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convolution of two aperiodic signal

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For discrete time signals of finite support, the length of the convolution is the sum of the lengths of the input signals plus 1. So convolving two finite support signals will always gives a finite support signal.
I dont know the case of signals with infinite support (periodic or not) I think that circular convolution may apply. You may take a look here:
https://en.wikipedia.org/wiki/Circular_convolution
 

Yes it is possible.Any aperiodic signal can be represented as a periodic signal of period 0-2 pi, where the 2 pi is the time when the signal has stopped being observed.Now this is the basic principle behind the formulation of Fourier Transform of an aperiodic signal.So since even an aoeriodic signal is also considered as a periodic signal of really huge period(as long as we observe),it is also a form of periodic signal.I hope it helps.
 

in oppenheim 'signals and systems' book, he give one example that the convolution of two aperiodic signal is able to form a periodic signal . how can we analyze with help of fourier transform
 

I don't know what that example is ,I have that book though.I can maybe help you if I know that example. There is a rule in convolution that the result of two signals convoluted is the product of their Fourier Transform.
 

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