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 <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>, the second-order term in the maximum rate expansion is of order <inline-formula> <tex-math notation="LaTeX">$\Theta (1/\rho)$ </tex-math></inline-formula> for any sampling rate <inline-formula> <tex-math notation="LaTeX">$\rho =O(1/\sqrt {n})$ </tex-math></inline-formula> (and <inline-formula> <tex-math notation="LaTeX">$\rho =\omega (1/n)$ </tex-math></inline-formula> for otherwise reliable communication is impossible). Instead, if <inline-formula> <tex-math notation="LaTeX">$\rho =\omega (1/\sqrt {n})$ </tex-math></inline-formula>, then the second-order term is the same as under full sampling and is given by a standard <inline-formula> <tex-math notation="LaTeX">$\Theta (\sqrt {n})$ </tex-math></inline-formula> term. However, if the delay constraint is only slightly relaxed to <inline-formula> <tex-math notation="LaTeX">$n(1+o(1))$ </tex-math></inline-formula>, then the above order transition (for <inline-formula> <tex-math notation="LaTeX">$\rho =O(1/\sqrt {n})$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$\rho =\omega (1/\sqrt {n})$ </tex-math></inline-formula>) vanishes and the second-order term remains the same as under full sampling for any <inline-formula> <tex-math notation="LaTeX">$\rho =\omega (1/n)$ </tex-math></inline-formula>.