Second-Order Asymptotics for Communication under Strong Asynchronism

The capacity under strong asynchronism was recently shown to be essentially unaffected by the imposed decoding delay - the elapsed time between when information is available at the transmitter and when it is decoded - and the output sampling rate. This paper shows that, in contrast with capacity, the second-order term in the maximum rate expansion is sensitive to both parameters. When the receiver must locate the sent codeword exactly and therefore achieve minimum delay equal to the blocklength n , the second-order term in the maximum rate expansion is of order \Theta (1/\rho) for any sampling rate \rho =O(1/\sqrt {n}) (and \rho =\omega (1/n) for otherwise reliable communication is impossible). Instead, if \rho =\omega (1/\sqrt {n}) , then the second-order term is the same as under full sampling and is given by a standard \Theta (\sqrt {n}) term. However, if the delay constraint is only slightly relaxed to n(1+o(1)) , then the above order transition (for \rho =O(1/\sqrt {n}) and \rho =\omega (1/\sqrt {n}) ) vanishes and the second-order term remains the same as under full sampling for any \rho =\omega (1/n).