Yes! On iOS 8 and above, the VPN will remain connected after you have selected the powered button (or slide the connection switch to the ON position). However, if you choose to change network connections (i.e. Wi-Fi to LTE or vice versa) the VPN connection may disconnect shortly but will then auto-reconnect once a new network connection has been established.
(Note: please be aware that the VPN needs a stable network connection to remain connected to the VPN if you are not connected to Wi-Fi or your Mobile Data the application will not connect you to the VPN)
One way to ensure that your iOS Device auto connects to the VPN is to enable “Connect on Demand” within your iOS VPN Settings, by going to General> VPN & Device Management> VPN> Private Internet Access. Select the blue “i” icon next to “Private Internet Access” and from there you will see the option to Connect on Demand.
On iOS 7, due to technical limitations, the VPN will disconnect after a time period of no traffic or when the screen goes to sleep.