holiday on the Arabian Peninsular when he was found dead.
The statement, released on Monday evening, reads: “We would like to thank you for the support and the loving words about our son and brother.
“We are so grateful for everyone who loved Tim’s music and have precious memories of his songs.
“Thank you for all the initiatives taken to honor Tim, with public gatherings, church bells ringing out his music, tributes at Coachella and moments of silence around the world.
“We are grateful for the privacy during this difficult time. Our wish is that it continues that way. With love, The Tim Bergling Family.”
Avicii’s parents, sister and brother arrived in Oman over the weekend.
“Two postmortems were carried out ... and we can confirm that there is no criminal suspicion in the death,” the statement read.
Stars from the music world paid tribute to the ‘Hey Brother’ hitmaker in the wake of Friday’s announcement.
Calvin Harris and Nile Rodgers were among the first to share their condolences, with the former writing on Twitter: “Devastating news about Avicii, a beautiful soul, passionate and extremely talented with so much more to do. My heart goes out to his family. God bless you Tim x.”
Avicii’s 2011 track ‘Levels’ launched him into the mainstream and he went on to have huge commercial success, with two albums charting in the UK Top 10.
After years on the road and over 800 live shows, he announced his decision to quit performing in March 2016.
Telling fans that he wanted to focus on music production, he said: “I know I am blessed to be able to travel all around the world and perform, but I have too little left for the life of a real person behind the artist.
“I will however never let go of music — I will continue to speak to my fans through it.”