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You most likely have a hotspot timeout setting enabled.
You can disable it by going to your Advanced Hotspot Settings, and editing the timeout value to never.
Note that in some versions of Android in order to access the Advanced Hotspot Settings you need to first access the WiFi Hotspot Settings and then press the phone's Menu button.
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Just a note for others out there, the setting is not very easy to find (at least on my Samsung Galaxy S3 mini Android 4.1.1).
Go to Wireless and network -> More settings... -> Tethering and portable hotspot -> Portable WiFi hotspot. Once there, you'll have to press the Menu button (bottom left of your phone) and up pops Timeout settings!
One can only guess why they decided to hide that setting in such a hard-to-find place....
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Maybe my phone does not have any option to prevent this, I cannot find any "Timeout settings" for wifi-hotspot of my android phone. I have found an alternative solution:
1. I turn off the security(pattern lock/password etc.) when I use wifi-hotspot and it works fine. For all the android mobile it should work I think: Go to "Settings" -> "Security" -> "Screen Lock" (you may have to enter existing pattern or password for the lock) and set security to "None".
2. Also set the sleep duration to a low number such as 2 minutes or 15 seconds etc. To change the sleep duration go to "Settings" -> "Display" -> "Sleep".
After this your mobile will hopefully not stop wifi-hotspot when it goes to sleep mode.
When you have done with your work Enable the security again later by going to the same menu and setting security to "Pattern Lock" or anything you need. And you can also change the sleep settings as you like.
I have faced this problem with my android phone that I am currently using. This solution works perfectly for my phone and I found this solution myself finding no solution in internet. Then I shared the solution here in stackoverflow.
My recent findings is that, maybe the second step(mentioned above) alone can also do the trick.
Answer
I had the same problem. And then I realized that I'd set up Tasker to turn the hotspot off if it detected my favorite wifi network nearby. I've since removed that Tasker option, and the hotspot has stayed on.
I did not simply forget about the Tasker setting. Rather, even when I was nowhere near my favorite wifi networks, the hotspot would turn off. Tasker needs to briefly turn on the phone's wifi every so often, in order to see if it's near my favorite wifi network, and in so doing, it must also turn off the hotspot; my guess is that, somehow, it was unable to turn the hotspot back on after it shut wifi down.
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There is also another possibility I installed a program named Du Speed Buster and it turn off hotspot in order to boost the system speed I had to unmark that to solve the problem check all of your programs, because it can be some program like it
Source https://android.stackexchange.com/questions/29547/why-does-my-hotspot-stop-after-a-period-of-time