Exponent Function for Stationary Memoryless Channels with Input Cost at Rates above the Capacity

We consider the stationaly memoryless channels with input cost. We prove that for transmission rates above the capacity the correct probability of decoding tends to zero exponentially as the block length $n$ of codes tends to infinity. In the case where both of channel input and output sets are finite, we determine the optimal exponent function on the above exponential decay of the correct probability. To derive this result we use a new technique called the recuresive method, which is based on the information spectrum approach. The recursive method utilize a certain recursive structure on the information spectrum quantities.